WEBVTT Kind: captions; Language: en-US

NOTE
Created on 2024-02-07T20:52:10.2470009Z by ClassTranscribe

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Good morning.

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Alright, so I'm going to just first

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finish up what I was, what I was going

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to cover at the end of the last lecture

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about Cannon.

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And then I'll talk about probabilities

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and Naive Bayes.

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And so I wanted to give an example of

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how K&N is used in practice.

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Here's one example of using it for face

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recognition.

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A lot of times when it's used in

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practice, there's a lot of feature

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learning that goes on ahead of the

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nearest neighbor.

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So nearest neighbor itself is really

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simple.

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It's efficacy depends on learning good

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representation so that.

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Data points that are near each other

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actually have similar labels.

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Here's one example.

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They want to try to be able to

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recognize whether two faces are the

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same person.

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And so the method is that you Detect

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facial features and then use those

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feature detections to align the image

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so that the face looks more frontal.

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Then they use a CNN convolutional

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neural network to train Features that

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will be good for recognizing faces.

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And the way they did that is that they

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first collected hundreds of Faces from

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a few thousand different people.

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I think it was their employees of

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Facebook.

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And they trained a classifier to say

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which, given a face, which of these

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people does the face belong to.

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And from that, they learn a

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REPRESENTATION.

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Those classifiers aren't very useful,

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because nobody's interested in seeing

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given a face.

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Which of the Facebook employees is that

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they want to know?

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Like, is it you want to know?

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Like, organize your photo album or see

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whether you've been tagged in another

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photo or something like that?

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And so then they throw out the

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Classifier and they just use the

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feature representation that was learned

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and use nearest neighbor to identify a

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person that's been detected in a

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photograph.

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So in their paper, they showed that

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this performs similarly to humans in

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this data set called label faces in the

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wild where you're trying to recognize

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celebrities.

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But it can be used for many things.

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So you can organize photo albums, you

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can detect faces and then you try to

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match Faces across the photos.

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So then you can organize like which

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photos have a particular person.

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Again, you can't identify celebrities

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or famous people by building up a

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database of faces of famous people.

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And you can also alert, alert somebody

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if somebody else uploads a photo of

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them.

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So you can see if somebody uploads a

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photo, then you can detect faces, you

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can see what their friends network is,

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see what other which of their faces

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have been uploaded and then Detect the

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other users whose faces have been

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uploaded and ask them for permission to

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like make this photo public.

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So this algorithm is actually used by

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Facebook.

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It has been for several years.

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They're limiting some of its use more

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recently, but they've been.

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But it's been used really heavily.

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And of course they have expanded

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training data because whenever anybody

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uploads photos then they can

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automatically detect them and add them

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to the database.

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So here the use of KN is important

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because KNN doesn't require any

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training.

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So every time somebody uploads a new

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face you can update the model just by

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adding this four 4096 dimensional

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feature vector that corresponds to the

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face and then use it in like based on

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the friend networks to.

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To recognize faces that are associated

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with somebody.

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I won't take time to discuss it now,

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but it's worth thinking about some of

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the consequences of the way that the

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algorithm was trained and the way that

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it's deployed.

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So for example.

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If you think about that, it was that

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the initial Features were learned on

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Facebook employees.

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That's not a very.

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That's not very representative

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demographic of the US the employees

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tend to be younger and.

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Probably skew towards male might skew

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towards certain ethnicities.

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And so the Algorithm may be much better

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at recognizing some kinds of Faces than

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other faces.

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And then, of course, there's lots and

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lots of ethical issues that surround

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the use of face recognition and its

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applications.

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Of course, like in many ways, this is

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used to help people maintain privacy.

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But even the use of recognition at all

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raises privacy concerns, and that's why

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they've limited the use to some extent.

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So just something to think about.

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So just to recap kann, the key

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assumptions of K&N are that K nearest

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neighbors that Samples with similar

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features will have similar output

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predictions.

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And for most of the Distance measures

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you implicitly assumes that all the

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dimensions are equally important.

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So it requires some kind of scaling or

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learning to be really effective.

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The parameters are just the data

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itself.

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You don't really have to learn any kind

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of statistics of the data.

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The data are the parameters.

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The designs are mainly the choice of K

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if you have higher K then it gets

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smoother Prediction.

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You can decide how you're going to

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combine predictions if K is greater

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than one, usually it's just voting or

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averaging.

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You can try to design the features and

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that's where things can get a lot more

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creative.

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And you can choose a Distance function.

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So this K&N is useful in many cases.

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So if you have very few examples per

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class then it can be applied even if

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you just have one.

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It can also work if you have many

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Examples per class.

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It's best if the features are all

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roughly equally important, because K&N

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itself doesn't really learn which

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features are important.

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It's good if the training data is

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changing frequently.

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In the face recognition Example face,

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there's no way that Facebook will

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collect everybody's Faces up front.

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People keep on joining and leaving the

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social network, and so they and they

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don't want to have to keep retraining

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models every time somebody uploads a

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image with a new face in it or tags a

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new face.

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And so the ability to instantly update

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your model is very important.

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You can apply it to classification or

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regression whether you have discrete or

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continuous values, and its most

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powerful when you do some feature

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learning as an upfront operation.

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So there's cases where it has its

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downsides though.

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One is that if you have a lot of

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examples that are available per class,

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then usually training a Logistic

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regressor other Linear Classifier will

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outperform because it's able to learn

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the importance of different Features.

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Also, K&N requires that you store all

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the training data and that may require

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a lot of storage and it requires a lot

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of computation, and that you have to

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compare each new input to all of the

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inputs in your training data.

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So in the case of Facebook for example,

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they don't need if somebody uploads, if

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they detect a face in somebody's image,

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they don't need to compare it to the

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other, like 2 billion Facebook users.

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They just would compare it to people in

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the person's social network, which will

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be a much smaller number of Faces.

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So they're able to limit the

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computation that way.

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And then finally, to recap what we

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learned on Thursday, there's a basic

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machine learning process, which is that

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you've got training data, validation

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data and TestData.

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Given the training data, which are

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pairs of Features and labels, you fit

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the parameters of your Model.

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Then you use the validation Model to

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check how good the Model is and maybe

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check many models.

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You choose the best one and then you

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get your final estimate of performance

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on the TestData.

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We talked about KNN, which is simple

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but effective Classifier and regressor

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that predicts the label of the most

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similar training Example.

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And then we talked about kind of

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patterns of error and what causes

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errors.

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So it's important to remember that as

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you get more training, more training

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samples, you would expect that fitting

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the training data gets harder.

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So your error will tend to go up while

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your error on the TestData will get

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lower because the training data better

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represents the TestData or better

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represents the full distribution.

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And there's many reasons why at the end

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of training your Algorithm, you're

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still going to have error in most

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cases.

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It could be that the problem is

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intrinsically difficult, or it's

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impossible to have 0 error.

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It could be that you're Model has

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limited power.

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It could be that your Model has plenty

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of power, but you have limited data so

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you can't Estimate the parameters

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exactly.

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And it could be that there's

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differences in the training test

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distribution.

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And then finally it's important to

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remember that this Model fitting, the

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model design and fitting is just one

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part of a larger processing collecting

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data and fitting it into an

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application.

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So both the cases of in Facebook's case

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for example they had pre training stage

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which is like training a classifier and

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then they use that in a different, they

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use it in a different way as a nearest

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neighbor recognizer on their pool of

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user data.

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And so they're kind of building a model

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using it.

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They're building a model one way and

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then using it in a different way.

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So often that's the case that you have

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to kind of be creative.

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About how you collect data and how you

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can get the model that you need to

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solve your application.

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Alright, so now I'm going to move on to

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the main topic of today's lecture,

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which is probabilities and the night

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based Classifier.

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So the knight based Classifier is

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unlike nearest neighbor, it's not.

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Usually like the final approach that

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somebody takes, but it's sometimes a

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piece of a piece of how somebody is

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estimating probabilities as part of

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their approach.

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And it's a good introduction to

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Probabilistic models.

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So with the nearest neighbor

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classifier, that's an instance based

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Classifier, which means that you're

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assigning labels just based on matching

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other instances.

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The instances the data are the Model.

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Now we're going to start talking about

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Probabilistic models.

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In a Probabilistic Model, you choose

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the label that is most likely given the

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Features.

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So that's kind of an intuitive thing to

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do if you want to know.

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Which if you're looking at an image and

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trying to classify it into a Digit, it

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makes sense that you would assign it to

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the Digit that is most likely given the

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Features given the pixel intensities.

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But of course, like the challenge is

00:12:38.170 --> 00:12:40.030
modeling this probability function, how

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do you Model the probability of the

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label given the data?

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So this is just a very compact way of

00:12:47.520 --> 00:12:48.135
writing that.

00:12:48.135 --> 00:12:50.270
So I have Y star is the predicted

00:12:50.270 --> 00:12:53.150
label, and that's equal to the argmax

00:12:53.150 --> 00:12:53.836
over Y.

00:12:53.836 --> 00:12:55.770
So it's the Y that maximizes

00:12:55.770 --> 00:12:56.950
probability of Y given X.

00:12:56.950 --> 00:12:59.250
So you assign the label that's most

00:12:59.250 --> 00:13:00.590
likely given the data.

00:13:03.170 --> 00:13:05.210
So I just want to do a very brief

00:13:05.210 --> 00:13:08.240
review of some probability things.

00:13:08.240 --> 00:13:10.730
Hopefully this looks familiar, but it's

00:13:10.730 --> 00:13:12.920
still useful to refresh on it.

00:13:13.720 --> 00:13:15.290
So first Joint and conditional

00:13:15.290 --> 00:13:16.260
probability.

00:13:16.260 --> 00:13:19.040
If you say probability of X&Y then that

00:13:19.040 --> 00:13:20.900
means the probability that both of

00:13:20.900 --> 00:13:24.180
those values are true at the same time,

00:13:24.180 --> 00:13:25.030
so.

00:13:26.330 --> 00:13:28.400
So if you say like the probability that

00:13:28.400 --> 00:13:29.290
it's sunny.

00:13:29.980 --> 00:13:32.540
And it's rainy, then that's probably a

00:13:32.540 --> 00:13:33.910
very low probability, because those

00:13:33.910 --> 00:13:35.700
usually don't happen at the same time.

00:13:35.700 --> 00:13:37.635
Both X&Y are true.

00:13:37.635 --> 00:13:40.396
That's equal to the probability of X

00:13:40.396 --> 00:13:42.179
given Y times probability of Y.

00:13:42.180 --> 00:13:45.725
So probability of X given Y is the

00:13:45.725 --> 00:13:48.700
probability that X is true given the

00:13:48.700 --> 00:13:50.956
known values of Y times the probability

00:13:50.956 --> 00:13:52.280
that Y is true.

00:13:52.970 --> 00:13:54.789
And that's also equal to probability of

00:13:54.790 --> 00:13:56.769
Y given X times probability of X.

00:13:56.770 --> 00:13:59.450
So you can take a Joint probability and

00:13:59.450 --> 00:14:01.580
turn it into a conditional probability

00:14:01.580 --> 00:14:04.370
times the probability of their meaning

00:14:04.370 --> 00:14:06.190
variables, the condition variables.

00:14:07.010 --> 00:14:08.660
And you can apply that down a chain.

00:14:08.660 --> 00:14:11.341
So probability of ABC is probability of

00:14:11.341 --> 00:14:13.531
a given BC times probability of B given

00:14:13.531 --> 00:14:14.900
C times probability of C.

00:14:17.320 --> 00:14:18.730
And then it's important to remember

00:14:18.730 --> 00:14:21.110
Bayes rule, which is a way of relating

00:14:21.110 --> 00:14:23.160
probability of X given Y and

00:14:23.160 --> 00:14:24.869
probability of Y given X.

00:14:25.520 --> 00:14:27.440
So of X given Y.

00:14:28.100 --> 00:14:30.516
Is equal to probability of Y given X

00:14:30.516 --> 00:14:32.222
times probability of X over probability

00:14:32.222 --> 00:14:35.090
of Y and you can get that by saying

00:14:35.090 --> 00:14:38.595
probability of X given Y is probability

00:14:38.595 --> 00:14:41.599
of X&Y over probability of Y.

00:14:41.600 --> 00:14:43.730
So what was done here is you multiply

00:14:43.730 --> 00:14:45.910
this by probability of Y and then

00:14:45.910 --> 00:14:47.771
divide it by probability of Y and

00:14:47.771 --> 00:14:49.501
probability of X given Y times

00:14:49.501 --> 00:14:51.519
probability of Y is probability of X&Y.

00:14:52.600 --> 00:14:54.390
And then the probability of X&Y is

00:14:54.390 --> 00:14:56.030
broken out into probability of Y given

00:14:56.030 --> 00:14:57.209
X times probability of X.

00:14:59.150 --> 00:15:01.040
So often it's the case that you want to

00:15:01.040 --> 00:15:03.484
kind of switch things you the label and

00:15:03.484 --> 00:15:06.339
you want to know the likelihood of the

00:15:06.339 --> 00:15:08.350
Features, but you have like a

00:15:08.350 --> 00:15:10.544
likelihood for that, but you want a

00:15:10.544 --> 00:15:11.830
likelihood the other way of the

00:15:11.830 --> 00:15:13.654
probability of the label given the

00:15:13.654 --> 00:15:13.868
Features.

00:15:13.868 --> 00:15:15.529
And so you use Bayes rule to kind of

00:15:15.530 --> 00:15:17.550
turn the tables on your likelihood

00:15:17.550 --> 00:15:17.950
function.

00:15:20.620 --> 00:15:25.810
So using using using these rules of

00:15:25.810 --> 00:15:26.530
probability.

00:15:27.210 --> 00:15:29.830
We can show that if I want to find the

00:15:29.830 --> 00:15:33.250
Y that maximizes the likelihood of the

00:15:33.250 --> 00:15:34.690
label given the data.

00:15:35.370 --> 00:15:38.490
That's equivalent to finding the Y that

00:15:38.490 --> 00:15:41.240
maximizes the likelihood of the data

00:15:41.240 --> 00:15:44.520
given the label times the probability

00:15:44.520 --> 00:15:45.210
of the label.

00:15:45.920 --> 00:15:47.690
So in other words, if you wanted to

00:15:47.690 --> 00:15:50.030
say, well, what is the probability that

00:15:50.030 --> 00:15:53.550
my face is Derek given my facial

00:15:53.550 --> 00:15:54.220
features?

