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+WEBVTT Kind: captions; Language: en-US
+
+NOTE
+Created on 2024-02-07T20:52:10.2470009Z by ClassTranscribe
+
+00:01:22.340 --> 00:01:22.750
+Good morning.
+
+00:01:24.260 --> 00:01:27.280
+Alright, so I'm going to just first
+
+00:01:27.280 --> 00:01:29.738
+finish up what I was, what I was going
+
+00:01:29.738 --> 00:01:31.660
+to cover at the end of the last lecture
+
+00:01:31.660 --> 00:01:32.980
+about Cannon.
+
+00:01:33.640 --> 00:01:36.550
+And then I'll talk about probabilities
+
+00:01:36.550 --> 00:01:37.540
+and Naive Bayes.
+
+00:01:38.260 --> 00:01:39.940
+And so I wanted to give an example of
+
+00:01:39.940 --> 00:01:41.930
+how K&N is used in practice.
+
+00:01:42.530 --> 00:01:44.880
+Here's one example of using it for face
+
+00:01:44.880 --> 00:01:45.920
+recognition.
+
+00:01:46.750 --> 00:01:48.480
+A lot of times when it's used in
+
+00:01:48.480 --> 00:01:50.030
+practice, there's a lot of feature
+
+00:01:50.030 --> 00:01:51.780
+learning that goes on ahead of the
+
+00:01:51.780 --> 00:01:52.588
+nearest neighbor.
+
+00:01:52.588 --> 00:01:54.510
+So nearest neighbor itself is really
+
+00:01:54.510 --> 00:01:55.125
+simple.
+
+00:01:55.125 --> 00:01:58.530
+It's efficacy depends on learning good
+
+00:01:58.530 --> 00:02:00.039
+representation so that.
+
+00:02:00.800 --> 00:02:02.640
+Data points that are near each other
+
+00:02:02.640 --> 00:02:04.410
+actually have similar labels.
+
+00:02:05.450 --> 00:02:07.385
+Here's one example.
+
+00:02:07.385 --> 00:02:10.550
+They want to try to be able to
+
+00:02:10.550 --> 00:02:12.330
+recognize whether two faces are the
+
+00:02:12.330 --> 00:02:13.070
+same person.
+
+00:02:13.820 --> 00:02:16.460
+And so the method is that you Detect
+
+00:02:16.460 --> 00:02:18.940
+facial features and then use those
+
+00:02:18.940 --> 00:02:21.630
+feature detections to align the image
+
+00:02:21.630 --> 00:02:23.300
+so that the face looks more frontal.
+
+00:02:24.060 --> 00:02:26.480
+Then they use a CNN convolutional
+
+00:02:26.480 --> 00:02:29.240
+neural network to train Features that
+
+00:02:29.240 --> 00:02:32.600
+will be good for recognizing faces.
+
+00:02:32.600 --> 00:02:34.360
+And the way they did that is that they
+
+00:02:34.360 --> 00:02:37.950
+first collected hundreds of Faces from
+
+00:02:37.950 --> 00:02:40.300
+a few thousand different people.
+
+00:02:40.300 --> 00:02:41.680
+I think it was their employees of
+
+00:02:41.680 --> 00:02:42.250
+Facebook.
+
+00:02:43.030 --> 00:02:46.420
+And they trained a classifier to say
+
+00:02:46.420 --> 00:02:48.970
+which, given a face, which of these
+
+00:02:48.970 --> 00:02:50.960
+people does the face belong to.
+
+00:02:52.030 --> 00:02:54.340
+And from that, they learn a
+
+00:02:54.340 --> 00:02:55.210
+REPRESENTATION.
+
+00:02:55.210 --> 00:02:57.030
+Those classifiers aren't very useful,
+
+00:02:57.030 --> 00:02:59.300
+because nobody's interested in seeing
+
+00:02:59.300 --> 00:03:00.230
+given a face.
+
+00:03:00.230 --> 00:03:01.843
+Which of the Facebook employees is that
+
+00:03:01.843 --> 00:03:02.914
+they want to know?
+
+00:03:02.914 --> 00:03:04.932
+Like, is it you want to know?
+
+00:03:04.932 --> 00:03:07.460
+Like, organize your photo album or see
+
+00:03:07.460 --> 00:03:08.800
+whether you've been tagged in another
+
+00:03:08.800 --> 00:03:09.960
+photo or something like that?
