Inference algorithm

To use a Bayesian network as a classifier, one simply calculates $argmax_yP(y\vert{\bf x})$ using the distribution $P(U)$ represented by the Bayesian network. Now note that

$$P(y\vert{\bf x}) = P(U)/P({\bf x})$$

$$\propto P(U)$$

$$= \prod_{u\in U}p(u\vert pa(u))%\\$$ (1)

And since all variables in ${\bf x}$ are known, we do not need complicated inference algorithms, but just calculate (1) for all class values.