00:15:54.950 --> 00:15:56.100
That's the top.

00:15:56.100 --> 00:15:58.323
That's equivalent to saying what's the

00:15:58.323 --> 00:16:00.400
probability that it's me without

00:16:00.400 --> 00:16:02.635
looking at the Features times the

00:16:02.635 --> 00:16:04.270
probability of my Features given that

00:16:04.270 --> 00:16:04.870
it's me?

00:16:04.870 --> 00:16:05.980
Those are the same.

00:16:06.330 --> 00:16:09.770
Those the why that maximizes that is

00:16:09.770 --> 00:16:11.150
going to be the same so.

00:16:12.990 --> 00:16:15.230
And the reason for that is derived down

00:16:15.230 --> 00:16:15.720
here.

00:16:15.720 --> 00:16:17.473
So I can take Y given X.

00:16:17.473 --> 00:16:20.686
So argmax of Y given X is the as argmax

00:16:20.686 --> 00:16:23.029
of Y given X times probability of X.

00:16:23.780 --> 00:16:26.000
And the reason for that is just that

00:16:26.000 --> 00:16:27.880
probability of X doesn't depend on Y.

00:16:27.880 --> 00:16:31.140
So I can multiply multiply this thing

00:16:31.140 --> 00:16:33.092
in the argmax by anything that doesn't

00:16:33.092 --> 00:16:35.410
depend on Y and it's going to be

00:16:35.410 --> 00:16:37.890
unchanged because it's just going to.

00:16:38.870 --> 00:16:41.460
The way that maximizes it will be the

00:16:41.460 --> 00:16:41.780
same.

00:16:43.410 --> 00:16:44.940
So then I turn that.

00:16:45.530 --> 00:16:47.810
I turned that into the Joint Y&X and

00:16:47.810 --> 00:16:48.940
then I broke it out again.

00:16:49.900 --> 00:16:51.300
Right, so the reason why this is

00:16:51.300 --> 00:16:54.430
important is that I can choose to

00:16:54.430 --> 00:16:57.562
either Model directly the probability

00:16:57.562 --> 00:17:00.659
of the label given the data, or I can

00:17:00.659 --> 00:17:02.231
choose the Model the probability of the

00:17:02.231 --> 00:17:03.129
data given the label.

00:17:03.910 --> 00:17:06.172
In a Naive Bayes, we're going to Model

00:17:06.172 --> 00:17:07.950
probability the data given the label,

00:17:07.950 --> 00:17:09.510
and then in the next class we'll talk

00:17:09.510 --> 00:17:11.425
about logistic regression where we try

00:17:11.425 --> 00:17:12.930
to directly Model the probability of

00:17:12.930 --> 00:17:14.000
the label given the data.

00:17:22.090 --> 00:17:24.760
All right, so let's just.

00:17:26.170 --> 00:17:29.400
Do a simple probability exercise just

00:17:29.400 --> 00:17:31.430
to kind of make sure that.

00:17:33.430 --> 00:17:34.730
That we get.

00:17:37.010 --> 00:17:38.230
So let's say.

00:17:39.620 --> 00:17:41.060
Here I have a feature.

00:17:41.060 --> 00:17:41.970
Doesn't really matter what the

00:17:41.970 --> 00:17:43.440
Features, but let's say that it's

00:17:43.440 --> 00:17:45.233
whether something is larger than £10

00:17:45.233 --> 00:17:48.210
and I collected a bunch of different

00:17:48.210 --> 00:17:50.530
animals, cats and dogs and measured

00:17:50.530 --> 00:17:50.770
them.

00:17:51.450 --> 00:17:53.130
And I want to train something that will

00:17:53.130 --> 00:17:54.510
tell me whether or not something is a

00:17:54.510 --> 00:17:54.810
cat.

00:17:55.730 --> 00:17:57.370
And so.

00:17:58.190 --> 00:18:00.985
Or a dog, and so I have like 40

00:18:00.985 --> 00:18:03.280
different cats and 45 different dogs,

00:18:03.280 --> 00:18:04.860
and I measured whether or not they're

00:18:04.860 --> 00:18:06.693
bigger than £10.

00:18:06.693 --> 00:18:10.270
So first, given this empirical

00:18:10.270 --> 00:18:12.505
distribution, given these samples that

00:18:12.505 --> 00:18:15.120
I have, what's the probability that Y

00:18:15.120 --> 00:18:15.810
is a cat?

00:18:22.430 --> 00:18:25.970
So it's actually 40 / 85 because it's

00:18:25.970 --> 00:18:26.960
going to be.

00:18:27.640 --> 00:18:29.030
Let me see if I can write on this.

00:18:36.840 --> 00:18:37.330
OK.

00:18:39.520 --> 00:18:40.460
That's not what I wanted.

00:18:43.970 --> 00:18:45.500
If I can get the pen to work.

00:18:48.610 --> 00:18:50.360
OK, it doesn't work that well.

00:18:55.010 --> 00:18:56.250
OK, forget that.

00:18:56.250 --> 00:18:57.420
Alright, I'll write it on the board.

00:18:57.420 --> 00:18:59.639
So it's 40 / 85.

00:19:01.780 --> 00:19:05.010
So it's 40 / 40 + 45.

00:19:05.920 --> 00:19:08.595
And that's because there's 40 cats and

00:19:08.595 --> 00:19:09.888
there's 45 dogs.

00:19:09.888 --> 00:19:13.040
So I take the count of all the cats and

00:19:13.040 --> 00:19:14.970
divide it by the count of all the data

00:19:14.970 --> 00:19:16.635
in total, all the cats and dogs.

00:19:16.635 --> 00:19:17.860
So that's 40 / 85.

00:19:18.580 --> 00:19:20.470
And what's the probability that Y is a

00:19:20.470 --> 00:19:22.810
cat given that X is false?

00:19:29.380 --> 00:19:31.510
So it's right?

00:19:31.510 --> 00:19:34.240
So it's 15 / 20 or 3 / 4.

00:19:34.240 --> 00:19:35.890
And that's because given that X is

00:19:35.890 --> 00:19:37.620
false, I'm just in this one column

00:19:37.620 --> 00:19:40.799
here, so it's 15 / 15 / 20.

00:19:42.090 --> 00:19:45.110
And what's the probability that X is

00:19:45.110 --> 00:19:46.650
false given that Y is a cat?

00:19:49.320 --> 00:19:51.570
Right, 15 / 480 because if I know that

00:19:51.570 --> 00:19:53.500
Y is a Cat, then I'm in the top row, so

00:19:53.500 --> 00:19:55.590
it's just 15 divided by all the cats,

00:19:55.590 --> 00:19:56.650
so 15 / 40.

00:19:58.320 --> 00:20:00.737
OK, and it's important to remember that

00:20:00.737 --> 00:20:03.119
Y given X is different than X given Y.

00:20:05.110 --> 00:20:08.276
Right, so some other simple rules of

00:20:08.276 --> 00:20:08.572
probability.

00:20:08.572 --> 00:20:11.150
One is the law of total probability.

00:20:11.150 --> 00:20:13.060
That is, if you sum over all the values

00:20:13.060 --> 00:20:16.020
of a variable, then the sum of those

00:20:16.020 --> 00:20:17.630
probabilities is equal to 1.

00:20:18.240 --> 00:20:20.450
And if this were a continuous variable,

00:20:20.450 --> 00:20:21.840
it would just be an integral over the

00:20:21.840 --> 00:20:23.716
domain of X over all the values of X

00:20:23.716 --> 00:20:26.180
and then the integral over P of X is

00:20:26.180 --> 00:20:26.690
equal to 1.

00:20:27.980 --> 00:20:29.470
Then I've got Marginalization.

00:20:29.470 --> 00:20:31.990
So if I have a joint probability of two

00:20:31.990 --> 00:20:34.150
variables and I want to get rid of one

00:20:34.150 --> 00:20:34.520
of them.

00:20:35.280 --> 00:20:37.630
Then I take this sum over all the

00:20:37.630 --> 00:20:39.290
values of 1 and the variables.

00:20:39.290 --> 00:20:41.052
In this case it's the sum over all the

00:20:41.052 --> 00:20:41.900
values of X.

00:20:42.570 --> 00:20:46.268
Of X&Y and that's going to be equal to

00:20:46.268 --> 00:20:46.910
P of Y.

00:20:53.440 --> 00:20:55.380
And then finally independence.

00:20:55.380 --> 00:20:59.691
So A is independent of B if and only if

00:20:59.691 --> 00:21:02.414
the probability of A&B is equal to the

00:21:02.414 --> 00:21:04.115
probability of a times the probability

00:21:04.115 --> 00:21:04.660
of B.

00:21:05.430 --> 00:21:07.974
Or another way to write it is that

00:21:07.974 --> 00:21:10.142
probability that what this implies is

00:21:10.142 --> 00:21:12.500
that probability of a given B is equal

00:21:12.500 --> 00:21:13.890
to probability of a.

00:21:13.890 --> 00:21:15.680
So if I just divide both sides by

00:21:15.680 --> 00:21:17.250
probability of B then I get that.

00:21:18.160 --> 00:21:20.855
Or probability of B given A equals

00:21:20.855 --> 00:21:22.010
probability of B.

00:21:22.010 --> 00:21:24.150
So these things are the top one.

00:21:24.150 --> 00:21:25.700
Might not be something that pops into

00:21:25.700 --> 00:21:26.420
your head right away.

00:21:26.420 --> 00:21:28.450
It's not necessarily as intuitive, but

00:21:28.450 --> 00:21:30.001
these are pretty intuitive that

00:21:30.001 --> 00:21:32.376
probability of a given B equals

00:21:32.376 --> 00:21:33.564
probability of a.

00:21:33.564 --> 00:21:36.050
So in other words, whether or not a is

00:21:36.050 --> 00:21:37.470
true doesn't depend on B at all.

00:21:38.720 --> 00:21:40.430
And whether or not B is true doesn't

00:21:40.430 --> 00:21:42.360
depend on A at all, and then you can

00:21:42.360 --> 00:21:44.810
easily get to the one up there just by

00:21:44.810 --> 00:21:47.410
multiplying here both sides by

00:21:47.410 --> 00:21:48.100
probability of a.

00:21:56.140 --> 00:21:59.180
Alright, so in some of the slides

00:21:59.180 --> 00:22:00.650
there's going to be a bunch of like

00:22:00.650 --> 00:22:02.760
indices, so I just wanted to try to be

00:22:02.760 --> 00:22:04.370
consistent in the way that I use them.

00:22:05.030 --> 00:22:07.674
And also like usually verbally say what

00:22:07.674 --> 00:22:10.543
the what the variables mean, but when I

00:22:10.543 --> 00:22:14.300
say XI mean the ith feature so I is a

00:22:14.300 --> 00:22:15.085
feature index.

00:22:15.085 --> 00:22:18.619
When I say XNI mean the nth sample, so

00:22:18.620 --> 00:22:20.520
north is the sample index and Lynn

00:22:20.520 --> 00:22:21.590
would be the nth label.

00:22:22.370 --> 00:22:24.993
So if I say X and I, then that's the

00:22:24.993 --> 00:22:26.760
ith feature of the nth label.

00:22:26.760 --> 00:22:29.763
So for digits for example, would be the

00:22:29.763 --> 00:22:33.720
ith pixel of the nth Digit Example.

00:22:35.070 --> 00:22:37.580
I used this delta here to indicate with

00:22:37.580 --> 00:22:39.900
some expression inside to indicate that

00:22:39.900 --> 00:22:42.780
it returns true or returns one if the

00:22:42.780 --> 00:22:44.850
expression inside it is true and 0

00:22:44.850 --> 00:22:45.410
otherwise.

00:22:46.200 --> 00:22:48.110
And I'll Use V for a feature value.

00:22:55.320 --> 00:22:57.900
So if I want to Estimate the

00:22:57.900 --> 00:22:59.830
probabilities of some function, I can

00:22:59.830 --> 00:23:00.578
just do it by counting.

00:23:00.578 --> 00:23:02.760
So if I want to say what is the

00:23:02.760 --> 00:23:04.950
probability that X equals some value

00:23:04.950 --> 00:23:07.600
and I have capital N Samples, then I

00:23:07.600 --> 00:23:09.346
can just take a sum over all the

00:23:09.346 --> 00:23:11.350
samples and count for how many of them

00:23:11.350 --> 00:23:14.030
XN equals V so that's kind of intuitive

00:23:14.030 --> 00:23:14.480
if I have.

00:23:15.870 --> 00:23:17.750
If I have a month full of days and I

00:23:17.750 --> 00:23:19.280
want to say what's the probability that

00:23:19.280 --> 00:23:21.610
one of those days is sunny, then I can

00:23:21.610 --> 00:23:23.809
just take a sum over all the I can

00:23:23.810 --> 00:23:25.370
count how many sunny days there were

00:23:25.370 --> 00:23:26.908
divided by the total number of days and

00:23:26.908 --> 00:23:27.930
that gives me an Estimate.

00:23:31.930 --> 00:23:35.340
But what if I have 100 variables?

00:23:35.340 --> 00:23:36.380
So if I have.

00:23:37.310 --> 00:23:39.220
For example, in the digits case I have

00:23:39.220 --> 00:23:42.840
784 different and pixel intensities.

00:23:43.710 --> 00:23:46.350
And there's no way I can count over all

00:23:46.350 --> 00:23:48.222
possible combinations of pixel

00:23:48.222 --> 00:23:49.000
intensities, right?

00:23:49.000 --> 00:23:51.470
Even if I were to turn them into binary

00:23:51.470 --> 00:23:56.070
values, there would be 2 to the 784

00:23:56.070 --> 00:23:58.107
different combinations of pixel

00:23:58.107 --> 00:23:58.670
intensities.

00:23:58.670 --> 00:24:01.635
So you would need like data samples

00:24:01.635 --> 00:24:03.520
that are equal to like number of atoms

00:24:03.520 --> 00:24:05.300
in the universe or something like that

00:24:05.300 --> 00:24:07.415
in order to even begin to Estimate it.

00:24:07.415 --> 00:24:08.900
And that would that would only be

00:24:08.900 --> 00:24:10.460
giving you very few samples per

00:24:10.460 --> 00:24:11.050
combination.

