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

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

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Its nodes are in variables and edges are direct impact.

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For example, here we have two invariables,

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rain and the fact that the grass is wet.

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And these edge shows as if there is rain,

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

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We can see a more complex graph.

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For example, here we added another invariable called sprinkler,

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

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Also the sprinkler will not work if there is rain.

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

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

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It can be written using the following formula.

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

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is probability given all the parents.

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On this graph, for example,

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

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

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So the joint probability of sprinkler,

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

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

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Those are the parents of this node.

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

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The rain is the only parent of this node.

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

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Since this node doesn't have any parents,

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we just write down the probability of the rain.

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And this is our final model.

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We can see a bit more complex ones.

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For example, you all know the Naive Bayes classifier.

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It's graphical model looks as follows.

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We have a class,

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C that directly impacts the values of the features,

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that is for different classes.

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The distribution of the features may be different.

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

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

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the probability of the current feature given the class.

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

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

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It is written as follows.

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

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that directly impacts the features.

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And this works around this random variable with a number of

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repetitions such that we have to repeat this sub-graph

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that is contained inside this box and times.

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And so this is exactly Kuhn's graphical model,

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the one we saw on the previous slide.