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Bayes' Theorem

If we combine the conditional probability with the marginal probability and the standard product rule, we have

p(X\vert Y)= \frac{p(X,Y)}{p(Y)},

which we can rewrite as

p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)},

which is Bayes' theorem. It allows us to evaluate the uncertainty in in X after we have observed Y . We can easily interchange X with Y .