A.1 Bayes' theorem

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

$$\bf x \bf x \bf x \bf x$$ (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

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

This is Bayes' theorem.