00:24:12.860 --> 00:24:15.407
So obviously, like jointly modeling a

00:24:15.407 --> 00:24:17.799
whole bunch of different, the

00:24:17.800 --> 00:24:19.431
probability of a whole bunch of

00:24:19.431 --> 00:24:20.740
different variables is usually

00:24:20.740 --> 00:24:23.490
impossible, and even approximating it,

00:24:23.490 --> 00:24:24.880
it's very challenging.

00:24:24.880 --> 00:24:26.260
You have to try to solve for the

00:24:26.260 --> 00:24:28.036
dependency structures and then solve

00:24:28.036 --> 00:24:30.236
for different combinations of variables

00:24:30.236 --> 00:24:30.699
and.

00:24:31.550 --> 00:24:33.740
And then worry about the dependencies

00:24:33.740 --> 00:24:35.040
that aren't fully accounted for.

00:24:35.880 --> 00:24:37.670
And so it's just really difficult to

00:24:37.670 --> 00:24:40.160
estimate the probability of all your

00:24:40.160 --> 00:24:41.810
Features given the label.

00:24:42.900 --> 00:24:43.610
Jointly.

00:24:44.440 --> 00:24:47.540
And so that's the Naive Bayes Model

00:24:47.540 --> 00:24:48.240
comes in.

00:24:48.240 --> 00:24:50.430
It makes us greatly simplifying

00:24:50.430 --> 00:24:51.060
assumption.

00:24:51.730 --> 00:24:54.132
Which is that all of the features are

00:24:54.132 --> 00:24:56.010
independent given the label, so it

00:24:56.010 --> 00:24:57.480
doesn't mean the Features are

00:24:57.480 --> 00:24:57.840
independent.

00:24:57.940 --> 00:25:00.200
Unconditionally, but they're

00:25:00.200 --> 00:25:02.370
independent given the label, so.

00:25:03.550 --> 00:25:05.716
So because of because they're

00:25:05.716 --> 00:25:06.149
independent.

00:25:06.150 --> 00:25:08.400
Remember that probability of A&B equals

00:25:08.400 --> 00:25:11.173
probability of a * b times probability

00:25:11.173 --> 00:25:12.603
B if they're independent.

00:25:12.603 --> 00:25:15.160
So probability of X that's like a Joint

00:25:15.160 --> 00:25:17.920
X, all the Features given Y is equal to

00:25:17.920 --> 00:25:20.501
the product over all the features of

00:25:20.501 --> 00:25:22.919
probability of each feature given Y.

00:25:24.880 --> 00:25:28.866
And so then I can make my Classifier as

00:25:28.866 --> 00:25:30.450
the Y star.

00:25:30.450 --> 00:25:32.880
The most likely label is the one that

00:25:32.880 --> 00:25:35.415
maximizes this joint probability of

00:25:35.415 --> 00:25:37.930
probability of X given Y times

00:25:37.930 --> 00:25:38.779
probability of Y.

00:25:39.810 --> 00:25:42.715
And this thing, the joint probability

00:25:42.715 --> 00:25:44.985
of X given Y would be really hard to

00:25:44.985 --> 00:25:45.240
Estimate.

00:25:45.240 --> 00:25:47.490
You need tons of data, but this is not

00:25:47.490 --> 00:25:49.120
so hard to Estimate because you're just

00:25:49.120 --> 00:25:50.590
estimating the probability of 1

00:25:50.590 --> 00:25:51.590
variable at a time.

00:25:57.200 --> 00:25:59.190
So for example if I.

00:25:59.810 --> 00:26:01.900
In the Digit Example, this would be

00:26:01.900 --> 00:26:03.860
saying that the I'm going to choose the

00:26:03.860 --> 00:26:07.310
label that maximizes the product of

00:26:07.310 --> 00:26:09.220
likelihoods of each of the pixel

00:26:09.220 --> 00:26:09.980
intensities.

00:26:10.690 --> 00:26:12.555
So I'm just going to consider each

00:26:12.555 --> 00:26:13.170
pixel.

00:26:13.170 --> 00:26:15.170
How likely is each pixel to have its

00:26:15.170 --> 00:26:16.959
intensity given the label?

00:26:16.960 --> 00:26:18.230
And then I choose the label that

00:26:18.230 --> 00:26:20.132
maximizes that, taking the product of

00:26:20.132 --> 00:26:21.760
all the all those likelihoods over the

00:26:21.760 --> 00:26:22.140
pixels.

00:26:23.210 --> 00:26:23.690
So.

00:26:24.650 --> 00:26:26.880
Obviously it's not a perfect Model,

00:26:26.880 --> 00:26:28.210
even if I know that.

00:26:28.210 --> 00:26:30.610
If I'm given that it's a three, knowing

00:26:30.610 --> 00:26:32.759
that one pixel has an intensity of 1

00:26:32.760 --> 00:26:33.920
makes it more likely that the

00:26:33.920 --> 00:26:35.815
neighboring pixel has a likelihood of

00:26:35.815 --> 00:26:36.240
1.

00:26:36.240 --> 00:26:37.630
On the other hand, it's not a terrible

00:26:37.630 --> 00:26:38.710
Model either.

00:26:38.710 --> 00:26:41.028
If I know that it's a 3, then I have a

00:26:41.028 --> 00:26:43.210
pretty good idea of the expected

00:26:43.210 --> 00:26:45.177
intensity of each pixel, so I have a

00:26:45.177 --> 00:26:46.503
pretty good idea of how likely each

00:26:46.503 --> 00:26:47.920
pixel is to be a one or a zero.

00:26:50.490 --> 00:26:51.780
In the case of the temperature

00:26:51.780 --> 00:26:53.760
Regression will make a slightly

00:26:53.760 --> 00:26:55.040
different assumption.

00:26:55.040 --> 00:26:57.736
So here we have continuous Features and

00:26:57.736 --> 00:26:59.320
a continuous Prediction.

00:27:00.030 --> 00:27:02.840
So we're going to assume that each

00:27:02.840 --> 00:27:05.490
feature predicts the temperature that

00:27:05.490 --> 00:27:07.690
we're trying to predict the tomorrow's

00:27:07.690 --> 00:27:10.160
Cleveland temperature with some offset

00:27:10.160 --> 00:27:10.673
and variance.

00:27:10.673 --> 00:27:13.100
So for example, if I know yesterday's

00:27:13.100 --> 00:27:14.670
Cleveland temperature, then tomorrow's

00:27:14.670 --> 00:27:16.633
Cleveland temperature is probably about

00:27:16.633 --> 00:27:19.300
the same, but with some variance around

00:27:19.300 --> 00:27:19.577
it.

00:27:19.577 --> 00:27:21.239
If I know the Cleveland temperature

00:27:21.240 --> 00:27:23.520
from three days ago, then tomorrow's is

00:27:23.520 --> 00:27:25.732
also expected to be about the same but

00:27:25.732 --> 00:27:26.525
with higher variance.

00:27:26.525 --> 00:27:28.596
If I know the temperature of Austin,

00:27:28.596 --> 00:27:30.590
TX, then probably Cleveland is a bit

00:27:30.590 --> 00:27:31.819
colder with some variance.

00:27:33.550 --> 00:27:34.940
And so I'm going to use just that

00:27:34.940 --> 00:27:37.100
combination of individual predictions

00:27:37.100 --> 00:27:38.480
to make my final prediction.

00:27:44.170 --> 00:27:48.680
So here is the Naive Bayes Algorithm.

00:27:49.540 --> 00:27:53.250
For training, I Estimate the parameters

00:27:53.250 --> 00:27:55.370
for each of my likelihood functions,

00:27:55.370 --> 00:27:57.290
the probability of each feature given

00:27:57.290 --> 00:27:57.910
the label.

00:27:58.940 --> 00:28:01.878
And I Estimate the parameters for my

00:28:01.878 --> 00:28:02.232
prior.

00:28:02.232 --> 00:28:06.640
The prior is like the my Estimate, my

00:28:06.640 --> 00:28:08.370
likelihood of the label when I don't

00:28:08.370 --> 00:28:10.180
know anything else, just before I look

00:28:10.180 --> 00:28:11.200
at anything.

00:28:11.200 --> 00:28:13.475
So the probability of the label.

00:28:13.475 --> 00:28:14.770
And that's usually really easy to

00:28:14.770 --> 00:28:15.140
Estimate.

00:28:17.020 --> 00:28:19.280
And then at Prediction time, I'm going

00:28:19.280 --> 00:28:22.970
to solve for the label that maximizes

00:28:22.970 --> 00:28:26.330
the probability of X&Y or the and which

00:28:26.330 --> 00:28:28.620
the Naive Bayes assumption is the

00:28:28.620 --> 00:28:31.110
product over I of probability of XI

00:28:31.110 --> 00:28:32.649
given Y times probability of Y.

00:28:36.470 --> 00:28:40.455
The Naive Naive Bayes is that it's just

00:28:40.455 --> 00:28:42.050
the independence assumption.

00:28:42.050 --> 00:28:45.150
It's not an insult to Thomas Bayes that

00:28:45.150 --> 00:28:46.890
he's an idiot or something.

00:28:46.890 --> 00:28:49.970
It's just that we're going to make this

00:28:49.970 --> 00:28:52.140
very simplifying assumption.

00:28:58.170 --> 00:29:00.550
So all right, so the first thing we

00:29:00.550 --> 00:29:02.710
have to deal with is how do we Estimate

00:29:02.710 --> 00:29:03.590
this probability?

00:29:03.590 --> 00:29:06.500
We want to get some probability of each

00:29:06.500 --> 00:29:08.050
feature given the data.

00:29:08.960 --> 00:29:10.990
And the basic principles are that you

00:29:10.990 --> 00:29:12.909
want to choose parameters.

00:29:12.910 --> 00:29:14.550
First you have to have a model for your

00:29:14.550 --> 00:29:16.610
likelihood, and then you have to

00:29:16.610 --> 00:29:19.394
maximize the parameters of that model

00:29:19.394 --> 00:29:21.908
that you have to, sorry, Choose the

00:29:21.908 --> 00:29:22.885
parameters of that Model.

00:29:22.885 --> 00:29:25.180
That makes your training data most

00:29:25.180 --> 00:29:25.600
likely.

00:29:25.600 --> 00:29:27.210
That's the main principle.

00:29:27.210 --> 00:29:29.780
So if I say somebody says maximum

00:29:29.780 --> 00:29:32.390
likelihood estimation or Emily, that's

00:29:32.390 --> 00:29:34.190
like straight up maximizes the

00:29:34.190 --> 00:29:37.865
probability of the data given your

00:29:37.865 --> 00:29:38.800
parameters in your model.

00:29:40.320 --> 00:29:42.480
Sometimes that can result in

00:29:42.480 --> 00:29:44.120
overconfident estimates.

00:29:44.120 --> 00:29:46.210
So for example if I just have like.

00:29:46.970 --> 00:29:47.800
If I.

00:29:48.430 --> 00:29:51.810
If I have like 2 measurements, let's

00:29:51.810 --> 00:29:53.470
say I want to know what's the average

00:29:53.470 --> 00:29:56.044
weight of a bird and I just have two

00:29:56.044 --> 00:29:58.480
birds, and I say it's probably like a

00:29:58.480 --> 00:29:59.585
Gaussian distribution.

00:29:59.585 --> 00:30:02.012
I can Estimate a mean and a variance

00:30:02.012 --> 00:30:05.970
from those two birds, but that Estimate

00:30:05.970 --> 00:30:07.105
could be like way off.

00:30:07.105 --> 00:30:09.100
So often it's a good idea to have some

00:30:09.100 --> 00:30:11.530
kind of Prior or to prevent the

00:30:11.530 --> 00:30:12.780
variance from going too low.

00:30:12.780 --> 00:30:14.740
So if I looked at two birds and I said

00:30:14.740 --> 00:30:16.860
and they both happen to be like 47

00:30:16.860 --> 00:30:17.510
grams.

00:30:17.870 --> 00:30:19.965
I probably wouldn't want to say that

00:30:19.965 --> 00:30:22.966
the mean is 47 and the variance is 0,

00:30:22.966 --> 00:30:25.170
because then I would be saying like if

00:30:25.170 --> 00:30:27.090
there's another bird that has 48 grams,

00:30:27.090 --> 00:30:28.550
that's like infinitely unlikely.

00:30:28.550 --> 00:30:29.880
It's a 0 probability.

00:30:29.880 --> 00:30:31.600
So often you want to have some kind of

00:30:31.600 --> 00:30:34.270
Prior over your variables as well in

00:30:34.270 --> 00:30:37.025
order to prevent likelihoods going to 0

00:30:37.025 --> 00:30:38.430
because you just didn't have enough

00:30:38.430 --> 00:30:40.120
data to correctly Estimate them.

00:30:40.930 --> 00:30:42.650
So it's like Warren Buffett says with

00:30:42.650 --> 00:30:43.230
investing.

00:30:43.850 --> 00:30:45.550
It's not just about maximizing the

00:30:45.550 --> 00:30:47.690
expectation, it's also about making

00:30:47.690 --> 00:30:48.890
sure there are no zeros.

00:30:48.890 --> 00:30:50.190
Because if you have a zero and your

00:30:50.190 --> 00:30:51.670
product of likelihoods, the whole thing

00:30:51.670 --> 00:30:52.090
is 0.

00:30:53.690 --> 00:30:55.995
And if you have a zero, return your

00:30:55.995 --> 00:30:57.900
whole investment at any point, your

00:30:57.900 --> 00:30:59.330
whole bank account is 0.

00:31:03.120 --> 00:31:06.550
All right, so we have so.

00:31:06.920 --> 00:31:08.840
How do we Estimate P of X given Y given

00:31:08.840 --> 00:31:09.340
the data?

00:31:09.340 --> 00:31:10.980
It's always based on maximizing the

00:31:10.980 --> 00:31:11.930
likelihood of the data.

00:31:12.690 --> 00:31:14.360
Over your parameters, but you have

00:31:14.360 --> 00:31:15.940
different solutions depending on your

00:31:15.940 --> 00:31:18.200
Model and.

00:31:18.370 --> 00:31:19.860
I guess it just depends on your Model.

