A common application of Jensen's inequality is to bound a more complicated expression by a simpler one, such as the log-likelihood of partially observed random variables. This is utilized in variational methods using the inequality $E_{Y \sim P(Y)}[-\log P(X \mid Y)] \geq -\log P(X)$, since $\int P(Y) P(X \mid Y) dY = P(X)$. In this context, $Y$ typically represents an unobserved random variable (such as cluster labels in clustering), $P(Y)$ is its estimated distribution, and $P(X \mid Y)$ is the generative model, with $P(X)$ being the distribution with $Y$ integrated out.