Professor B: Alright , so e we have to figure out some eh eh uh basically display hack or something to do this because anyway I I let me consi suggest that 's a s not a first - order consideration , we have two first - order considerations which is what are the uh influences A , A , and B how do they get combined mathematically , how do we display them is an issue , but um
Grad C: I don't , yeah I just don't think this has been designed to support something like that .
Grad D: Yeah . Yeah , I I mean , it might soon , if this is gonna be used in a serious way like java base then it might soon be necessary to uh start modifying it for our purposes .
Professor B: Right . Yeah , and Um I that seems like a perfectly feasible thing to get into , but um we have to know what we want first . OK , so why don't you tell us a little bit about decision nodes and what what the choices might be for these ?
Grad D: So Ah , sorry . I guess that 's
Grad C: You can technically wear that as you 're talking .
Grad D: Yeah , it 's right , I guess I can do that .
Grad A: Darn .
Professor B: Put it in your , yeah .
Grad D: I guess this board works fine . So um recall the basic problem which is that um you have a belief - net and you have like a lot of different nodes all contributing to one node . Right ? So as we discussed specifying this kind of thing is a big pain and it 's so will take a long time to write down because for example if these S have three possibilities each and this has three possibilities then you know you have two hundred and forty - three possibilities which is already a lot of numbers to write down . So what um helps us in our situation is that these all have values in the same set , right ? These are all like saying EV or A , right ? So it 's not just a generalized situation like I mean basically we wanna just take a combination of we wanna view each of these as experts ea who are each of them is making a decision based on some factors and we wanna sort of combine their decisions and create you know , um sorta weighted combination .
Grad E: Hmm . ROVER , the ROVER decision .
Grad D: The what decision ?
Grad E: ROVER . All of their outputs combined to make a decision .
Grad A: Hmm .
Grad D: Yeah . Yeah . So the problem is to specify the uh so the conditional property of this given all those , right ? That 's the way belief - nets are defined , like each node given its parents , right ? So um that 's what we want , we want for example P of um let 's call this guy Y and let 's call these X - one , X - two XN , right . So we want probability that Y equals , you know , for example um E given that these guys are I 'll just refer to this as like X um hat or something , uh the co like all of them ? Given that for example the data says you know , A , V , A , E , or something right ?
Professor B: Yep .
Grad D: So we would like to do this kind of combination .
Professor B: Alright , so um Is that uh I yeah , I just wanna make sure everybody is with us before he goes on .
Grad A: I think so , yeah .
Professor B: It 's it 's cl e is is it clear what he wants to compute ?
Grad A: Mm - hmm .
Grad D: Right . So , right . So Basically um what we don't wanna do is to for every single combination of E and V and A and every single letter E , s give a number
Grad A: Mm - hmm .
Grad D: because that 's obviously not desirable . What we wanna do is find some principled way of um saying what each of these is and we want it to be a valid probability distribution , so we want it to um add up to one , right ?
Grad A: Hmm .
Grad D: So those are the two things that we uh need . So what uh I guess , what Jerry suggested earlier was basically that we , you know view these guys as voting and we just take the uh we essentially take um averages , right ? So for example here two people have voted for A , one has voted for V , and one has voted for E , so we could say that the probabilities are , you know , probability of being E is one over four , because one person voted for E out of four and similarly , probability of so this is probability of E s and then probability of A given all that is um two out of four and probability of V is one out of four . Right ? So that 's step that 's the uh yeah that 's the that 's the basic uh thing . Now
Grad E: Um Yeah .
Grad D: Is that all OK ?
Grad E: And that one outcome , that 's
Professor B: What ?
Grad E: it 's X X - one voted for A X - two voted for V
Grad A: Mm - hmm .
Grad E: and so forth ?
Professor B: Y right . Yep .
Grad D: Yeah .
Grad E: Yeah .
Professor B: S so this assumes symmetry and equal weights and all this sort of things , which may or may not be a good assumption ,
Grad E: That 's the outcome .
Grad A: Mm - hmm . Right .