+
+00:03:10.630 --> 00:03:12.050
+And so then they throw out the
+
+00:03:12.050 --> 00:03:13.980
+Classifier and they just use the
+
+00:03:13.980 --> 00:03:16.280
+feature representation that was learned
+
+00:03:16.280 --> 00:03:21.070
+and use nearest neighbor to identify a
+
+00:03:21.070 --> 00:03:22.510
+person that's been detected in a
+
+00:03:22.510 --> 00:03:23.090
+photograph.
+
+00:03:24.830 --> 00:03:26.540
+So in their paper, they showed that
+
+00:03:26.540 --> 00:03:28.565
+this performs similarly to humans in
+
+00:03:28.565 --> 00:03:30.470
+this data set called label faces in the
+
+00:03:30.470 --> 00:03:31.970
+wild where you're trying to recognize
+
+00:03:31.970 --> 00:03:32.560
+celebrities.
+
+00:03:34.140 --> 00:03:35.770
+But it can be used for many things.
+
+00:03:35.770 --> 00:03:37.516
+So you can organize photo albums, you
+
+00:03:37.516 --> 00:03:40.360
+can detect faces and then you try to
+
+00:03:40.360 --> 00:03:41.970
+match Faces across the photos.
+
+00:03:41.970 --> 00:03:44.175
+So then you can organize like which
+
+00:03:44.175 --> 00:03:46.320
+photos have a particular person.
+
+00:03:47.070 --> 00:03:49.950
+Again, you can't identify celebrities
+
+00:03:49.950 --> 00:03:51.860
+or famous people by building up a
+
+00:03:51.860 --> 00:03:54.919
+database of faces of famous people.
+
+00:03:55.870 --> 00:03:58.110
+And you can also alert, alert somebody
+
+00:03:58.110 --> 00:04:00.100
+if somebody else uploads a photo of
+
+00:04:00.100 --> 00:04:00.330
+them.
+
+00:04:00.330 --> 00:04:02.922
+So you can see if somebody uploads a
+
+00:04:02.922 --> 00:04:05.364
+photo, then you can detect faces, you
+
+00:04:05.364 --> 00:04:07.830
+can see what their friends network is,
+
+00:04:07.830 --> 00:04:10.056
+see what other which of their faces
+
+00:04:10.056 --> 00:04:12.220
+have been uploaded and then Detect the
+
+00:04:12.220 --> 00:04:14.330
+other users whose faces have been
+
+00:04:14.330 --> 00:04:16.580
+uploaded and ask them for permission to
+
+00:04:16.580 --> 00:04:17.930
+like make this photo public.
+
+00:04:19.750 --> 00:04:22.020
+So this algorithm is actually used by
+
+00:04:22.020 --> 00:04:22.560
+Facebook.
+
+00:04:22.560 --> 00:04:24.340
+It has been for several years.
+
+00:04:24.340 --> 00:04:28.640
+They're limiting some of its use more
+
+00:04:28.640 --> 00:04:30.544
+recently, but they've been.
+
+00:04:30.544 --> 00:04:32.010
+But it's been used really heavily.
+
+00:04:32.680 --> 00:04:34.410
+And of course they have expanded
+
+00:04:34.410 --> 00:04:36.365
+training data because whenever anybody
+
+00:04:36.365 --> 00:04:37.940
+uploads photos then they can
+
+00:04:37.940 --> 00:04:40.353
+automatically detect them and add them
+
+00:04:40.353 --> 00:04:42.360
+to the database.
+
+00:04:42.360 --> 00:04:45.150
+So here the use of KN is important
+
+00:04:45.150 --> 00:04:47.220
+because KNN doesn't require any
+
+00:04:47.220 --> 00:04:47.490
+training.
+
+00:04:47.490 --> 00:04:49.295
+So every time somebody uploads a new
+
+00:04:49.295 --> 00:04:50.930
+face you can update the model just by
+
+00:04:50.930 --> 00:04:54.430
+adding this four 4096 dimensional
+
+00:04:54.430 --> 00:04:56.646
+feature vector that corresponds to the
+
+00:04:56.646 --> 00:05:00.230
+face and then use it in like based on
+
+00:05:00.230 --> 00:05:02.550
+the friend networks to.
+
+00:05:02.910 --> 00:05:04.840
+To recognize faces that are associated
+
+00:05:04.840 --> 00:05:05.410
+with somebody.
