Now, you might wonder why I would bother decomposing the utility function instead
of having just a single big monolithic utility function that is affected by all
of these different factors. Well, that is exactly why because it is,
in fact, affected by all of these different factors.
So, whereas, here, I have a utility function that has one argument.
And another one has one argument and one has two arguments.
If I had put them all together, I would have a single monolithic utility function
that has four arguments. And that would be much harder to elicit
because of the exponential combinatorial growth in the number of possible
combinations that I need to consider. So this is a decomposition of a function.
In this case, the utility function, into factors in exactly the same way that we,
for example, like to decompose a probability distribution.
As a product of factors so another interesting extension of the influence
diagram representation allows us to capture is the notion of the information
available to the agent when they make their decision.
So, let's look at this example over here, which elaborates our entrepreneur
example. And here, we come in with the assumption
that the entrepreneur has the opportunity to conduct to survey regarding the
overall market demand for widgets before deciding whether to found a company.
Now the survey is not entirely reliable. And so, over here, we have the CPD, that
tells us. How likely different survey results are
to come up, given different values of the true market demand.
Now the point is that, having conducted the survey, the agent can base his
decision on the value of the survey. We denote that with the presence of an
edge such as this which indicates that the.
Founder. That the entrepreneur can make different
decisions, depending on how the value of the survey turns out.
So from a formal perspective, that means that what the agent's allowed to is to,
choose a decision rule, which we're going to denote delta, at the action node.
So the, and what a decision rule is, is effectively a CPD.
It tells the agent given an assignment of the values to the parents of A, which are
the val, the variables that the agent can observe prior to making a decision.
The agent, based on that, can decide on the probability distribution of the
action that it takes in that particular situation.
So here for example we would have a CPD which tells us the probability of
founding a company, which in this case is a binary valued variable, given the three
possible values of the survey variable. So you might say, why would I need to
have a CPD? why would the agent ever want to make a
random choice between two different actions?
And it turns out, in fact, that in a single agent decision making situation.
There's no value to allowing that extra expressive power, because of the
terministic CPD would work just as well. Nevertheless for reasons that will become
clear on the next slide, it's actually useful to think about this as a CPD, even
though in most cases it'll actually be the terministic at reasoning samples that
we're talking about. So now that we've given the agent that
expressive power how do you formulate the decision problem that they need to make?
That is what do they get to choose and how do they choose it?
So given a decision rule delta for an action variable A if we inject that into
the decision network now all of the variables in our network, both A and the
remaining variable X, variables X all have a CPD associated with them.
So now we have effectively defined a joint probability distribution.
Over, all the variables X union. The variable A because each of them has a
CPD, and so this. Is, the, this probability distribution.
And the agent expected utility, once they've selected the decision rule delta
A is simply this expectation over here. We average out over all possible
scenarios, where scenarios and assignment to the chance variables X as well as to H
is action A. And but we're averaging out is the
agent's utility in that scenario. And so that's the overall expectation
that the agent prior to anything going on can expect to gain given a decision rule
delta A. So obviously the agents want, the agents
wants to choose the decision rule delta A.
That is going to give him the maximum value for this expression, that denotes
his expected utility. And so this is the optimization problem
that the agent is trying to solve. So how do we go about finding the
decision rule that maximizes the agent's expected utility?
Well, let's look at this first in the context of the simple example, and then
we'll do the general case. So over here, we see, once more, the
expected utility equation for a given decision rule.
And now we're trying to optimize this delta A.
So let's write down the expected utility in this particular example.
So the bayesian network in this case, has now two original CPD's.
There is P of M, that comes in from the M variable.
There's P of S given M, which comes out from the survey variable and we have the
CPD that is as yet to be selected, which specifies the decisional at the at the
action F. And then multiplied with all that is the
utility which depends on the decision to found the company.
And on the true market value. Well, this should look awfully familiar.
It is simply a product of factors. Some of these factors are probabilistic
factors. And one of them, the U, is a different
numerical factor. But it's still a factor which depends on
a scope of two variables, in this case F and M.
So now we can go ahead and apply the same kinds of analysis that we've done in
previous computations. Specifically we can, since we're trying
to optimize over this expression over here the delta F of F given S.
We're going to push in everything that we don't currently need to manipulate, which
is just F and S which are the two things that appear in the factor, the decision
role. And so specifically we're going push in M
and we with in the summation, over M, all of the factors.
that depends on M. And so, that gives us a sum over M, P of
M, P of S given M, U of F, comma M. And if we marginalize out M, what we end
up with, if we multiply these three factors and marginalize that M, is a
factor that we're going to call mu of FS. And the reason that it's called mew is to
suggest that it has utility component U, so mew and U.
And now we have actually a fairly simple expression that the agent is trying to
optimize. It's a summation over all possible values
of S and F over here, of the Delta, the decision rule that the agent takes over
F, given the situation as given the observation, multiplied by this factor
that I just computed. And now if our goal is not to optimize
this expression, what, the agent ought to do is to pick the variance factor, U of
FF, the highest value. the S that gives it the highest value in
each circumstance S. So this is a little bit of strat so let's
look at this in the context of an actual numerical example to see this.
So here are the three factors that we have in this network to begin with.
This is the CPD for M over here. The CPD for S given M in the middle, and
here is the utility factor U, it comes from here.
And now if we're going to go through this expression, through this computation, and
we're going to compute. This expression over here, this is what
we get. It's a factor that gives us this value
for each value of the variable of the action F.
And each value of the survey variable S. And if we want to maximize the summation.
