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Next: A.2 Maximum likelihood Up: A. Statistical tools Previous: A. Statistical tools

A.1 Bayes' theorem

The probability p($ \bf x$, j) of observing both $ \bf x$ and j can be written in two ways,

p($\displaystyle \bf x$, j) = p($\displaystyle \bf x$| j)P(j) = P(j|$\displaystyle \bf x$)p($\displaystyle \bf x$) . (A.1)

p($ \bf x$) is the probability of $ \bf x$ (independent of j), P(j) is the probability of j (independent of $ \bf x$), p($ \bf x$| j) is the probability of $ \bf x$ under the condition that j is given, and P(j|$ \bf x$) is the probability of j under the condition that $ \bf x$ is given. Reorganizing (A.1) about P(j|$ \bf x$) gives

P(j|$\displaystyle \bf x$) = $\displaystyle {\frac{{p({\bf x}\vert j)P(j)}}{{p({\bf x})}}}$ . (A.2)

This is Bayes' theorem.

Heiko Hoffmann