So far we focused solely on the global structure of the distribution. The fact that you can take it and factorize it as a product of factors that correspond to subsets of variables. but it turns out that you also have other types of structure that you might want to encode and that is actually really important for real world applications. So to motivate that, let's look at the tabular representation of conditional probability distributions which is what we've used universally until now in the examples that we've given. So a tabular representation is one of these examples where we have, you know, a row for each assignment. So this is, just a reminder, this is G with parents I and v. And here we have a row for each assignment of appearance that give us explicitly enumerating all of the entry's that correspond to the probabilities of the variable G. So this is great because it will what's the problem and then it sounds like a perfectly reasonable and very understandable representation. Well, that's great, but now let's consider a more realistic example. For instance, in a medical application we might have a variable corresponding to cough. Well, there is lots and lots of things that might make you cough. You might have pneumonia or the flu or tuberculosis, or bronchitis, or just the common cold. And, by the time you finish enumerating all of the many things that might make you cough, you realize that the variable, usually, might have as many as ten, or fifteen, or even twenty parents for a variable such as fever. So that's so when you think about what the implications of that are, relative to a tabular CPD. You realize that if we have k parents. And let's assume for the moment that they're all binary just for simplicity then the number of entries in the CPD grows as two to the k, or of two to the k, depending on the number of variable, value of the child. So, unfortunately these situations are more common than not and that means that tabular representations are really not suitable for a lot of real world applications. So we have to think beyond that. So, fortunately, there is nothing in the definition of a Bayesian network that imposes on us a fully specified table as the only representation of a conditional probability distribution. The only thing we need is that the CPV, p of x, given y1 up to yk, needs to specify a distribution. Over x for each assignment, y, one to y, k. And it can do that completely implicitly. It can do that as a little bit of c code that looks at y, one to y, k and, and prints out a distribution over x. Now fortunately we don't usually have to sorry or in fact we can use any function parameterized or as a C routine or anything to specify a factor over the scope X Y1 up to Yk such that well it has to be a probability distribution over X, so it has to sum, so when you sum up all the values of X. for given assignment Y1 up to YP has the sum to one. Anything that satisfies these criteria, these constraints, is a legitimate representation of a CP. Like I said. Fortunately, we don't usually have to resort to C code, to specify CPDs. the theory, the framework of statistics has defined for us a multitude of different representations of a conditional probability distribution given a set of, of conditioning variables. so some examples include deterministic CPDs where x is a is a deterministic function of y 1 up to y k. We've already seen a couple of examples of that. you can think, we can define, and we will define CPDs that have the form of what's called a decision tree or a regression tree framework that some of you have seen before. you can think of CPDs that are logistic functions or more generally, log linear models. We're going to talk about things that are noisy or noisy. And which are noisy versions of deterministic CPDs. And then, in the continuous case, we also have, frameworks that allow us to represent the probability distribution of a continuous variable on a set of continuous or discrete parents. And that is really critical because, obviously, when you have a variable that takes on a continuum of values, you. And possibly write down the table that lists every single one of them. Now one of the things that are intertwined with the notion of structure within a CPD is a useful notion called context specific independence and it turns out that this notion arises in some of thee, representations, of CPD's that we're going to talk about. Context specific independence is a type of independence that is co, were we have a set of variables X, a set of variables Y, a set of variables Z, and then we have a particular assignment C which is, an assignment to some set of variables, big C. So this is conditioning. This is a, an independent statement that only holds for particular values of the conditioning variable C. As opposed to all values of the, of the conditioning variable C. Now, the definition of this is exactly as we have seen before. So, except that if you remember before, we had Z stays when we were doing conditional independence given Z, we had the Z stays on the right hand side of the conditioning bar, everywhere. Well now, Z and C stay on the right hand side of the conditioning bar in all of these forms of the independent state. So let's look at y context specific independence might arise when we have particular internal structure within a CPD. So, let's imagine. Let's, let's consider the case that x is a deterministic or of y1 and y2. And the question is. What form of context specific independence holds when we have X being a deterministic, X being this deterministic or? So let's look at these statements one at a time and here we have that Y2 is false. And when Y, what happens when Y2 if false? Well, when Y2 is false, X is the same as Y1. In which case obviously their not going to be independent. On the other hand, what if Y2 is true. Y was true. I don't care about Y1 anymore, because if Y2 is true, X is true. So, here we have a notion of context specific independence. What about this one? I'm now asking about y1 being independent of y2 giving two possible values of x. What happens when x is false? What do I know about y1 and y2. They're both false. Well if they're both false, then they're independent of each other. Because if you tell me that one of them is false, I already know the other one's false, so they're independent. And so this is another, condition, context specific independence that holds here, does it hold if I tell you that x is true? No, because I don't know which of y1 and y2 made x true. And so this last one doesn't hold and so here are two context specific independencies that hold in the context of deterministic cpd that wouldn't hold. Neither of these would hold for general purpose cpd, would necessarily hold for general purpose cpd, where, x depending on y one and y two.