00:31:20.520 --> 00:31:24.180
So for binomial, a binomial is just if

00:31:24.180 --> 00:31:25.790
you have a binary variable, then

00:31:25.790 --> 00:31:27.314
there's some probability that the

00:31:27.314 --> 00:31:29.450
variable is 1 and 1 minus that

00:31:29.450 --> 00:31:31.790
probability that the variable is 0.

00:31:31.790 --> 00:31:36.126
So Theta Ki is the probability that X I

00:31:36.126 --> 00:31:38.510
= 1 given y = K.

00:31:39.510 --> 00:31:40.590
And you can write it.

00:31:40.590 --> 00:31:42.349
It's kind of a weird way.

00:31:42.350 --> 00:31:43.700
I mean it looks like a weird way to

00:31:43.700 --> 00:31:44.390
write it.

00:31:44.390 --> 00:31:46.190
But if you think about it, if XI equals

00:31:46.190 --> 00:31:48.760
one, then the probability is Theta Ki.

00:31:49.390 --> 00:31:51.630
And if XI equals zero, then the

00:31:51.630 --> 00:31:54.160
probability is 1 minus Theta Ki so.

00:31:54.800 --> 00:31:55.440
Makes sense?

00:31:56.390 --> 00:31:58.390
And if I want to Estimate this, all I

00:31:58.390 --> 00:32:00.530
have to do is count over all my data

00:32:00.530 --> 00:32:01.180
Samples.

00:32:01.180 --> 00:32:06.410
How many times does xni equal 1 and y =

00:32:06.410 --> 00:32:06.880
K?

00:32:07.530 --> 00:32:09.310
Divided by the total number of times

00:32:09.310 --> 00:32:10.490
that Y and equals K.

00:32:11.610 --> 00:32:13.290
And then here it is in Python.

00:32:13.290 --> 00:32:15.620
So it's just a sum over all my data.

00:32:15.620 --> 00:32:18.170
I'm looking at the ith feature here,

00:32:18.170 --> 00:32:20.377
checking how many times these equal 1

00:32:20.377 --> 00:32:23.585
and the label is equal to K divided by

00:32:23.585 --> 00:32:25.170
the number of times the label is equal

00:32:25.170 --> 00:32:25.580
to K.

00:32:27.240 --> 00:32:28.780
And if I have a multinomial, it's

00:32:28.780 --> 00:32:31.100
basically the same thing except that I

00:32:31.100 --> 00:32:35.342
sum over the number of times that X and

00:32:35.342 --> 00:32:37.990
I = V, where V could be say, zero to 10

00:32:37.990 --> 00:32:38.840
or something like that.

00:32:39.740 --> 00:32:42.490
And otherwise it's the same.

00:32:42.490 --> 00:32:46.040
So I can Estimate if I have 10

00:32:46.040 --> 00:32:49.576
different variables and I Estimate

00:32:49.576 --> 00:32:52.590
Theta KIV for all 10 variables, then

00:32:52.590 --> 00:32:54.410
the sum of those Theta kives should be

00:32:54.410 --> 00:32:54.624
one.

00:32:54.624 --> 00:32:56.540
So one of those is a constrained

00:32:56.540 --> 00:32:56.910
variable.

00:32:58.820 --> 00:33:00.420
And it will workout that way if you

00:33:00.420 --> 00:33:01.270
Estimate it this way.

00:33:05.970 --> 00:33:08.733
So if we have a continuous variable by

00:33:08.733 --> 00:33:11.730
the way, like, these can be fairly

00:33:11.730 --> 00:33:15.360
easily derived just by writing out the

00:33:15.360 --> 00:33:18.720
likelihood terms and taking a partial

00:33:18.720 --> 00:33:21.068
derivative with respect to the variable

00:33:21.068 --> 00:33:22.930
and setting that equal to 0.

00:33:22.930 --> 00:33:24.810
But it does take like a page of

00:33:24.810 --> 00:33:26.940
equations, so I decided not to subject

00:33:26.940 --> 00:33:27.379
you to it.

00:33:28.260 --> 00:33:30.190
Since since, solving for these is not

00:33:30.190 --> 00:33:30.920
the point right now.

00:33:32.920 --> 00:33:34.730
And so.

00:33:34.800 --> 00:33:36.000
Are.

00:33:36.000 --> 00:33:38.620
Let's say X is a continuous variable.

00:33:38.620 --> 00:33:40.740
Maybe I want to assume that XI is a

00:33:40.740 --> 00:33:44.052
Gaussian given some label, where the

00:33:44.052 --> 00:33:45.770
label is a discrete variable.

00:33:47.220 --> 00:33:51.023
So Gaussians, if you took hopefully you

00:33:51.023 --> 00:33:52.625
took probably your statistics and you

00:33:52.625 --> 00:33:53.940
probably ran into Gaussians all the

00:33:53.940 --> 00:33:54.230
time.

00:33:54.230 --> 00:33:55.820
Gaussians come up a lot for many

00:33:55.820 --> 00:33:56.550
reasons.

00:33:56.550 --> 00:33:58.749
One of them is that if you add a lot of

00:33:58.750 --> 00:34:01.125
random variables together, then if you

00:34:01.125 --> 00:34:02.839
add enough of them, then it will end up

00:34:02.840 --> 00:34:03.000
there.

00:34:03.000 --> 00:34:04.280
Some of them will end up being a

00:34:04.280 --> 00:34:05.320
Gaussian distribution.

00:34:07.080 --> 00:34:09.415
So there's lots of things end up being

00:34:09.415 --> 00:34:09.700
Gaussians.

00:34:09.700 --> 00:34:11.500
Gaussians is a really common noise

00:34:11.500 --> 00:34:13.536
model, and it also is like really easy

00:34:13.536 --> 00:34:14.320
to work with.

00:34:14.320 --> 00:34:16.060
Even though it looks complicated.

00:34:16.060 --> 00:34:17.820
When you take the log of it ends up

00:34:17.820 --> 00:34:19.342
just being a quadratic, which is easy

00:34:19.342 --> 00:34:20.010
to minimize.

00:34:22.250 --> 00:34:24.460
So there's the Gaussian expression on

00:34:24.460 --> 00:34:24.950
the top.

00:34:26.550 --> 00:34:28.420
And I.

00:34:29.290 --> 00:34:30.610
So let me get my.

00:34:33.940 --> 00:34:34.490
There it goes.

00:34:34.490 --> 00:34:37.060
OK, so here's the Gaussian expression

00:34:37.060 --> 00:34:39.260
one over square of 2π Sigma Ki.

00:34:39.260 --> 00:34:42.075
So the parameters here are M UI which

00:34:42.075 --> 00:34:43.830
is mu Ki which is the mean.

00:34:44.980 --> 00:34:47.700
For the KTH label and the ith feature

00:34:47.700 --> 00:34:49.946
in Sigma, Ki is the stair deviation for

00:34:49.946 --> 00:34:52.080
the Keith label and the Ith feature.

00:34:52.900 --> 00:34:54.700
And so the higher the standard

00:34:54.700 --> 00:34:57.090
deviation is, the bigger the Gaussian

00:34:57.090 --> 00:34:57.425
is.

00:34:57.425 --> 00:34:59.920
So if you look at these plots here, the

00:34:59.920 --> 00:35:02.150
it's kind of blurry the.

00:35:02.770 --> 00:35:05.540
The red curve or the actually the

00:35:05.540 --> 00:35:07.130
yellow curve has like the biggest

00:35:07.130 --> 00:35:08.880
distribution, the broadest distribution

00:35:08.880 --> 00:35:10.510
and it has the highest variance or

00:35:10.510 --> 00:35:12.010
highest standard deviation.

00:35:14.070 --> 00:35:15.780
So this is the MLE, the maximum

00:35:15.780 --> 00:35:17.240
likelihood estimate of the mean.

00:35:17.240 --> 00:35:19.809
It's just the sum of all the X's

00:35:19.810 --> 00:35:21.850
divided by the number of X's.

00:35:21.850 --> 00:35:25.109
Or, sorry, it's a sum over all the X's.

00:35:26.970 --> 00:35:30.190
For which Y n = K divided by the total

00:35:30.190 --> 00:35:31.900
number of times that Y n = K.

00:35:32.790 --> 00:35:34.845
Because I'm estimating the conditional

00:35:34.845 --> 00:35:36.120
conditional mean.

00:35:36.760 --> 00:35:41.570
So it's the sum over all the X's time.

00:35:41.570 --> 00:35:44.060
This will be where Y and equals K

00:35:44.060 --> 00:35:45.670
divided by the count of y = K.

00:35:46.320 --> 00:35:48.050
And they're staring deviation squared.

00:35:48.050 --> 00:35:50.650
Or the variance is the sum over all the

00:35:50.650 --> 00:35:53.340
differences of the X and the mean

00:35:53.340 --> 00:35:56.890
squared where Y and equals K divided by

00:35:56.890 --> 00:35:58.890
the number of times that y = K.

00:35:59.640 --> 00:36:01.180
And you have to estimate the mean

00:36:01.180 --> 00:36:02.480
before you Estimate the steering

00:36:02.480 --> 00:36:02.950
deviation.

00:36:02.950 --> 00:36:05.100
And if you take a statistics class,

00:36:05.100 --> 00:36:07.980
you'll probably like prove that this is

00:36:07.980 --> 00:36:09.945
an OK thing to do, that you're relying

00:36:09.945 --> 00:36:11.720
on one Estimate in order to get the

00:36:11.720 --> 00:36:12.720
other Estimate.

00:36:12.720 --> 00:36:14.420
But it does turn out it's OK.

00:36:16.670 --> 00:36:20.220
Alright, so in our homework for the

00:36:20.220 --> 00:36:22.890
temperature Regression, we're going to

00:36:22.890 --> 00:36:26.095
assume that Y minus XI is a Gaussian,

00:36:26.095 --> 00:36:27.930
so we have two continuous variables.

00:36:28.900 --> 00:36:29.710
So.

00:36:30.940 --> 00:36:34.847
The idea is that the temperature of

00:36:34.847 --> 00:36:38.565
some city on someday predicts the

00:36:38.565 --> 00:36:41.530
temperature of Cleveland on some other

00:36:41.530 --> 00:36:41.850
day.

00:36:42.600 --> 00:36:44.600
With some offset and some variance.

00:36:45.830 --> 00:36:48.190
And that is pretty easy to Model.

00:36:48.190 --> 00:36:51.020
So here's Sigma I is then the stair

00:36:51.020 --> 00:36:53.770
deviation of that offset Prediction and

00:36:53.770 --> 00:36:54.910
MU I is the offset.

00:36:55.560 --> 00:36:58.230
And I just have Y minus XI minus MU I

00:36:58.230 --> 00:37:00.166
squared here instead of Justice XI

00:37:00.166 --> 00:37:02.590
minus MU I squared, which would be if I

00:37:02.590 --> 00:37:03.960
just said XI is a Gaussian.

00:37:05.170 --> 00:37:08.820
And the mean is just why the sum of Y

00:37:08.820 --> 00:37:11.603
minus XI divided by north, where north

00:37:11.603 --> 00:37:12.870
is the total number of Samples.

00:37:13.990 --> 00:37:14.820
Because why?

00:37:14.820 --> 00:37:16.618
Is not discrete, so I'm not counting

00:37:16.618 --> 00:37:20.100
over certain over only values X where Y

00:37:20.100 --> 00:37:21.625
is equal to some value, I'm counting

00:37:21.625 --> 00:37:22.550
over all the values.

00:37:23.410 --> 00:37:25.280
And the Syrian deviation or their

00:37:25.280 --> 00:37:28.590
variance is Y minus XI minus MU I

00:37:28.590 --> 00:37:29.630
squared divided by north.

00:37:30.480 --> 00:37:32.300
And here's the Python.

00:37:33.630 --> 00:37:35.840
Here I just use the mean and steering

00:37:35.840 --> 00:37:37.630
deviation functions to get it, but it's

00:37:37.630 --> 00:37:40.470
also not a very long formula if I were

00:37:40.470 --> 00:37:41.340
to write it all out.

00:37:44.020 --> 00:37:46.830
And then X&Y were jointly Gaussian.

00:37:46.830 --> 00:37:49.660
So if I say that I need to jointly

00:37:49.660 --> 00:37:52.850
Model them, then one way to do it is

00:37:52.850 --> 00:37:53.600
by.

00:37:54.460 --> 00:37:56.510
By saying that probability of XI given

00:37:56.510 --> 00:38:00.660
Y is the joint probability of XI and Y.

00:38:00.660 --> 00:38:03.070
So now I have a 2 variable Gaussian

00:38:03.070 --> 00:38:06.780
with A2 variable mean and a two by two

00:38:06.780 --> 00:38:07.900
covariance matrix.

00:38:08.920 --> 00:38:11.210
Divided by the probability of Y, which

00:38:11.210 --> 00:38:12.700
is a 1D Gaussian.

00:38:12.700 --> 00:38:14.636
Just the Gaussian over probability of

00:38:14.636 --> 00:38:14.999
Y.

00:38:15.000 --> 00:38:16.340
And if you were to write out all the

00:38:16.340 --> 00:38:18.500
math for it would simplify into some

00:38:18.500 --> 00:38:21.890
other Gaussian equation, but it's

00:38:21.890 --> 00:38:23.360
easier to think about it this way.

00:38:27.660 --> 00:38:28.140
Alright.

00:38:28.140 --> 00:38:31.660
And then what if XI is continuous but

00:38:31.660 --> 00:38:32.770
it's not Gaussian?

00:38:33.920 --> 00:38:35.750
And why is discrete?

00:38:35.750 --> 00:38:37.763
There's one simple thing I can do is I

00:38:37.763 --> 00:38:40.770
can just first turn X into a discrete.

00:38:40.860 --> 00:38:41.490


00:38:42.280 --> 00:38:45.060
Into a discrete function, so.

00:38:46.810 --> 00:38:48.640
For example if.

00:38:49.590 --> 00:38:52.260
Let me venture with my pen again, but.

00:39:08.410 --> 00:39:08.810
Can't do it.

00:39:08.810 --> 00:39:09.170
I want.

00:39:15.140 --> 00:39:15.490
OK.

00:39:16.820 --> 00:39:20.930
So for example, X has a range from.

00:39:21.120 --> 00:39:22.130
From zero to 1.

00:39:22.810 --> 00:39:26.332
That's the case for our intensities of

00:39:26.332 --> 00:39:28.340
the pixel, intensities of amnesty.