+
+00:05:07.530 --> 00:05:11.270
+I won't take time to discuss it now,
+
+00:05:11.270 --> 00:05:13.473
+but it's worth thinking about some of
+
+00:05:13.473 --> 00:05:15.710
+the consequences of the way that the
+
+00:05:15.710 --> 00:05:17.888
+algorithm was trained and the way that
+
+00:05:17.888 --> 00:05:18.620
+it's deployed.
+
+00:05:18.620 --> 00:05:19.600
+So for example.
+
+00:05:20.510 --> 00:05:22.680
+If you think about that, it was that
+
+00:05:22.680 --> 00:05:24.650
+the initial Features were learned on
+
+00:05:24.650 --> 00:05:26.030
+Facebook employees.
+
+00:05:26.030 --> 00:05:27.440
+That's not a very.
+
+00:05:28.070 --> 00:05:29.630
+That's not very representative
+
+00:05:29.630 --> 00:05:32.120
+demographic of the US the employees
+
+00:05:32.120 --> 00:05:35.000
+tend to be younger and.
+
+00:05:35.490 --> 00:05:38.446
+Probably skew towards male might skew
+
+00:05:38.446 --> 00:05:40.210
+towards certain ethnicities.
+
+00:05:40.820 --> 00:05:43.210
+And so the Algorithm may be much better
+
+00:05:43.210 --> 00:05:45.030
+at recognizing some kinds of Faces than
+
+00:05:45.030 --> 00:05:46.016
+other faces.
+
+00:05:46.016 --> 00:05:47.628
+And then, of course, there's lots and
+
+00:05:47.628 --> 00:05:49.495
+lots of ethical issues that surround
+
+00:05:49.495 --> 00:05:51.830
+the use of face recognition and its
+
+00:05:51.830 --> 00:05:52.610
+applications.
+
+00:05:53.930 --> 00:05:55.550
+Of course, like in many ways, this is
+
+00:05:55.550 --> 00:05:58.150
+used to help people maintain privacy.
+
+00:05:58.150 --> 00:06:00.080
+But even the use of recognition at all
+
+00:06:00.080 --> 00:06:03.120
+raises privacy concerns, and that's why
+
+00:06:03.120 --> 00:06:04.860
+they've limited the use to some extent.
+
+00:06:06.470 --> 00:06:08.060
+So just something to think about.
+
+00:06:09.980 --> 00:06:13.430
+So just to recap kann, the key
+
+00:06:13.430 --> 00:06:16.480
+assumptions of K&N are that K nearest
+
+00:06:16.480 --> 00:06:18.260
+neighbors that Samples with similar
+
+00:06:18.260 --> 00:06:19.730
+features will have similar output
+
+00:06:19.730 --> 00:06:20.695
+predictions.
+
+00:06:20.695 --> 00:06:23.290
+And for most of the Distance measures
+
+00:06:23.290 --> 00:06:25.590
+you implicitly assumes that all the
+
+00:06:25.590 --> 00:06:27.200
+dimensions are equally important.
+
+00:06:27.200 --> 00:06:29.820
+So it requires some kind of scaling or
+
+00:06:29.820 --> 00:06:31.500
+learning to be really effective.
+
+00:06:33.540 --> 00:06:35.620
+The parameters are just the data
+
+00:06:35.620 --> 00:06:36.080
+itself.
+
+00:06:36.080 --> 00:06:37.870
+You don't really have to learn any kind
+
+00:06:37.870 --> 00:06:40.526
+of statistics of the data.
+
+00:06:40.526 --> 00:06:42.270
+The data are the parameters.
+
+00:06:43.820 --> 00:06:46.160
+The designs are mainly the choice of K
+
+00:06:46.160 --> 00:06:48.130
+if you have higher K then it gets
+
+00:06:48.130 --> 00:06:49.360
+smoother Prediction.
+
+00:06:50.340 --> 00:06:51.730
+You can decide how you're going to
+
+00:06:51.730 --> 00:06:54.400
+combine predictions if K is greater
+
+00:06:54.400 --> 00:06:56.750
+than one, usually it's just voting or
+
+00:06:56.750 --> 00:06:57.280
+averaging.
+
+00:06:58.610 --> 00:07:00.920
+You can try to design the features and
+
+00:07:00.920 --> 00:07:03.450
+that's where things can get a lot more
+
+00:07:03.450 --> 00:07:03.930
+creative.
