But [COUGH]. Today's topic is an important extension on the language on graphical models. And it's intended to deal with the very large class of cases. Where what we'd like to do is not just write down one kind of graphical model for a particular application. But rather, come up with something that is a general purpose representation that allows us to solve multiple problems using the same exact model. So to understand what that means, in a somewhat more concrete setting. Let's go back. Now for genetic inheritance that we've discussed previously it's arguably the earliest sample of Bayesian network reasoning, before Bayesian network were invented. And here we have as input a pedigree. Which is a family tree. And we're interested in reasoning about a particular trait. And for each pedigreeing each trait we can construct a Bayesian network and the Bayesian network might look like this. But clearly if you have a somewhat different family tree. So suddenly you had another three cousins and a great grandfather join the family tree or if you had a different family altogether, you would still want to use the same sort of ideas. The same pieces of what we used in the, in the first network to construct this other network because there's clearly a lot of commonalities between them. We have what you might call sharing between models. But in addition to that, you also have, in this example as a fairly obvious sharing within the model. So for example, this CPD that tells us how, Selma's genotype affects Selma's blood type presumably is the same process by which Marge's genotype affects Marge's blood type. And the same for Maggie and Lisa and Bart and Homer and everybody. So so we have this tremendous amount of sharing of both dependency models and parameters. Similarly you might argue. You might looking at this realise that the genetic inheritance model by which Bart's genotype is determined by the genotype of his two parents is the same inheritance model that applies to Lisa, to Maggie, to Marge, the Selma, and so on. So once again we have, a lot of parameters that are shared, not just between, but also within the model. So we'd like to have, some way of constructing models that have this large amounts of shared structure, that allows us to, both, construct very large models from a, sparse parameterization, and also to construct entire families of models from a single concise representation. This is not the only such application. This is probably the most commonly used type of graphical models are those that have shared structure and shared parameters. So here is another example that we've seen previously. This is for natural language, processing a sequence model for in this case, trying to identify name density recognition so. A very common task for which graphical models have been used. And here also we have a sequence model. And we have again shared shared pieces for example the parameters that relate the latent variable. In this case, what type of what type of variable, what type of part, what type of entity it is. Is it a person? Is it a location, and so on. There is a set of parameters here, and they're going to be independent of the place in the sequence. In which we find the word because, not because it's necessarily an exactly correct model. I mean one could clearly imagine cases where the position and the sequence might make a difference but because it's often a very useful, simplifying assumption specifically because it allows us to A, use parameters, re-use parameters, and B, allow us to apply the same model to sequences of varying length without having to worry about what is my fifteen word model versus what's my eight word model. And so this is a, another case where you have We have a tremendous amount of of Reuse of parameters. We've already seen similarly the example of image segmentation. Clearly we don't want to have a separate model for every superpixel in the image. There's hundreds of superpixels so we have sharing across [INAUDIBLE] super pixels or pixels. So, for example the model here that, that relates the class label of a super pixel to the image features of that super pixel is generally going to be shared. As are the parameters that involve adjacent super pixels. So these edge potentials are also going to be shared across. In this case, pairs of super pixels. These pairs of adjacent superpixels. And once again, we have sharing across, models as well. Because we are going to have one such model for image a. And obviously, we don't want to construct a separate model for every image. So, once again, we have sharing between and who then, a model. so lets look at one example a little bit more concretely because it's an example that we're going to use in some of the later analysis. So go, now lets return to our university example where we have student who takes a class and gets a grade and that grade depends on the student's difficulty of, I'm sorry, difficulty of the course, not on intelligence of the student. Now this is all fine if we're interested reading in just about an individual student, but now let's imagine that we want to think about an entire university. So now we have a difficulty variable, so these are all difficulty variables. The different courses, in this case, C1 up to CN are different courses that exist in our university. And conversely on the other side, we have multiple students and this is a set of intelligence variables that are indexed by different students. So, we have the intelligence of student one up to the intelligence of student m. Now note that these are different random variables. They can and generally will take different values from each other. but they all share a probabilistic model. And that's sort of the kind of sharing that we have in mind. And what we see here is that the grade of a student within a course, which are these variables done here, depend on the difficulty of the relevant course and the intelligence of the relevant student. So for example, the grade of student one in course one depends on the difficulty of course one and on the intelligence of student student one. And, once again we have, sharing of the both the structure and the parameters across these different grade variables. So that they all have, the same kind of dependency structure, and the same CPD. another example is that of robot localization, and this is another example of, in this case, a time series. the robot moves through time from one position to another, and although the position. At time t, is different changes over time. We expected the dynamics of the robot are fixed. We will talk more about that later. So that gives us a graphical model that, again, looks a little bit. this is one example of such a graphical model. Where we see the B position, the robot pose over here, these X variables depend on, for example, the previous pose, on whatever control action the robot took. And we're assuming, that once again, we have sharing of these parameters, where each extentiation of this variable. So, what then gives rise to is a class of models that are represented in terms of template variables, where a template variable is something that we end up replicating in many cases again and again within a single model as well as across models, and so the replication is indexed by the, by the fact that the variable, you can think of it as a function that takes arguments, and the arguments for example might be, high in points as in this example over here. So here we have a location, variable that's index by time point or sonar reading with index by time point. We have in this case, a genotype variable and a phenotype variable that's indexed by a particular person. a pic, a label, a class label that's indexed by a pixel, and similarly the difficulty of the intelligence and the grades that are indexed, and in this case, different combinations of indices course student or course student pairs. And the template model is a, language that tels us how template variables can be, the dependency models fro template variables and how concrete instantiations of, variables, what are called ground variables. Like the ones that are actually indexed by a particular time point or person how they inherit the dependency model [INAUDIBLE]. And there is a whole range of such languages that have been developed in the special purpose way for different applications so dynamic Bayesian networks are intended for dealing with temporal processes, for example. Where we have replication over time. We have a whole range of language for object relation models, both directed models and undirected models where we have multiple objects such as people or students or courses such as pixels so people. Horses, pixels, and lots of other things that can be related to each other in different ways, and how do you represent the dependency model over that ensemble in a coherent way?