00:39:29.180 --> 00:39:31.830
I can just set a threshold for example

00:39:31.830 --> 00:39:38.230
of 0.5 and if X is greater than 05 then

00:39:38.230 --> 00:39:40.369
I'm going to say that it's equal to 1.

00:39:41.030 --> 00:39:43.860
NFX is less than five, then I'm going

00:39:43.860 --> 00:39:45.050
to say it's equal to 0.

00:39:45.050 --> 00:39:46.440
So now I turn my continuous

00:39:46.440 --> 00:39:49.350
distribution into a binary distribution

00:39:49.350 --> 00:39:51.040
and now I can just Estimate it using

00:39:51.040 --> 00:39:52.440
the Bernoulli equation.

00:39:53.100 --> 00:39:54.910
Or I could turn X into 10 different

00:39:54.910 --> 00:39:57.280
values by just multiplying X by 10 and

00:39:57.280 --> 00:39:58.050
taking the floor.

00:39:58.050 --> 00:39:59.560
So now the values are zero to 9.

00:40:01.490 --> 00:40:04.150
So that's one that's actually the one

00:40:04.150 --> 00:40:06.110
of the easiest way to deal with the

00:40:06.110 --> 00:40:08.190
continuous variable that's not

00:40:08.190 --> 00:40:08.850
Gaussian.

00:40:12.900 --> 00:40:15.950
Sometimes X will be like text, so for

00:40:15.950 --> 00:40:18.800
example it could be like blue, orange

00:40:18.800 --> 00:40:19.430
or green.

00:40:20.080 --> 00:40:22.070
And then you just need to Map those

00:40:22.070 --> 00:40:25.390
different text tokens into integers.

00:40:25.390 --> 00:40:26.441
So I might say blue.

00:40:26.441 --> 00:40:28.654
I'm going to say I'm going to Map blue

00:40:28.654 --> 00:40:30.620
into zero, orange into one, green into

00:40:30.620 --> 00:40:32.580
two, and then I can just Solve by

00:40:32.580 --> 00:40:33.060
counting.

00:40:36.610 --> 00:40:38.830
And then finally I need to also

00:40:38.830 --> 00:40:40.380
Estimate the probability of Y.

00:40:41.060 --> 00:40:42.990
One common thing to do is just to say

00:40:42.990 --> 00:40:45.880
that Y is equally likely to be all the

00:40:45.880 --> 00:40:46.860
possible labels.

00:40:47.550 --> 00:40:49.440
And that can be a good thing to do,

00:40:49.440 --> 00:40:51.169
because maybe our training distribution

00:40:51.170 --> 00:40:52.870
isn't even, but you don't think you're

00:40:52.870 --> 00:40:54.310
training distribution will be the same

00:40:54.310 --> 00:40:55.790
as the test distribution.

00:40:55.790 --> 00:40:58.340
So then you say that probability of Y

00:40:58.340 --> 00:41:00.470
is uniform even though it's not uniform

00:41:00.470 --> 00:41:00.920
in training.

00:41:01.630 --> 00:41:03.530
If it's uniform, you can just ignore it

00:41:03.530 --> 00:41:05.910
because it won't have any effect on

00:41:05.910 --> 00:41:07.060
which Y is most likely.

00:41:07.980 --> 00:41:09.860
FY is discrete and non uniform.

00:41:09.860 --> 00:41:11.810
You can just solve it by counting how

00:41:11.810 --> 00:41:14.050
many times is Y equal 1 divided by all

00:41:14.050 --> 00:41:16.850
my data is the probability of Y equal

00:41:16.850 --> 00:41:17.070
1.

00:41:17.790 --> 00:41:19.450
If it's continuous, you can Model it as

00:41:19.450 --> 00:41:21.660
a Gaussian or chop it up into bins and

00:41:21.660 --> 00:41:23.000
then turn it into a classification

00:41:23.000 --> 00:41:23.360
problem.

00:41:25.690 --> 00:41:26.050
Right.

00:41:28.290 --> 00:41:31.550
So I'll give you your minute or two,

00:41:31.550 --> 00:41:32.230
Stretch break.

00:41:32.230 --> 00:41:33.650
But I want you to think about this

00:41:33.650 --> 00:41:34.370
while you do that.

00:41:35.390 --> 00:41:38.100
So suppose I want to classify a fruit

00:41:38.100 --> 00:41:40.230
based on description and my Features

00:41:40.230 --> 00:41:42.389
are weight, color, shape and whether

00:41:42.390 --> 00:41:44.190
it's a hard whether the outside is

00:41:44.190 --> 00:41:44.470
hard.

00:41:45.330 --> 00:41:47.960
And so first, here's some examples of

00:41:47.960 --> 00:41:49.100
those Features.

00:41:49.100 --> 00:41:50.750
See if you can figure out which fruit

00:41:50.750 --> 00:41:51.990
correspond to these Features.

00:41:52.630 --> 00:41:56.150
And second, what might be a good set of

00:41:56.150 --> 00:41:58.080
models to use for probability of XI

00:41:58.080 --> 00:41:59.730
given fruit for those four Features?

00:42:01.210 --> 00:42:03.620
So you have two minutes to think about

00:42:03.620 --> 00:42:05.630
it and Oregon Stretch or use the

00:42:05.630 --> 00:42:07.240
bathroom or check your e-mail or

00:42:07.240 --> 00:42:07.620
whatever.

00:44:24.040 --> 00:44:24.730
Alright.

00:44:26.640 --> 00:44:31.100
So first, what is the top 1.5 pounds

00:44:31.100 --> 00:44:31.640
red round?

00:44:31.640 --> 00:44:33.750
Yes, OK, good.

00:44:33.750 --> 00:44:34.870
That's what I was thinking.

00:44:34.870 --> 00:44:37.930
What's the 2nd 115 pounds?

00:44:39.070 --> 00:44:39.810
Avocado.

00:44:39.810 --> 00:44:41.260
That's a huge avocado.

00:44:43.770 --> 00:44:44.660
What is it?

00:44:46.290 --> 00:44:48.090
Watermelon watermelons, what I was

00:44:48.090 --> 00:44:48.450
thinking.

00:44:49.170 --> 00:44:52.140
.1 pounds purple round and not hard.

00:44:53.330 --> 00:44:54.980
I was thinking of a Grape.

00:44:54.980 --> 00:44:55.980
OK, good.

00:44:57.480 --> 00:44:58.900
There wasn't really, there wasn't

00:44:58.900 --> 00:45:00.160
necessarily a right answer.

00:45:00.160 --> 00:45:01.790
It's just kind of what I was thinking.

00:45:02.800 --> 00:45:05.642
Alright, and then how do you Model the

00:45:05.642 --> 00:45:07.700
probability of the feature given the

00:45:07.700 --> 00:45:08.450
fruit for each of these?

00:45:08.450 --> 00:45:09.550
So let's say the weight.

00:45:09.550 --> 00:45:11.172
What would be a good model for

00:45:11.172 --> 00:45:13.270
probability of XI given the label?

00:45:15.080 --> 00:45:17.420
Gaussian would, Gaussian would probably

00:45:17.420 --> 00:45:18.006
be a good choice.

00:45:18.006 --> 00:45:19.820
It has each of these probably has some

00:45:19.820 --> 00:45:21.250
expectation, maybe a Gaussian

00:45:21.250 --> 00:45:22.130
distribution around it.

00:45:24.000 --> 00:45:26.490
Alright, what about the color red,

00:45:26.490 --> 00:45:27.315
green, purple?

00:45:27.315 --> 00:45:28.440
What could I do for that?

00:45:31.440 --> 00:45:35.610
So I could use a multinomial so I can

00:45:35.610 --> 00:45:37.210
just turn it into discrete very

00:45:37.210 --> 00:45:39.410
discrete numbers, integer numbers and

00:45:39.410 --> 00:45:41.480
then count and the shape.

00:45:50.470 --> 00:45:52.470
So if there's assuming that there's

00:45:52.470 --> 00:45:54.470
other shapes, I don't know if there are

00:45:54.470 --> 00:45:55.880
star fruit for example.

00:45:56.790 --> 00:45:58.940
And then multinomial.

00:45:58.940 --> 00:46:00.640
But either way I'll turn it in discrete

00:46:00.640 --> 00:46:04.090
variables and count and the yes nodes.

00:46:05.540 --> 00:46:07.010
So that will be Binomial.

00:46:08.240 --> 00:46:08.540
OK.

00:46:14.840 --> 00:46:18.500
All right, so now we know how to

00:46:18.500 --> 00:46:20.770
Estimate probability of X given Y.

00:46:20.770 --> 00:46:23.065
Now after I go through all that work on

00:46:23.065 --> 00:46:25.178
the training data and I get new test

00:46:25.178 --> 00:46:25.512
sample.

00:46:25.512 --> 00:46:27.900
Now I want to know what's the most

00:46:27.900 --> 00:46:29.620
likely label of that test sample.

00:46:31.200 --> 00:46:31.660
So.

00:46:32.370 --> 00:46:33.860
I can write this in two ways.

00:46:33.860 --> 00:46:36.615
One is I can write Y is the argmax over

00:46:36.615 --> 00:46:38.735
the product of probability of XI given

00:46:38.735 --> 00:46:39.959
Y times probability of Y.

00:46:40.990 --> 00:46:44.334
Or I can write it as the argmax of the

00:46:44.334 --> 00:46:46.718
log of that, which is just the argmax

00:46:46.718 --> 00:46:48.970
of Y of the sum over I of log of

00:46:48.970 --> 00:46:50.904
probability of XI given Yi plus log of

00:46:50.904 --> 00:46:51.599
probability of Y.

00:46:52.570 --> 00:46:55.130
And I can do that because the thing

00:46:55.130 --> 00:46:57.798
that maximizes X also maximizes log of

00:46:57.798 --> 00:46:59.280
X and vice versa.

00:46:59.280 --> 00:47:01.910
And that's actually a really useful

00:47:01.910 --> 00:47:04.270
property because often the logs are

00:47:04.270 --> 00:47:05.745
probabilities are a lot simpler.

00:47:05.745 --> 00:47:08.790
And for example, if I took for example

00:47:08.790 --> 00:47:10.434
at the Gaussian, if I take the log of

00:47:10.434 --> 00:47:11.950
the Gaussian, then it just becomes a

00:47:11.950 --> 00:47:12.760
squared term.

00:47:13.640 --> 00:47:16.400
And the other thing is that these

00:47:16.400 --> 00:47:18.350
probability of Xis might be.

00:47:18.470 --> 00:47:21.553
If I have a lot of them, if I have like

00:47:21.553 --> 00:47:23.723
500 of them and they're on average like

00:47:23.723 --> 00:47:26.320
.1, that would be like .1 to the 500,

00:47:26.320 --> 00:47:27.530
which is going to go outside in

00:47:27.530 --> 00:47:28.690
numerical precision.

00:47:28.690 --> 00:47:30.740
So if you try to Compute this product

00:47:30.740 --> 00:47:32.290
directly, you're probably going to get

00:47:32.290 --> 00:47:34.470
0 or some kind of wonky value.

00:47:35.190 --> 00:47:37.320
And so it's much better to take the sum

00:47:37.320 --> 00:47:39.265
of the logs than to take the product of

00:47:39.265 --> 00:47:40.060
the probabilities.

00:47:42.650 --> 00:47:44.290
Right, so, but I can compute the

00:47:44.290 --> 00:47:45.830
probability of X&Y or the log

00:47:45.830 --> 00:47:48.004
probability of X&Y for each value of Y

00:47:48.004 --> 00:47:49.630
and then choose the value with maximum

00:47:49.630 --> 00:47:50.240
likelihood.

00:47:50.240 --> 00:47:51.686
That will work in the case of the

00:47:51.686 --> 00:47:53.409
digits because I only have 10 digits.

00:47:54.420 --> 00:47:56.940
And so I can check for each possible

00:47:56.940 --> 00:48:00.365
Digit, how likely is the sum of log

00:48:00.365 --> 00:48:01.958
probability of XI given Yi plus

00:48:01.958 --> 00:48:03.770
probability log probability of Y.

00:48:03.770 --> 00:48:06.980
And then I choose the Digit Digit label

00:48:06.980 --> 00:48:08.570
that makes this most likely.

00:48:11.240 --> 00:48:12.580
That's pretty simple.

00:48:12.580 --> 00:48:14.110
In the case of Y is discrete.

00:48:14.900 --> 00:48:16.415
And again, I just want to emphasize

00:48:16.415 --> 00:48:18.983
that this thing of turning product of

00:48:18.983 --> 00:48:21.070
probabilities into a sum of log

00:48:21.070 --> 00:48:23.250
probabilities is really, really widely

00:48:23.250 --> 00:48:23.760
used.

00:48:23.760 --> 00:48:27.610
Almost anytime you Solve for anything

00:48:27.610 --> 00:48:29.140
with probabilities, it involves that

00:48:29.140 --> 00:48:29.380
step.

00:48:31.840 --> 00:48:34.420
Now if Y is continuous, it's a bit more

00:48:34.420 --> 00:48:36.610
complicated and I.

00:48:37.440 --> 00:48:39.890
So I have the derivation here for you.

00:48:39.890 --> 00:48:42.166
So this is for the case.

00:48:42.166 --> 00:48:44.859
I'm going to use as an example the case

00:48:44.860 --> 00:48:47.470
where I'm modeling probability of Y

00:48:47.470 --> 00:48:51.400
minus XI of 1 dimensional Gaussian.

00:48:53.280 --> 00:48:56.260
And anytime you solve this kind of

00:48:56.260 --> 00:48:58.320
thing you're going to go through, you

00:48:58.320 --> 00:48:59.580
would go through the same derivation.

00:48:59.580 --> 00:49:00.280
If it's not.

00:49:00.280 --> 00:49:03.180
Just like a simple matter of if you

00:49:03.180 --> 00:49:05.000
don't have discrete wise, if you have

00:49:05.000 --> 00:49:06.360
continuous wise, then you have to find

00:49:06.360 --> 00:49:08.320
the Y that actually maximizes this

00:49:08.320 --> 00:49:10.760
because you can't check all possible

00:49:10.760 --> 00:49:12.310
values of a continuous variable.

00:49:14.180 --> 00:49:15.390
So it's not.