+
+00:07:04.680 --> 00:07:06.770
+And you can choose a Distance function.
+
+00:07:08.900 --> 00:07:12.370
+So this K&N is useful in many cases.
+
+00:07:12.370 --> 00:07:14.520
+So if you have very few examples per
+
+00:07:14.520 --> 00:07:16.605
+class then it can be applied even if
+
+00:07:16.605 --> 00:07:17.320
+you just have one.
+
+00:07:18.080 --> 00:07:20.290
+It can also work if you have many
+
+00:07:20.290 --> 00:07:21.560
+Examples per class.
+
+00:07:22.200 --> 00:07:24.910
+It's best if the features are all
+
+00:07:24.910 --> 00:07:26.960
+roughly equally important, because K&N
+
+00:07:26.960 --> 00:07:28.540
+itself doesn't really learn which
+
+00:07:28.540 --> 00:07:29.449
+features are important.
+
+00:07:31.570 --> 00:07:33.910
+It's good if the training data is
+
+00:07:33.910 --> 00:07:34.585
+changing frequently.
+
+00:07:34.585 --> 00:07:37.520
+In the face recognition Example face,
+
+00:07:37.520 --> 00:07:38.830
+there's no way that Facebook will
+
+00:07:38.830 --> 00:07:41.160
+collect everybody's Faces up front.
+
+00:07:41.160 --> 00:07:43.030
+People keep on joining and leaving the
+
+00:07:43.030 --> 00:07:45.480
+social network, and so they and they
+
+00:07:45.480 --> 00:07:47.080
+don't want to have to keep retraining
+
+00:07:47.080 --> 00:07:49.850
+models every time somebody uploads a
+
+00:07:49.850 --> 00:07:52.005
+image with a new face in it or tags a
+
+00:07:52.005 --> 00:07:52.615
+new face.
+
+00:07:52.615 --> 00:07:54.990
+And so the ability to instantly update
+
+00:07:54.990 --> 00:07:56.330
+your model is very important.
+
+00:07:58.160 --> 00:07:59.850
+You can apply it to classification or
+
+00:07:59.850 --> 00:08:01.740
+regression whether you have discrete or
+
+00:08:01.740 --> 00:08:04.570
+continuous values, and its most
+
+00:08:04.570 --> 00:08:06.020
+powerful when you do some feature
+
+00:08:06.020 --> 00:08:08.180
+learning as an upfront operation.
+
+00:08:10.130 --> 00:08:12.210
+So there's cases where it has its
+
+00:08:12.210 --> 00:08:13.330
+downsides though.
+
+00:08:13.330 --> 00:08:15.650
+One is that if you have a lot of
+
+00:08:15.650 --> 00:08:18.250
+examples that are available per class,
+
+00:08:18.250 --> 00:08:20.360
+then usually training a Logistic
+
+00:08:20.360 --> 00:08:23.690
+regressor other Linear Classifier will
+
+00:08:23.690 --> 00:08:26.200
+outperform because it's able to learn
+
+00:08:26.200 --> 00:08:27.990
+the importance of different Features.
+
+00:08:28.950 --> 00:08:32.125
+Also, K&N requires that you store all
+
+00:08:32.125 --> 00:08:34.692
+the training data and that may require
+
+00:08:34.692 --> 00:08:38.153
+a lot of storage and it requires a lot
+
+00:08:38.153 --> 00:08:40.145
+of computation, and that you have to
+
+00:08:40.145 --> 00:08:42.200
+compare each new input to all of the
+
+00:08:42.200 --> 00:08:43.750
+inputs in your training data.
+
+00:08:43.750 --> 00:08:45.525
+So in the case of Facebook for example,
+
+00:08:45.525 --> 00:08:47.745
+they don't need if somebody uploads, if
+
+00:08:47.745 --> 00:08:49.780
+they detect a face in somebody's image,
+
+00:08:49.780 --> 00:08:51.520
+they don't need to compare it to the
+
+00:08:51.520 --> 00:08:53.410
+other, like 2 billion Facebook users.
+
+00:08:53.410 --> 00:08:55.176
+They just would compare it to people in
+
+00:08:55.176 --> 00:08:56.570
+the person's social network, which will
+
+00:08:56.570 --> 00:08:58.900
+be a much smaller number of Faces.