Professor B: so that
Grad D: Yeah . Yeah . So step two is um right . So we 've assumed equal weights whereas it might turn out that you know , some w be that for example , what the um the actual the uh verbal content of what the person said , like what uh what might be uh somehow more uh important than the uh
Grad C: X - one matters more i than X - two or
Grad D: Right . Sure , so we don't wanna like give them all equal weight so currently we 've been giving them all weight one fourth so we could replace this by uh W - one , W - two , W - three , and W - four
Grad A: Hmm .
Grad D: right ? And in order for this to be a valid probability distribution for each um X - hat , we just need that the W 's sum to one . So they can be for example , you know you you could have point one , point three , point two , and point four , say .
Grad E: That 's one .
Grad D: And that 'd be one . So that um also seems to work fine . And uh
Grad C: So I jus just to make sure I understand this , so in this case um we would still compute the average ?
Grad D: You 'd compute the weighted average , so the probability of E would be uh
Grad C: OK , so so it 'd be so in this case the probability that Y equals A would be uh {comment} W one times Grad A: Point three .
Grad C: or A or let 's see , one full quarter times point one
Grad D: Not one quarter ,
Grad A: No .
Grad D: so these numbers have been replaced with point one , point three , point two , and point four . So you can view these as gone .
Grad C: OK .
Grad D: Probability of
Grad C: OK .
Grad D: Yeah . Yeah . OK . So , alright . So this is uh step two . So the next possibility is that um we 've given just a single weight to each expert , right , whereas it might be the case that um in certain situations one of the experts is more uh reliable and in certain situations the other expert is more reliable . So the way this is handled is by what 's called a mixture of experts , so what you can have is you augment these diagrams like this so you have a new thing called " H " , OK ? This is a hidden variable . And what this is is it gets its input from X - one , X - two , X - three , and X - four , and what it does is it decides which of the experts is to be trusted in this particular situation . Right ? And then these guys all come here . OK . So this is sightly uh more complicated . So what 's going on is that um this H node looks at these four values of those guys and it decides in given these values which of these isn't likely to be more reliable or most reliable . So H produces some you know , it produces a number , either one , two , three , or four , in our situation , right ? Now this guy he looks at the value of H say it 's two , and then he just selects the uh thing . That 's all there is to say , I guess about it . Right , so you can have a mixture that
Grad E: Mm - hmm .
Grad D: Right .
Grad A: So so the function of the thing that comes out of H is very different from the function of the other inputs . It 's driving how the other four are interpreted . OK .
Grad D: Yeah . Yeah .
Grad C: So H passes a vector on to the next node ?
Grad D: It could .
Grad C: It could ? A vector of the weights as the se
Grad D: Yeah , it could
Grad C: oh .
Grad D: Sorry ?
Grad A: Well a vector with three zero 's and one one , right ?
Grad C: Oh it 's basically to tell the bottom node which one of the situations that it 's in or which one of the weighting systems
Grad D: Right , so I mean the way you desc
Grad C: W I was just , if you wanted to pay attention to more than one you could pass a w a weighting s system though too , couldn't you ? OK .
Grad A: Um Does H have to have another input to tell it alpha , beta , whatever , or is the that 's determined by what the experts are saying , like the type of situ OK . Hmm . OK . OK . I mean It it just seems that like without that that outside input that you 've got a situation where , you know , like if if uh X - one says no , you know , a low value coming out of X - on or i if X - one says no then ignore X - one , you know , I mean that seems like that 'd be weird ,
Grad D: Yeah , well could be things like if X - two and X - three say yes then i ignore X - one also .
Grad A: right ? Oh , OK . OK . Alright , right .
Grad C: Oh The situations that H has , are they built into the net or OK , so they they could either be hand coded or learned or OK .
Grad D: Yeah .
Grad C: Based on training data , OK .
Grad D: Yeah . Yes .
Grad C: So you specify one of these things for every one of those possi possible situations . Oh yeah .

Summary: Professor B first emphasized the necessity of combining influences mathematically and asked Grad D to further explain the potential solutions. Then the students discussed the computation methods including weighted combination and mixture of experts, based on the mechanism of belief-nets. But there was no consensus on which method should be applied in belief-nets.