Instead of creating a sample and then rejecting it, it is possible to mix sampling with inference to reason about the probability that a sample would be rejected. In importance sampling methods, each sample has a weight, and the sample average is computed using the weighted average of samples. Likelihood weighting is a form of importance sampling where the variables are sampled in the order defined by a belief network, and evidence is used to update the weights. The weights reflect the probability that a sample would not be rejected.

Figure 8.30 shows the details of the likelihood weighting for computing $P(Q\mid e)$ for query variable $Q$ and evidence $e$. The $for$ loop (from line 15) creates a sample containing a value for all of the variables. Each observed variable changes the weight of the sample by multiplying by the probability of the observed value given the assignment of the parents in the sample. The variables not observed are sampled according the probability of the variable given its parents in the sample. Note that the variables are sampled in an order to ensure that the parents of a variable have been assigned in the sample before the variable is selected.

To extract the distribution of the query variable $Q$, the algorithm maintain an array $counts$, such that $counts[v]$ is the sum of the weights of the samples where $Q=v$. This algorithm can also be adapted to the case where the query is some complicated condition on the values; we just have to count the cases where the condition is true and those where the condition is false.