+
+00:08:58.970 --> 00:09:01.240
+So they're able to limit the
+
+00:09:01.240 --> 00:09:02.190
+computation that way.
+
+00:09:05.940 --> 00:09:08.760
+And then finally, to recap what we
+
+00:09:08.760 --> 00:09:12.180
+learned on Thursday, there's a basic
+
+00:09:12.180 --> 00:09:14.420
+machine learning process, which is that
+
+00:09:14.420 --> 00:09:16.170
+you've got training data, validation
+
+00:09:16.170 --> 00:09:17.260
+data and TestData.
+
+00:09:18.160 --> 00:09:19.980
+Given the training data, which are
+
+00:09:19.980 --> 00:09:22.730
+pairs of Features and labels, you fit
+
+00:09:22.730 --> 00:09:25.060
+the parameters of your Model.
+
+00:09:25.060 --> 00:09:26.950
+Then you use the validation Model to
+
+00:09:26.950 --> 00:09:28.670
+check how good the Model is and maybe
+
+00:09:28.670 --> 00:09:29.805
+check many models.
+
+00:09:29.805 --> 00:09:31.960
+You choose the best one and then you
+
+00:09:31.960 --> 00:09:33.590
+get your final estimate of performance
+
+00:09:33.590 --> 00:09:34.410
+on the TestData.
+
+00:09:36.790 --> 00:09:39.670
+We talked about KNN, which is simple
+
+00:09:39.670 --> 00:09:42.040
+but effective Classifier and regressor
+
+00:09:42.040 --> 00:09:44.140
+that predicts the label of the most
+
+00:09:44.140 --> 00:09:45.540
+similar training Example.
+
+00:09:46.770 --> 00:09:49.110
+And then we talked about kind of
+
+00:09:49.110 --> 00:09:51.110
+patterns of error and what causes
+
+00:09:51.110 --> 00:09:51.580
+errors.
+
+00:09:51.580 --> 00:09:53.780
+So it's important to remember that as
+
+00:09:53.780 --> 00:09:56.069
+you get more training, more training
+
+00:09:56.070 --> 00:09:57.830
+samples, you would expect that fitting
+
+00:09:57.830 --> 00:09:58.962
+the training data gets harder.
+
+00:09:58.962 --> 00:10:01.500
+So your error will tend to go up while
+
+00:10:01.500 --> 00:10:03.390
+your error on the TestData will get
+
+00:10:03.390 --> 00:10:05.535
+lower because the training data better
+
+00:10:05.535 --> 00:10:07.010
+represents the TestData or better
+
+00:10:07.010 --> 00:10:08.430
+represents the full distribution.
+
+00:10:09.770 --> 00:10:11.840
+And there's many reasons why at the end
+
+00:10:11.840 --> 00:10:13.250
+of training your Algorithm, you're
+
+00:10:13.250 --> 00:10:14.720
+still going to have error in most
+
+00:10:14.720 --> 00:10:15.220
+cases.
+
+00:10:15.880 --> 00:10:17.400
+It could be that the problem is
+
+00:10:17.400 --> 00:10:20.940
+intrinsically difficult, or it's
+
+00:10:20.940 --> 00:10:22.590
+impossible to have 0 error.
+
+00:10:22.590 --> 00:10:24.232
+It could be that you're Model has
+
+00:10:24.232 --> 00:10:24.845
+limited power.
+
+00:10:24.845 --> 00:10:27.370
+It could be that your Model has plenty
+
+00:10:27.370 --> 00:10:29.015
+of power, but you have limited data so
+
+00:10:29.015 --> 00:10:30.710
+you can't Estimate the parameters
+
+00:10:30.710 --> 00:10:31.290
+exactly.
+
+00:10:32.050 --> 00:10:33.100
+And it could be that there's
+
+00:10:33.100 --> 00:10:34.550
+differences in the training test
+
+00:10:34.550 --> 00:10:35.280
+distribution.
+
+00:10:37.020 --> 00:10:38.980
+And then finally it's important to
+
+00:10:38.980 --> 00:10:41.315
+remember that this Model fitting, the
+
+00:10:41.315 --> 00:10:42.980
+model design and fitting is just one
+
+00:10:42.980 --> 00:10:44.750
+part of a larger processing collecting
+
+00:10:44.750 --> 00:10:46.600
+data and fitting it into an
+
+00:10:46.600 --> 00:10:47.610
+application.
