So broadening from this from this particular mathematical derivation,

what we have here is that if we were to think about the model where we have a

joint density of Y and X together as a kind of naive Bayes model, because

normally we have pairwise terms that relate the Y and the Xi so you can think

of it as a bunch. If, if we weren't doing this conditional

normalization, we would have a model that looks like this.

We would have, you know, the Y and we would have a pairwise feature that

relates Y to X1, Y to X2,

up to Y to Xn. And, that would be effectively like a

naive Bayes model because, since we don't have any features, pairwise features that

represents, connect the Xis to each other, we would effectively have a model

where the Xi's are independent given the Y.

So, if I weren't doing a conditional model, if I were to just use this set of

potentials. It represents the joint distribution of X

and Y, it would be a effectively like a naive

Bayes model and it would make very strong independence assumptions.

But because a modeling, this is a conditional distribution,

like this, this is the conditional distribution.

I've effectively removed from this analysis any notion of a correlations

between the Xs and I'm just modeling how the Xs come together to effect the

probability of Y. And so, that's really the difference

between a naive Bayes model and, and a logistic lagression model.

And that same intuition extends to much richer of classes of, of models, where I

don't just have binary variables in a single Y,

but rather a very rich set of Ys and Xs. And nevertheless, this ability to sort of

ignore the distribution over the features and focus on the target variables allows

me to exploit allows me to sort of ignore

correlations between rich features and not worry about whether they're

independent of each other or not. So, for example, going back to are notion

of CRFs for image segmentation here, we typically have very rich features of the

variables. so for example, we think about individual

node potentials that relate the features Xi to the class label Yi,

we don't worry about how correlated the features are.

We can have color histograms, texture features.

You can have discriminative patches, like looking through the eye of the cow, for

example. And all of these are going to be really

correlated with each other, but I don't care.

You can even look at features that are outside of the superpixel.

You can say, oh, well, if it's, you know, green underneath in a completely

different superpixel maybe it's more likely to be a cow or a sheep because

they tend to be on grass. That's cool, too.

These are definitely correlated, because you're counting the same feature for my

superpixel as well as for a different superpixel.

That's fine. I don't care, because I am not worried

about the correlations between the superpixels.

So the correlations don't matter. You can even, and this is very commonly

done, train your favorite discriminative classifier, a support vector machine,

boosting, random force to anything that you like to predict the probability of Yi

given a whole bunch of image features X. And that's fine, too.

And in fact, that is how one achieves high performance on most of these tasks

by training very strong classifiers for in most cases, your note potentials.

That is the predictors for individual variable and then adding on top of that

pairwise features or, or not just pairwise but a higher order feature is