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This is the definition of conditional probability + Total probability decomposition formula: $p(y|x) = \frac{p(y,x}{p(x)} = \frac{p(x,y)}{\sum_{y'}p(x,y')}$. The idea is to use some unsupervised learning algorithm to learn the distribution $p(x,y)$ for every possible value of $y$, and by using the previous formula you can find $p(y|x)$.


Intuitively, this is similar to the case when you are making predictions but you don't have all the necessary information to make the most accurate prediction or maybe there isn't a single accurate prediction, so you have a set of possible predictions (rather than a single prediction). For example, if you hadn't seen the last Liverpool game (in the ...

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