+
+00:10:47.610 --> 00:10:51.230
+So both the cases of in Facebook's case
+
+00:10:51.230 --> 00:10:54.160
+for example they had pre training stage
+
+00:10:54.160 --> 00:10:56.663
+which is like training a classifier and
+
+00:10:56.663 --> 00:10:58.852
+then they use that in a different, they
+
+00:10:58.852 --> 00:11:01.370
+use it in a different way as a nearest
+
+00:11:01.370 --> 00:11:05.320
+neighbor recognizer on their pool of
+
+00:11:05.320 --> 00:11:06.010
+user data.
+
+00:11:07.070 --> 00:11:10.384
+And so they're kind of building a model
+
+00:11:10.384 --> 00:11:11.212
+using it.
+
+00:11:11.212 --> 00:11:13.700
+They're building a model one way and
+
+00:11:13.700 --> 00:11:15.150
+then using it in a different way.
+
+00:11:15.150 --> 00:11:16.660
+So often that's the case that you have
+
+00:11:16.660 --> 00:11:17.590
+to kind of be creative.
+
+00:11:18.360 --> 00:11:20.580
+About how you collect data and how you
+
+00:11:20.580 --> 00:11:23.800
+can get the model that you need to
+
+00:11:23.800 --> 00:11:24.860
+solve your application.
+
+00:11:28.010 --> 00:11:30.033
+Alright, so now I'm going to move on to
+
+00:11:30.033 --> 00:11:31.640
+the main topic of today's lecture,
+
+00:11:31.640 --> 00:11:34.880
+which is probabilities and the night
+
+00:11:34.880 --> 00:11:35.935
+based Classifier.
+
+00:11:35.935 --> 00:11:39.690
+So the knight based Classifier is
+
+00:11:39.690 --> 00:11:41.220
+unlike nearest neighbor, it's not.
+
+00:11:41.990 --> 00:11:44.020
+Usually like the final approach that
+
+00:11:44.020 --> 00:11:46.080
+somebody takes, but it's sometimes a
+
+00:11:46.080 --> 00:11:49.460
+piece of a piece of how somebody is
+
+00:11:49.460 --> 00:11:51.210
+estimating probabilities as part of
+
+00:11:51.210 --> 00:11:51.870
+their approach.
+
+00:11:52.690 --> 00:11:55.610
+And it's a good introduction to
+
+00:11:55.610 --> 00:11:56.630
+Probabilistic models.
+
+00:11:59.220 --> 00:12:02.525
+So with the nearest neighbor
+
+00:12:02.525 --> 00:12:04.670
+classifier, that's an instance based
+
+00:12:04.670 --> 00:12:05.960
+Classifier, which means that you're
+
+00:12:05.960 --> 00:12:07.800
+assigning labels just based on matching
+
+00:12:07.800 --> 00:12:08.515
+other instances.
+
+00:12:08.515 --> 00:12:11.160
+The instances the data are the Model.
+
+00:12:12.260 --> 00:12:14.590
+Now we're going to start talking about
+
+00:12:14.590 --> 00:12:15.910
+Probabilistic models.
+
+00:12:15.910 --> 00:12:18.290
+In a Probabilistic Model, you choose
+
+00:12:18.290 --> 00:12:21.060
+the label that is most likely given the
+
+00:12:21.060 --> 00:12:21.630
+Features.
+
+00:12:21.630 --> 00:12:23.390
+So that's kind of an intuitive thing to
+
+00:12:23.390 --> 00:12:25.510
+do if you want to know.
+
+00:12:26.520 --> 00:12:28.690
+Which if you're looking at an image and
+
+00:12:28.690 --> 00:12:30.390
+trying to classify it into a Digit, it
+
+00:12:30.390 --> 00:12:32.074
+makes sense that you would assign it to
+
+00:12:32.074 --> 00:12:34.000
+the Digit that is most likely given the
+
+00:12:34.000 --> 00:12:35.940
+Features given the pixel intensities.
+
+00:12:36.610 --> 00:12:38.170
+But of course, like the challenge is
+
+00:12:38.170 --> 00:12:40.030
+modeling this probability function, how
+
+00:12:40.030 --> 00:12:42.590
+do you Model the probability of the
+
+00:12:42.590 --> 00:12:44.000
+label given the data?
+
+00:12:45.340 --> 00:12:47.520
+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.
+