Conditional probability table

In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to demonstrate conditional probability of a single variable with respect to the others. For example, assume there are three random variables $x_{1},x_{2},x_{3}$ where each has $K$ states. Then, the conditional probability table of $x_{1}$ provides the conditional probability values for $P(x_{1}=a_{k}\mid x_{2},x_{3})$for each of the K possible values $a_{k}$ of the variable $x_{1}$ and for each possible combination of values of $x_{2},\,x_{3}.$ This table has $K^{3}$ cells. In general, for $M$ variables $x_{1},x_{2},\ldots ,x_{M}$ with $K$ states for each of them, the CPT for any one of them has size $K^{M}.$

With only two variables, a CPT can be put into matrix form. For example, the values of $P(x_{1}=a_{k}\mid x_{2}=b_{j})=T_{kj},$ with k and j ranging over K values, create a K×K matrix. This matrix is a stochastic matrix since the columns sum to 1; i.e. $\sum _{k}T_{kj}=1$ for all j.