00:49:16.540 --> 00:49:17.451
It's a lot.

00:49:17.451 --> 00:49:18.362
It's a lot.

00:49:18.362 --> 00:49:20.350
It's a fair number of equations, but

00:49:20.350 --> 00:49:23.420
it's not anything super complicated.

00:49:23.420 --> 00:49:24.940
Let me see if I can get my cursor up

00:49:24.940 --> 00:49:25.960
there again, OK?

00:49:26.710 --> 00:49:29.560
Alright, so first I take the partial

00:49:29.560 --> 00:49:32.526
derivative of the log probability of

00:49:32.526 --> 00:49:34.780
X&Y with respect to Y and set it equal

00:49:34.780 --> 00:49:35.190
to 0.

00:49:35.190 --> 00:49:36.890
So you might remember from calculus

00:49:36.890 --> 00:49:38.720
like if you want to find the min or Max

00:49:38.720 --> 00:49:39.580
of some value.

00:49:40.290 --> 00:49:43.109
Then take the partial with respect to

00:49:43.110 --> 00:49:44.750
some variable.

00:49:44.750 --> 00:49:47.340
You take the partial derivative with

00:49:47.340 --> 00:49:48.800
respect to that variable and set it

00:49:48.800 --> 00:49:49.539
equal to 0.

00:49:50.680 --> 00:49:51.360
And.

00:49:53.080 --> 00:49:55.020
So here I did that.

00:49:55.020 --> 00:49:58.100
Now I've plugged in this Gaussian

00:49:58.100 --> 00:50:00.200
distribution and taken the log.

00:50:01.050 --> 00:50:02.510
And I kind of like there's some

00:50:02.510 --> 00:50:04.020
invisible steps here, because there's

00:50:04.020 --> 00:50:06.410
some terms like the log of one over

00:50:06.410 --> 00:50:07.940
square of 2π Sigma.

00:50:08.580 --> 00:50:10.069
That just don't.

00:50:10.069 --> 00:50:12.290
Those terms don't matter because they

00:50:12.290 --> 00:50:13.080
don't involve Y.

00:50:13.080 --> 00:50:14.743
So the partial derivative of those

00:50:14.743 --> 00:50:16.215
terms with respect to Y is 0.

00:50:16.215 --> 00:50:19.090
So I just didn't include them.

00:50:19.750 --> 00:50:21.815
So these are the terms that include Y

00:50:21.815 --> 00:50:23.590
and I've already taken the log.

00:50:23.590 --> 00:50:25.550
This was originally east to the -, 1

00:50:25.550 --> 00:50:27.839
half whatever is shown here, and the

00:50:27.839 --> 00:50:30.360
log of X of X is equal to X.

00:50:31.840 --> 00:50:33.490
And so I get this guy.

00:50:34.450 --> 00:50:36.530
Now I broke it out into different

00:50:36.530 --> 00:50:39.320
terms, so I did the quadratic of Y

00:50:39.320 --> 00:50:41.190
minus XI minus MU I ^2.

00:50:42.420 --> 00:50:44.100
Mainly so that I don't have to use the

00:50:44.100 --> 00:50:45.620
chain rule and I can keep my

00:50:45.620 --> 00:50:46.740
derivatives really Simple.

00:50:47.830 --> 00:50:51.959
So here I just broke that out to y ^2 y

00:50:51.960 --> 00:50:54.130
axis YMUI.

00:50:54.130 --> 00:50:55.530
And again, I don't need to worry about

00:50:55.530 --> 00:50:57.779
the MU I squared over Sigma I squared

00:50:57.780 --> 00:50:59.750
because it doesn't involve Y so I just

00:50:59.750 --> 00:51:00.230
left it out.

00:51:02.140 --> 00:51:03.990
I.

00:51:04.100 --> 00:51:07.021
Take the derivative with respect to Y.

00:51:07.021 --> 00:51:09.468
So the derivative of y ^2 is 2 Y.

00:51:09.468 --> 00:51:10.976
So this half goes away.

00:51:10.976 --> 00:51:14.080
Derivative of YX is just X.

00:51:15.070 --> 00:51:18.000
So this should be a subscript I.

00:51:18.730 --> 00:51:21.120
And then I did the same for these guys

00:51:21.120 --> 00:51:21.330
here.

00:51:22.500 --> 00:51:25.740
It's just basic algebra, so I just try

00:51:25.740 --> 00:51:27.610
to group the terms that involve Y and

00:51:27.610 --> 00:51:29.480
the terms that don't involve Yi, put

00:51:29.480 --> 00:51:30.840
the terms that don't involve Y and the

00:51:30.840 --> 00:51:33.370
right side, and then finally I divide

00:51:33.370 --> 00:51:36.830
the coefficient of Y and I get this guy

00:51:36.830 --> 00:51:37.150
here.

00:51:38.030 --> 00:51:41.269
So at the end Y is equal to 1 over the

00:51:41.270 --> 00:51:44.408
sum over all the features of 1 / sqrt.

00:51:44.408 --> 00:51:46.690
I mean one over Sigma I ^2.

00:51:47.420 --> 00:51:50.580
Plus one over Sigma y ^2 which is the

00:51:50.580 --> 00:51:52.160
standard deviation of the Prior of Y.

00:51:52.160 --> 00:51:53.906
Or if I just assumed uniform likelihood

00:51:53.906 --> 00:51:55.520
of Yi wouldn't need that term.

00:51:56.610 --> 00:51:59.400
And then that's times the sum over all

00:51:59.400 --> 00:52:02.700
the features of that feature value.

00:52:02.700 --> 00:52:03.930
This should be subscript I.

00:52:04.940 --> 00:52:10.430
Plus MU I divided by Sigma I ^2 plus mu

00:52:10.430 --> 00:52:13.811
Y, the Prior mean of Y divided by Sigma

00:52:13.811 --> 00:52:14.539
y ^2.

00:52:16.150 --> 00:52:18.940
And so this is just a, it's actually

00:52:18.940 --> 00:52:19.849
just a weighted.

00:52:19.850 --> 00:52:22.823
If you say that one over Sigma I

00:52:22.823 --> 00:52:26.035
squared is Wei, it's like a weight for

00:52:26.035 --> 00:52:27.565
that prediction of the ith feature.

00:52:27.565 --> 00:52:29.830
This is just a weighted average of the

00:52:29.830 --> 00:52:31.720
predictions from all the Features

00:52:31.720 --> 00:52:33.250
that's weighted by one over the

00:52:33.250 --> 00:52:35.573
steering deviation squared or one over

00:52:35.573 --> 00:52:36.190
the variance.

00:52:37.590 --> 00:52:40.421
And so I have one over the sum over I

00:52:40.421 --> 00:52:45.683
of WI plus WY times, the sum X plus mu

00:52:45.683 --> 00:52:49.722
I XI plus MU I times, Wei plus mu Y

00:52:49.722 --> 00:52:50.100
times.

00:52:50.100 --> 00:52:50.670
Why?

00:52:51.630 --> 00:52:53.240
Amy sounds similar, unfortunately.

00:52:54.780 --> 00:52:56.430
So it's just the weighted average of

00:52:56.430 --> 00:52:57.910
all the predictions of the individual

00:52:57.910 --> 00:52:58.174
features.

00:52:58.174 --> 00:53:00.093
And it makes sense that it kind of

00:53:00.093 --> 00:53:01.624
makes sense intuitively that the weight

00:53:01.624 --> 00:53:02.650
is 1 over the variance.

00:53:02.650 --> 00:53:04.490
So if you have really high variance,

00:53:04.490 --> 00:53:05.790
then the weight is small.

00:53:05.790 --> 00:53:08.155
So if, for example, maybe the

00:53:08.155 --> 00:53:09.839
temperature in Sacramento is a really

00:53:09.840 --> 00:53:11.513
bad predictor for the temperature in

00:53:11.513 --> 00:53:12.984
Cleveland, so it will have high

00:53:12.984 --> 00:53:14.840
variance and it gets a little weight,

00:53:14.840 --> 00:53:16.460
while the temperature in Cleveland the

00:53:16.460 --> 00:53:19.130
previous day is much more highly

00:53:19.130 --> 00:53:20.849
predictive, has lower variance, so

00:53:20.850 --> 00:53:21.639
it'll get more weight.

00:53:32.280 --> 00:53:35.380
So let me pause here.

00:53:35.380 --> 00:53:38.690
So any questions about?

00:53:39.670 --> 00:53:43.255
Estimating the likelihoods P of X given

00:53:43.255 --> 00:53:47.970
Y, or solving for the Y that makes.

00:53:47.970 --> 00:53:49.880
That's most likely given your

00:53:49.880 --> 00:53:50.500
likelihoods.

00:53:52.460 --> 00:53:54.470
And obviously if I'm happy to work

00:53:54.470 --> 00:53:56.610
through this in office hours as well in

00:53:56.610 --> 00:53:59.940
the TAS should also if you want to like

00:53:59.940 --> 00:54:01.100
spend more time working through the

00:54:01.100 --> 00:54:01.530
equations.

00:54:03.920 --> 00:54:04.930
I just want to pause.

00:54:04.930 --> 00:54:07.830
I know it's a lot of math to soak up.

00:54:09.870 --> 00:54:13.260
And really, it's not that memorizing

00:54:13.260 --> 00:54:14.370
these things isn't important.

00:54:14.370 --> 00:54:15.860
It's really the process that you just

00:54:15.860 --> 00:54:17.385
set the partial derivative with respect

00:54:17.385 --> 00:54:20.140
to Y, set it to zero, and then you do

00:54:20.140 --> 00:54:20.540
the.

00:54:21.250 --> 00:54:23.120
Do the partial derivative and solve the

00:54:23.120 --> 00:54:23.510
algebra.

00:54:26.700 --> 00:54:28.050
All right, I'll go on then.

00:54:28.050 --> 00:54:31.990
So far, this is pure maximum likelihood

00:54:31.990 --> 00:54:32.530
estimation.

00:54:32.530 --> 00:54:34.920
I'm not, I'm not imposing any kinds of

00:54:34.920 --> 00:54:36.470
Priors over my parameters.

00:54:37.570 --> 00:54:39.600
In practice, you do want to impose a

00:54:39.600 --> 00:54:41.010
Prior in your parameters to make sure

00:54:41.010 --> 00:54:42.220
you don't have any zeros.

00:54:43.750 --> 00:54:46.380
Otherwise, like if some in the digits

00:54:46.380 --> 00:54:48.809
case for example the test sample had a

00:54:48.810 --> 00:54:50.470
dot in an unlikely place.

00:54:50.470 --> 00:54:52.662
If I had just had like a one and some

00:54:52.662 --> 00:54:54.030
unlikely pixel, all the probabilities

00:54:54.030 --> 00:54:55.630
would be 0 and you wouldn't know what

00:54:55.630 --> 00:54:57.620
the label is because of that one stupid

00:54:57.620 --> 00:54:57.970
pixel.

00:54:58.730 --> 00:55:01.040
So you want to have some kind of Prior?

00:55:01.730 --> 00:55:03.425
To avoid these zero probabilities.

00:55:03.425 --> 00:55:06.260
So the most common case if you're

00:55:06.260 --> 00:55:08.760
estimating a distribution of discrete

00:55:08.760 --> 00:55:10.430
variables like a multinomial or

00:55:10.430 --> 00:55:13.010
Binomial, is to just initialize with

00:55:13.010 --> 00:55:13.645
some count.

00:55:13.645 --> 00:55:16.180
So you just say for example alpha

00:55:16.180 --> 00:55:16.880
equals one.

00:55:17.610 --> 00:55:20.110
And now I say the probability of X I =

00:55:20.110 --> 00:55:21.620
V given y = K.

00:55:22.400 --> 00:55:24.950
Is Alpha plus the count of how many

00:55:24.950 --> 00:55:27.740
times XI equals V and y = K.

00:55:28.690 --> 00:55:31.865
Divided by the all the different values

00:55:31.865 --> 00:55:35.300
of alpha plus account of XI equals that

00:55:35.300 --> 00:55:37.610
value in y = K probably for clarity I

00:55:37.610 --> 00:55:39.700
should have used something other than B

00:55:39.700 --> 00:55:41.630
in the denominator, but hopefully

00:55:41.630 --> 00:55:42.230
that's clear enough.

00:55:43.060 --> 00:55:46.170
Here's the and then here's the Python

00:55:46.170 --> 00:55:47.070
for that, so it's just.

00:55:47.880 --> 00:55:50.350
Sum of all the values where XI equals V

00:55:50.350 --> 00:55:52.470
and y = K Plus some alpha.

00:55:53.300 --> 00:55:54.980
So if alpha equals zero, then I don't

00:55:54.980 --> 00:55:55.710
have any Prior.

00:55:56.840 --> 00:56:00.450
And then I'm just dividing by the sum

00:56:00.450 --> 00:56:04.270
of times at y = K and there will be.

00:56:04.850 --> 00:56:06.540
The number of alphas will be equal to

00:56:06.540 --> 00:56:08.150
the number of different values, so this

00:56:08.150 --> 00:56:10.510
is like a little bit of a shortcut, but

00:56:10.510 --> 00:56:11.330
it's the same thing.

00:56:12.860 --> 00:56:14.760
If I have a continuous variable and

00:56:14.760 --> 00:56:15.060
I've.

00:56:15.730 --> 00:56:17.010
Modeled it with the Gaussian.

00:56:17.010 --> 00:56:18.470
Then the usual thing to do is just to

00:56:18.470 --> 00:56:20.180
add a small value to your steering

00:56:20.180 --> 00:56:21.420
deviation or your variance.

00:56:22.110 --> 00:56:24.320
And you might want to make that value

00:56:24.320 --> 00:56:27.650
if N is unknown, then make it dependent

00:56:27.650 --> 00:56:29.300
on north so that if you have a huge

00:56:29.300 --> 00:56:31.395
number of samples then the effect of

00:56:31.395 --> 00:56:33.880
the Prior will go down, which is what

00:56:33.880 --> 00:56:34.170
you want.

00:56:36.140 --> 00:56:39.513
So for example, you can say that the

00:56:39.513 --> 00:56:41.990
stern deviation is whatever this

00:56:41.990 --> 00:56:44.770
whatever the MLE estimate of the stern

00:56:44.770 --> 00:56:47.340
deviation is, plus some small value

00:56:47.340 --> 00:56:49.730
sqrt 1 over the length of north.

