# What exactly is the point of computing a lower bound for the log partition function in variational methods in probabilistic graphical models?

Variational methods are applied when we are interested in a probability distribution $$P$$ but only have a tractably computable unnormalized form $$\tilde{P}$$ of $$P$$. Knowing the partition function $$Z = \sum_x \tilde{P}(X)$$ is desirable because it would allow us to cheaply compute $$P(x)=\tilde{P}(x)/Z$$.

Pretty much every single text on variational inference for probabilistic graphical models (for example Jaakola's "Tutorial on Variational Approximation Methods") describes the method as computing a probability distribution $$Q$$ that maximizes:

$$J(Q) = \log Z - KL(Q || P)$$

where $$Q$$ is a tractable distribution (meaning, it is easy to compute $$Q(x)$$). Because the $$KL$$ distance is always non-negative, $$J(Q)$$ is always a lower bound of $$\log Z$$. Therefore, finding a $$Q^*$$ that maximizes $$J(Q)$$ (or, equivalently, minimizes $$KL$$ since $$\log Z$$ is a constant in $$Q$$) yields the closest lower bound $$J(Q^*)$$ of $$\log Z$$ within the family to which $$Q$$ belongs.

This is often announced to much fanfare as if it were an obvious and useful goal.

However, I don't find it so obvious that a closest lower bound of $$\log Z$$ is a desirable goal. Sure, it gives us a lower bound $$Z'$$ of $$Z$$, but what is it useful for? Using it to compute an approximation $$P'$$ to $$P$$ by defining $$P'(x) = \tilde{P}/Z'$$ is an immediate idea but it does not sound that great because we wouldn't even have $$P'$$ to be a true distribution since it is not even guaranteed to sum up to 1.

So, why is computing this lower bound a useful thing?

PS: Another possible explanation is that the lower bound is not really the goal, but $$Q^*$$ is the goal, taken to be the closest approximation to $$P$$ within a tractable family of distributions. Fair enough, but then the whole thing could be explained much more succinctly and directly as a minimization of $$KL(Q || P)$$ instead of doing that as a step to minimize $$J(Q)$$. $$J(Q)$$ wouldn't even have to be defined in order to describe the method as finding $$Q^*$$ the closest approximation to $$P$$.