>>In this video, we'll see how to define a probabilistic model.

The most convenient way to do this is called the Bayesian Network. It is a graph.

Its nodes are in variables and edges are direct impact.

For example, here we have two invariables,

rain and the fact that the grass is wet.

And these edge shows as if there is rain,

if it's raining, then the grass will be wet for sure.

We can see a more complex graph.

For example, here we added another invariable called sprinkler,

and the grass maybe wet either because the sprinkler is working or because it is raining.

Also the sprinkler will not work if there is rain.

From the graph, we can write down the probabilistic model.

The probabilistic model is a joint probability over all random variables.

It can be written using the following formula.

The joint probability over all variables equals to the product for each variable,

is probability given all the parents.

On this graph, for example,

the parents of the node grass are sprinkler and the rain.

Let's try to write down the probabilistic model for this graph.

So the joint probability of sprinkler,

rain, and the grass equals to the product of two terms.

The first one is the probability of grass is worse given the sprinkler and the rain.

Those are the parents of this node.

Next, multiplier is the probability of the sprinkler given the rain.

The rain is the only parent of this node.

And finally, we write down the probability of the rain.

Since this node doesn't have any parents,

we just write down the probability of the rain.

And this is our final model.

We can see a bit more complex ones.

For example, you all know the Naive Bayes classifier.

It's graphical model looks as follows.

We have a class,

C that directly impacts the values of the features,

that is for different classes.

The distribution of the features may be different.

And the joint distribution can be written using the following formula.

It is the probability of the class times the product over all features,

the probability of the current feature given the class.

However, this notation is a bit interesting since we have a lot of equal sub-graphs.

A bit more convenient way to write down this graph is called a plate notation.

It is written as follows.

So we have a random variable that corresponds to the class,

that directly impacts the features.

And this works around this random variable with a number of

repetitions such that we have to repeat this sub-graph

that is contained inside this box and times.

And so this is exactly Kuhn's graphical model,

the one we saw on the previous slide.

How to define a model

Bayesian Methods for Machine Learning

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