00:56:50.420 --> 00:56:51.350
Of X, sorry.

00:57:00.440 --> 00:57:02.670
So what the Prior does is it.

00:57:02.810 --> 00:57:05.995
In the case of the discrete variables,

00:57:05.995 --> 00:57:09.110
the Prior is trying to push your

00:57:09.110 --> 00:57:11.152
Estimate towards a uniform likelihood.

00:57:11.152 --> 00:57:13.000
In fact, in both cases it's pushing it

00:57:13.000 --> 00:57:14.280
towards a uniform likelihood.

00:57:15.400 --> 00:57:18.670
So if you had a really large alpha,

00:57:18.670 --> 00:57:20.550
then let's say.

00:57:22.090 --> 00:57:23.440
Let's say that.

00:57:24.620 --> 00:57:25.850
Or I don't know if I can think of

00:57:25.850 --> 00:57:26.170
something.

00:57:28.140 --> 00:57:29.550
Let's say you have a population of

00:57:29.550 --> 00:57:30.900
students and you're trying to estimate

00:57:30.900 --> 00:57:32.510
the probability that a student is male.

00:57:33.520 --> 00:57:36.570
If I say alpha equals 1000, then I'm

00:57:36.570 --> 00:57:37.860
going to need like an awful lot of

00:57:37.860 --> 00:57:40.156
students before I budge very far from a

00:57:40.156 --> 00:57:42.070
5050 chance that a student is male or

00:57:42.070 --> 00:57:42.620
female.

00:57:42.620 --> 00:57:44.057
Because I'll start with saying there's

00:57:44.057 --> 00:57:46.213
1000 males and 1000 females, and then

00:57:46.213 --> 00:57:48.676
I'll count all the males and add them

00:57:48.676 --> 00:57:50.832
to 1000, count all the females, add

00:57:50.832 --> 00:57:53.370
them to 1000, and then I would take the

00:57:53.370 --> 00:57:55.210
male plus 1000 count and divide it by

00:57:55.210 --> 00:57:57.660
2000 plus the total population.

00:57:59.130 --> 00:58:00.860
If Alpha is 0, then I'm going to get

00:58:00.860 --> 00:58:03.410
just my raw empirical Estimate.

00:58:03.410 --> 00:58:06.810
So if I had like 3 students and I say

00:58:06.810 --> 00:58:09.090
alpha equals zero, and I have two males

00:58:09.090 --> 00:58:11.140
and a female, then I'll say 2/3 of them

00:58:11.140 --> 00:58:11.550
are male.

00:58:12.410 --> 00:58:14.670
If I say alpha is 1 and I have two

00:58:14.670 --> 00:58:17.110
males and a female, then I would say

00:58:17.110 --> 00:58:20.490
that my probability of male is 3 / 5

00:58:20.490 --> 00:58:24.100
because it's 2 + 1 / 3 + 2.

00:58:27.060 --> 00:58:28.330
Their deviation it's the same.

00:58:28.330 --> 00:58:30.240
It's like trying to just broaden your

00:58:30.240 --> 00:58:32.600
variance from what you would Estimate

00:58:32.600 --> 00:58:33.580
directly from the data.

00:58:36.500 --> 00:58:39.260
So I think I will not ask you all these

00:58:39.260 --> 00:58:41.210
probabilities because they're kind of

00:58:41.210 --> 00:58:43.220
you've shown the ability to count

00:58:43.220 --> 00:58:44.810
before mostly.

00:58:46.550 --> 00:58:47.640
And.

00:58:47.850 --> 00:58:50.060
So here's for example, the probability

00:58:50.060 --> 00:58:54.509
of X 1 = 0 and y = 0 is 2 out of four.

00:58:54.510 --> 00:58:56.050
I can get that just by looking down

00:58:56.050 --> 00:58:56.670
these rows.

00:58:56.670 --> 00:58:58.870
It takes a little bit of time, but

00:58:58.870 --> 00:59:02.786
there's four times that y = 0 and out

00:59:02.786 --> 00:59:06.660
of those two times X 1 = 0 and so this

00:59:06.660 --> 00:59:07.440
is 2 out of four.

00:59:08.090 --> 00:59:08.930
And the same.

00:59:08.930 --> 00:59:11.260
I can use the same counting method to

00:59:11.260 --> 00:59:13.120
get all of these other probabilities

00:59:13.120 --> 00:59:13.410
here.

00:59:15.770 --> 00:59:19.450
So just to check that everyone's awake,

00:59:19.450 --> 00:59:22.970
if I, what is the probability of Y?

00:59:23.840 --> 00:59:27.370
And X 1 = 1 and X 2 = 1.

00:59:28.500 --> 00:59:30.019
So can you get it from?

00:59:30.019 --> 00:59:32.560
Can you get it from this guy under an

00:59:32.560 --> 00:59:33.450
independence?

00:59:33.450 --> 00:59:35.670
So get it from this under an under an I

00:59:35.670 --> 00:59:36.540
Bayes assumption.

00:59:41.350 --> 00:59:43.240
Let's say I should say probability of Y

00:59:43.240 --> 00:59:43.860
equal 1.

00:59:45.380 --> 00:59:47.910
Probability of y = 1 given X 1 = 1 and

00:59:47.910 --> 00:59:48.930
X 2 = 1.

00:59:57.500 --> 01:00:00.560
And you don't worry about simplifying

01:00:00.560 --> 01:00:02.610
your numerator and denominator.

01:00:03.530 --> 01:00:05.110
What are the things that get multiplied

01:00:05.110 --> 01:00:05.610
together?

01:00:10.460 --> 01:00:14.350
Not sort of, partly that's in there.

01:00:15.220 --> 01:00:17.880
Raise your hand if you think the

01:00:17.880 --> 01:00:18.560
answer.

01:00:19.550 --> 01:00:21.130
I just want to give everyone time.

01:00:24.650 --> 01:00:27.962
But I mean probability of y = 1 given X

01:00:27.962 --> 01:00:29.960
1 = 1 and X 2 = 1.

01:00:39.830 --> 01:00:41.220
A Naive Bayes assumption.

01:01:24.310 --> 01:01:25.800
The raise your hand if you.

01:01:26.490 --> 01:01:27.030
Finished.

01:01:56.450 --> 01:01:57.740
But don't tell me the answer yet.

01:02:18.470 --> 01:02:19.260
Equals one.

01:02:23.210 --> 01:02:23.420
Alright.

01:02:23.420 --> 01:02:24.830
Did anybody get it yet?

01:02:24.830 --> 01:02:25.950
Raise your hand if you did.

01:02:25.950 --> 01:02:26.910
I just don't want to.

01:02:28.170 --> 01:02:29.110
Give it too early.

01:03:46.370 --> 01:03:46.960
Alright.

01:03:48.170 --> 01:03:52.029
Example, some people have gotten it, so

01:03:52.030 --> 01:03:53.950
let me I'll start going through it.

01:03:53.950 --> 01:03:55.480
All right, so the Naive Bayes

01:03:55.480 --> 01:03:56.005
assumption.

01:03:56.005 --> 01:03:57.760
So this would be.

01:03:58.060 --> 01:03:58.250
OK.

01:04:00.690 --> 01:04:02.960
OK, probability it's actually my touch

01:04:02.960 --> 01:04:03.230
screen.

01:04:03.230 --> 01:04:04.400
I think is kind of broken.

01:04:05.250 --> 01:04:09.560
Probability of X1 given Y times

01:04:09.560 --> 01:04:14.815
probability X2 given Y sorry equals

01:04:14.815 --> 01:04:15.200
one.

01:04:16.630 --> 01:04:19.050
Times probability of Y equal 1.

01:04:19.910 --> 01:04:21.950
Right, so it's the product of the

01:04:21.950 --> 01:04:23.180
probabilities of the Features.

01:04:23.180 --> 01:04:24.730
Give them label times the probability

01:04:24.730 --> 01:04:25.240
of the label.

01:04:26.500 --> 01:04:29.990
And so that will be probability of XYX.

01:04:31.030 --> 01:04:32.819
1 = 1.

01:04:33.850 --> 01:04:37.317
Given probability of Yi mean given y =

01:04:37.317 --> 01:04:38.260
1 is 3/4.

01:04:42.110 --> 01:04:46.010
And probably the X 2 = 1 given y = 1 is

01:04:46.010 --> 01:04:46.750
3/4.

01:04:49.250 --> 01:04:52.550
And the probability that y = 1 is two

01:04:52.550 --> 01:04:53.940
quarters or 1/2.

01:04:58.570 --> 01:05:00.180
So it's 930 seconds.

01:05:01.120 --> 01:05:01.390
Right.

01:05:02.580 --> 01:05:05.846
And the probability that y = 0 given X

01:05:05.846 --> 01:05:08.059
1 = 1 and Y 1 = 1.

01:05:09.800 --> 01:05:11.840
I mean sorry, the probability of y = 0

01:05:11.840 --> 01:05:14.480
given the X is equal equal 1.

01:05:15.620 --> 01:05:16.770
Is.

01:05:18.600 --> 01:05:19.190
Let's see.

01:05:20.250 --> 01:05:23.780
So that would be 2 fourths times 2

01:05:23.780 --> 01:05:24.300
fourths.

01:05:25.180 --> 01:05:26.320
Times 2 fourths.

01:05:27.260 --> 01:05:31.300
So if X 1 = 1 and X2 equal 1, then it's

01:05:31.300 --> 01:05:33.540
more likely that Y is equal to 1 than

01:05:33.540 --> 01:05:35.070
that Y is equal to 0.

01:05:41.720 --> 01:05:46.750
If I had if I use my Prior, this is how

01:05:46.750 --> 01:05:48.055
the probabilities would change.

01:05:48.055 --> 01:05:51.060
So if I say alpha equals one, you can

01:05:51.060 --> 01:05:52.900
see that the probabilities get less

01:05:52.900 --> 01:05:53.510
Peaky.

01:05:53.510 --> 01:05:56.422
So I went from 1/4 to 261 quarter and

01:05:56.422 --> 01:05:58.951
3/4 to 2/6 and four six for example.

01:05:58.951 --> 01:06:02.316
So 1/3 and 2/3 is more uniform than 1/4

01:06:02.316 --> 01:06:03.129
and 3/4.

01:06:05.050 --> 01:06:07.040
And then if the initial estimate was

01:06:07.040 --> 01:06:09.020
1/2, the final Estimate will still be

01:06:09.020 --> 01:06:11.620
1/2 because it's because this Prior is

01:06:11.620 --> 01:06:13.650
just trying to push things towards 1/2.

01:06:20.780 --> 01:06:24.220
So I want to give one example of a use

01:06:24.220 --> 01:06:24.550
case.

01:06:24.550 --> 01:06:25.685
So I've actually.

01:06:25.685 --> 01:06:28.360
I mean I want to say like I used Naive

01:06:28.360 --> 01:06:30.630
Bayes, but I use that assumption pretty

01:06:30.630 --> 01:06:31.440
often.

01:06:31.440 --> 01:06:33.480
For example if I wanted to Estimate a

01:06:33.480 --> 01:06:35.210
distribution of RGB colors.

01:06:36.740 --> 01:06:38.410
I would first convert it to a different

01:06:38.410 --> 01:06:39.860
color space, but let's just say I want

01:06:39.860 --> 01:06:41.780
to Estimate distribution of LGBT RGB

01:06:41.780 --> 01:06:42.390
colors.

01:06:42.390 --> 01:06:45.055
Then even though it's 3 dimensions, is

01:06:45.055 --> 01:06:45.690
a pretty.

01:06:45.690 --> 01:06:47.920
You need like a lot of data to estimate

01:06:47.920 --> 01:06:48.610
that distribution.

01:06:48.610 --> 01:06:50.700
And So what I might do is I'll say,

01:06:50.700 --> 01:06:52.820
well, I'm going to assume that RG and B

01:06:52.820 --> 01:06:54.645
are independent and so the probability

01:06:54.645 --> 01:06:57.350
of RGB is just the probability of R

01:06:57.350 --> 01:06:58.808
times probability of G times

01:06:58.808 --> 01:06:59.524
probability B.

01:06:59.524 --> 01:07:01.600
And I compute a histogram for each of

01:07:01.600 --> 01:07:04.940
those, and I use that to get my as my

01:07:04.940 --> 01:07:06.230
likelihood Estimate.

01:07:06.560 --> 01:07:08.520
So it's like really commonly used in

01:07:08.520 --> 01:07:10.120
that kind of setting where you want to

01:07:10.120 --> 01:07:11.770
Estimate the distribution of multiple

01:07:11.770 --> 01:07:13.380
variables and there's just no way to

01:07:13.380 --> 01:07:13.810
get a Joint.

01:07:13.810 --> 01:07:17.100
The only options you really have are to

01:07:17.100 --> 01:07:18.410
make something the Naive Bayes

01:07:18.410 --> 01:07:21.330
assumption or to do a mixture of

01:07:21.330 --> 01:07:23.416
Gaussians, which we'll talk about later

01:07:23.416 --> 01:07:24.320
in the semester.

01:07:26.380 --> 01:07:27.940
Right, But here's the case where it's

01:07:27.940 --> 01:07:29.450
used for object detection.

01:07:29.450 --> 01:07:32.280
So this was by Schneiderman Kanadi and

01:07:32.280 --> 01:07:35.500
it was the most accurate face and car

01:07:35.500 --> 01:07:36.520
detector for a while.

01:07:37.450 --> 01:07:39.720
They detector is based on wavelet

01:07:39.720 --> 01:07:41.420
coefficients which are just like local

01:07:41.420 --> 01:07:42.610
intensity differences.

01:07:43.320 --> 01:07:46.010
And the.

01:07:46.090 --> 01:07:48.880
The It's a Probabilistic framework, so

01:07:48.880 --> 01:07:51.070
they're trying to say whether if you

01:07:51.070 --> 01:07:54.107
Extract a window of Features from the

01:07:54.107 --> 01:07:56.386
image, some Features over some part of

01:07:56.386 --> 01:07:56.839
the image.

01:07:57.450 --> 01:07:59.020
And Extract all the wavelet

01:07:59.020 --> 01:08:00.330
coefficients.

01:08:00.330 --> 01:08:02.390
Then you want to say that it's a face

01:08:02.390 --> 01:08:03.950
if the probability of those

01:08:03.950 --> 01:08:05.853
coefficients is greater given that it's

01:08:05.853 --> 01:08:08.390
a face, than given that's not a face

01:08:08.390 --> 01:08:10.330
times the probability that's a face

01:08:10.330 --> 01:08:11.730
over the probability that's not a face.

01:08:12.430 --> 01:08:14.680
So it's this basic Probabilistic Model.

01:08:14.680 --> 01:08:16.740
And again, the probability modeling.

01:08:16.740 --> 01:08:17.920
The probability of all those

01:08:17.920 --> 01:08:19.370
coefficients is way too hard.

01:08:20.330 --> 01:08:23.290
On the other hand, modeling all the

01:08:23.290 --> 01:08:25.560
Features as independent given the label

01:08:25.560 --> 01:08:26.950
is a little bit too much of a

01:08:26.950 --> 01:08:28.410
simplifying assumption.

01:08:28.410 --> 01:08:30.270
So they use this algorithm that they

01:08:30.270 --> 01:08:33.220
call semi Naive Bayes which is proposed

01:08:33.220 --> 01:08:34.040
earlier.

01:08:35.220 --> 01:08:37.946
Where you just you Model the

01:08:37.946 --> 01:08:39.803
probabilities of little groups of

01:08:39.803 --> 01:08:41.380
features and then you say that the

01:08:41.380 --> 01:08:43.166
total probability is the probability

01:08:43.166 --> 01:08:44.830
the product or the probabilities of

01:08:44.830 --> 01:08:45.849
these groups of Features.

01:08:46.710 --> 01:08:47.845
So they call these patterns.

01:08:47.845 --> 01:08:50.160
So first you do some look at the mutual

01:08:50.160 --> 01:08:51.870
information, you have ways of measuring

01:08:51.870 --> 01:08:54.050
the dependence of different variables,

01:08:54.050 --> 01:08:56.470
and you cluster the Features together

01:08:56.470 --> 01:08:58.280
based on their dependencies.

01:08:58.920 --> 01:09:00.430
And then for little clusters of

01:09:00.430 --> 01:09:02.149
Features, 3 Features.

01:09:03.060 --> 01:09:05.800
You Estimate the probability of the

01:09:05.800 --> 01:09:08.500
Joint combination of these features and

01:09:08.500 --> 01:09:11.230
then the total probability of all the

01:09:11.230 --> 01:09:11.620
Features.

01:09:11.620 --> 01:09:12.920
I'm glad this isn't worker.

01:09:12.920 --> 01:09:14.788
The total probability of all the

01:09:14.788 --> 01:09:16.660
features is the product of the

01:09:16.660 --> 01:09:18.270
probabilities of each of these groups

01:09:18.270 --> 01:09:18.840
of Features.

01:09:19.890 --> 01:09:21.140
And so you Model.

01:09:21.140 --> 01:09:23.616
Likely a set of features are given that

01:09:23.616 --> 01:09:25.270
it's a face, and how likely they are

01:09:25.270 --> 01:09:27.790
given that it's not a face or given a

01:09:27.790 --> 01:09:29.280
random patch from an image.

01:09:29.930 --> 01:09:32.260
And then that can be used to classify

01:09:32.260 --> 01:09:33.060
images as face.

01:09:33.060 --> 01:09:33.896
You're not face.

01:09:33.896 --> 01:09:35.560
And you would Estimate this separately

01:09:35.560 --> 01:09:37.120
for cars and for each orientation of

01:09:37.120 --> 01:09:38.110
car question.

01:09:43.310 --> 01:09:45.399
So the question was what beat the 2005

01:09:45.400 --> 01:09:45.840
model?

01:09:45.840 --> 01:09:47.750
I'm not really sure that there was

01:09:47.750 --> 01:09:50.180
something that beat it in 2006, but

01:09:50.180 --> 01:09:53.820
that when Dalal Triggs SVM based

01:09:53.820 --> 01:09:55.570
detector came out.

01:09:56.200 --> 01:09:57.680
And I think it might have been, I

01:09:57.680 --> 01:10:00.617
didn't look it up so I'm not sure, but

01:10:00.617 --> 01:10:02.930
I was, I'm pretty confident it was the

01:10:02.930 --> 01:10:04.947
most accurate up to 2005, but not

01:10:04.947 --> 01:10:06.070
confident after that.

01:10:07.250 --> 01:10:10.430
And now it took a while for face

01:10:10.430 --> 01:10:12.650
detection to get more accurate than

01:10:12.650 --> 01:10:15.630
most famous face detector was actually

01:10:15.630 --> 01:10:18.330
the Viola joins detector, which was

01:10:18.330 --> 01:10:20.515
popular because it was really fast.

01:10:20.515 --> 01:10:24.046
This thing man at a couple frames per

01:10:24.046 --> 01:10:26.414
second, but Viola Jones ran at 15

01:10:26.414 --> 01:10:28.560
frames per second in 2001.

01:10:30.310 --> 01:10:31.960
But Viola Jones wasn't quite as

01:10:31.960 --> 01:10:32.460
accurate.

01:10:35.210 --> 01:10:37.840
Alright, so Summary of Naive bees.

01:10:38.180 --> 01:10:38.790
And.

01:10:39.940 --> 01:10:41.740
So the key assumption is that the

01:10:41.740 --> 01:10:43.460
Features are independent given the

01:10:43.460 --> 01:10:43.870
labels.

01:10:46.730 --> 01:10:48.110
The parameters are just the

01:10:48.110 --> 01:10:50.173
probabilities, are the parameters of

01:10:50.173 --> 01:10:51.990
each of these probability functions,

01:10:51.990 --> 01:10:53.908
the probability of each feature given Y

01:10:53.908 --> 01:10:55.750
and probability of Y and justice.

01:10:55.750 --> 01:10:57.250
Like in the Simple fruit example I

01:10:57.250 --> 01:10:59.405
gave, you can use different models for

01:10:59.405 --> 01:10:59.976
different features.

01:10:59.976 --> 01:11:02.340
Some of the features could be discrete

01:11:02.340 --> 01:11:04.120
values and some could be continuous

01:11:04.120 --> 01:11:04.560
values.

01:11:04.560 --> 01:11:05.520
That's not a problem.

01:11:08.520 --> 01:11:10.150
You have to choose which probability

01:11:10.150 --> 01:11:11.510
function you're going to use for each

01:11:11.510 --> 01:11:11.940
feature.

01:11:14.450 --> 01:11:16.250
Nine days can be useful if you have

01:11:16.250 --> 01:11:18.080
limited training data, because you only

01:11:18.080 --> 01:11:19.560
have to Estimate these one-dimensional

01:11:19.560 --> 01:11:21.150
distributions, which you can do from

01:11:21.150 --> 01:11:22.370
relatively few Samples.

01:11:23.000 --> 01:11:24.420
And if the features are not highly

01:11:24.420 --> 01:11:26.540
interdependent, and it can also be

01:11:26.540 --> 01:11:27.970
useful as a baseline if you want

01:11:27.970 --> 01:11:29.766
something that's fast to code, train

01:11:29.766 --> 01:11:30.579
and test.

01:11:30.580 --> 01:11:32.900
So as you do your homework, I think out

01:11:32.900 --> 01:11:34.860
of the methods, Naive Bayes has the

01:11:34.860 --> 01:11:37.140
lowest training plus test time.

01:11:37.140 --> 01:11:40.139
Logistic regression is going to be

01:11:40.140 --> 01:11:42.618
roughly tied for test time, but it

01:11:42.618 --> 01:11:43.680
takes an awful lot.

01:11:43.680 --> 01:11:45.980
Well, it takes longer to train.

01:11:45.980 --> 01:11:48.379
KNN takes no time to train, but takes a

01:11:48.380 --> 01:11:49.570
whole lot longer to test.

01:11:54.630 --> 01:11:56.830
So when not to use?

01:11:56.830 --> 01:11:58.760
Usually Logistic or linear regression

01:11:58.760 --> 01:12:01.070
will work better if you have enough

01:12:01.070 --> 01:12:01.440
data.

01:12:02.230 --> 01:12:05.510
And the reason is that under most

01:12:05.510 --> 01:12:07.860
probability the exponential

01:12:07.860 --> 01:12:09.790
distribution of probability models

01:12:09.790 --> 01:12:11.940
which include Binomial, multinomial and

01:12:11.940 --> 01:12:12.530
Gaussian.

01:12:13.640 --> 01:12:15.657
You can rewrite Naive Bayes as a linear

01:12:15.657 --> 01:12:18.993
function of the input features, but the

01:12:18.993 --> 01:12:21.740
linear function is highly constrained

01:12:21.740 --> 01:12:23.750
based on this, estimating likelihoods

01:12:23.750 --> 01:12:25.650
for each feature separately.

01:12:25.650 --> 01:12:27.500
Where linear and logistic regression,

01:12:27.500 --> 01:12:28.970
which we'll talk about next Thursday,

01:12:28.970 --> 01:12:30.815
are not constrained, you can solve for

01:12:30.815 --> 01:12:32.300
the full range of coefficients.

01:12:33.440 --> 01:12:35.050
The other issue is that it doesn't

01:12:35.050 --> 01:12:37.890
provide a very good confidence Estimate

01:12:37.890 --> 01:12:39.720
because it over counts the influence of

01:12:39.720 --> 01:12:40.880
dependent variables.

01:12:40.880 --> 01:12:42.860
If you repeat a feature of many times,

01:12:42.860 --> 01:12:44.680
it's going to count it every time, and

01:12:44.680 --> 01:12:47.215
so it will tend to have too much weight

01:12:47.215 --> 01:12:48.930
and give you bad confidence estimates.

01:12:51.010 --> 01:12:55.100
9 Bayes is easy and fast to train, Fast

01:12:55.100 --> 01:12:56.130
for inference.

01:12:56.130 --> 01:12:57.400
You can use it with different kinds of

01:12:57.400 --> 01:12:58.040
variables.

01:12:58.040 --> 01:12:59.220
It doesn't account for feature

01:12:59.220 --> 01:13:00.730
interaction, doesn't provide good

01:13:00.730 --> 01:13:01.670
confidence estimates.

01:13:02.390 --> 01:13:04.210
And it's best when used with discrete

01:13:04.210 --> 01:13:06.270
variables, those that can be fit well

01:13:06.270 --> 01:13:08.830
by a Gaussian, or if you use kernel

01:13:08.830 --> 01:13:10.690
density estimation, which is something

01:13:10.690 --> 01:13:11.840
that we'll talk about later in this

01:13:11.840 --> 01:13:13.580
semester, a more general like

01:13:13.580 --> 01:13:15.080
continuous distribution function.

01:13:17.210 --> 01:13:19.560
And justice, as a reminder, don't pack

01:13:19.560 --> 01:13:21.730
up until I'm done, but this will be the

01:13:21.730 --> 01:13:22.570
second to last slide.

01:13:24.220 --> 01:13:25.890
So things remember.

01:13:27.140 --> 01:13:28.950
So Probabilistic models are really

01:13:28.950 --> 01:13:30.837
large class of machine learning

01:13:30.837 --> 01:13:31.160
methods.

01:13:31.160 --> 01:13:32.590
There are many different kinds of

01:13:32.590 --> 01:13:34.690
machine learning methods that are based

01:13:34.690 --> 01:13:36.480
on estimating the likelihoods of the

01:13:36.480 --> 01:13:38.170
label given the data or the data given

01:13:38.170 --> 01:13:38.730
the label.

01:13:39.580 --> 01:13:41.630
Naive Bayes assumes that Features are

01:13:41.630 --> 01:13:45.430
independent given the label, and it's

01:13:45.430 --> 01:13:46.860
easy and fast to estimate the

01:13:46.860 --> 01:13:48.920
parameters and reduces the risk of

01:13:48.920 --> 01:13:50.480
overfitting when you have limited data.

01:13:52.270 --> 01:13:52.590
It's.

01:13:52.590 --> 01:13:55.190
You don't usually have to derive how to

01:13:55.190 --> 01:13:57.910
solve for the likelihood parameters,

01:13:57.910 --> 01:13:59.660
but you can do it if you want to by

01:13:59.660 --> 01:14:00.954
taking the partial derivative.

01:14:00.954 --> 01:14:03.540
Usually it's usually you would be using

01:14:03.540 --> 01:14:06.140
a common a common kind of Model and you

01:14:06.140 --> 01:14:07.290
can just look up the Emily.

01:14:09.490 --> 01:14:11.160
The Prediction involves finding the way

01:14:11.160 --> 01:14:13.190
that maximizes the probability of the

01:14:13.190 --> 01:14:15.150
data and the label, either by trying

01:14:15.150 --> 01:14:17.250
all the possible values of Y or solving

01:14:17.250 --> 01:14:18.230
the partial derivative.

01:14:19.270 --> 01:14:21.535
And finally, Maximizing log probability

01:14:21.535 --> 01:14:24.060
of I is equivalent to Maximizing

01:14:24.060 --> 01:14:25.360
probability of.

01:14:25.520 --> 01:14:27.310
Sorry, Maximizing log probability of

01:14:27.310 --> 01:14:30.270
X&Y is equivalent to maximizing the

01:14:30.270 --> 01:14:32.250
probability of X&Y, and it's usually

01:14:32.250 --> 01:14:34.000
much easier, so it's important to

01:14:34.000 --> 01:14:34.390
remember that.

01:14:35.970 --> 01:14:36.180
Right.

01:14:36.180 --> 01:14:37.840
And then next class I'm going to talk

01:14:37.840 --> 01:14:40.030
about logistic regression and linear

01:14:40.030 --> 01:14:40.700
regression.

01:14:41.530 --> 01:14:44.870
And one more thing is I posted a review

01:14:44.870 --> 01:14:49.310
questions and answers to the 1st 2

01:14:49.310 --> 01:14:51.440
cannon and this lecture on the web

01:14:51.440 --> 01:14:52.050
page.

01:14:52.050 --> 01:14:53.690
You don't have to do them but they're

01:14:53.690 --> 01:14:55.410
good review for the exam or just the

01:14:55.410 --> 01:14:56.820
check your knowledge after each

01:14:56.820 --> 01:14:57.200
lecture.

01:14:57.890 --> 01:14:58.750
Thank you.

01:15:11.030 --> 01:15:11.320
I.