Evidence lower bound

In variational Bayesian methods, the evidence lower bound (often abbreviated ELBO, also sometimes called the variational lower bound or negative variational free energy) is a useful lower bound on the log-likelihood of some observed data.

Terminology and notation

Let $X$ and $Z$ be random variables, jointly-distributed with distribution $p_\theta$. For example, $p_\theta( X)$ is the marginal distribution of $X$, and $p_\theta( Z \mid X)$ is the conditional distribution of $Z$ given $X$. Then, for any sample $x\sim p_\theta$, and any distribution $q_\phi$, we have\ln p_\theta(x) \ge \mathbb \mathbb E_\left \ln\frac \rightThe left-hand side is called the ''evidence'' for $x$, and the right-hand side is called the ''evidence lower bound for $x$'', or ''ELBO''. We refer to the above inequality as the ''ELBO inequality''.

In the terminology of variational Bayesian methods, the distribution $p_\theta( X)$ is called the ''evidence''. Some authors use the term ''evidence'' to mean $\ln p_\theta( X)$, and others authors call $\ln p_\theta( X)$ the ''log-evidence'', and some use the terms ''evidence'' and ''log-evidence'' interchangeably.

There is no generally fixed notation for the ELBO. In this article we useL(\phi, \theta; x) := \mathbb E_\left \ln\frac \right

Motivation

Variational Bayesian inference

Suppose we have an observable random variable $X$, and we want to find its true distribution $p^*$. This would allow us to generate data by sampling, and estimate probabilities of future events. In general, it is impossible to find $p^*$ exactly, forcing us to search for a good approximation''.''

That is, we define a sufficiently large parametric family $\_$ of distributions, then solve for $\min_\theta L(p_\theta, p^*)$ for some loss function $L$. One possible way to solve this is by considering small variation from $p_\theta$ to $p_$, and solve for $L(p_\theta, p^*) - L(p_, p^*) =0$. This is a problem in the calculus of variations, thus it is called the variational method.

Since there are not many explicitly parametrized distribution families (all the classical distribution families, such as the normal distribution, the Gumbel distribution, etc, are far too simplistic to model the true distribution), we consider ''implicitly parametrized'' probability distributions:

* First, define a simple distribution $p(z)$ over a latent random variable $Z$. Usually a normal distribution or a uniform distribution suffices.
* Next, define a family of complicated functions $f_\theta$ (such as a deep neural network) parametrized by $\theta$.
* Finally, define a way to convert any $f_\theta(z)$ into a simple distribution over the observable random variable $X$. For example, let $f_\theta(z) = (f_1(z), f_2(z))$ have two outputs, then we can define the corresponding distribution over $X$ to be the normal distribution $\mathcal N(f_1(z), e^)$.

This defines a family of joint distributions $p_\theta$ over $(X, Z)$. It is very easy to sample $(x, z) \sim p_\theta$: simply sample $z\sim p$, then compute $f_\theta(z)$, and finally sample $x \sim p_\theta(\cdot , z)$ using $f_\theta(z)$.

In other words, we have a generative model for both the observable and the latent.
Now, we consider a distribution $p_\theta$ good, if it is a close approximation of $p^*$:$p_\theta(X) \approx p^*(X)$since the distribution on the right side is over $X$ only, the distribution on the left side must marginalize the latent variable $Z$ away.

In general, it's impossible to perform the integral $p_\theta(x) = \int p_\theta(x, z)p(z)dz$, forcing us to perform another approximation.

Since $p_\theta(x) = \frac$, it suffices to find a good approximation of $p_\theta(z, x)$. So define another distribution family $q_\phi(z, x)$ and use it to approximate $p_\theta(z, x)$. This is a discriminative model for the latent.

The entire situation is summarized in the following table:

In Bayesian language, $X$ is the observed evidence, and $Z$ is the latent/unobserved. The distribution $p$ over $Z$ is the ''prior distribution'' over $Z$, $p_\theta(x, z)$ is the likelihood function, and $p_\theta(z, x)$ is the ''posterior'' ''distribution'' over $Z$.

Given an observation $x$, we can ''infer'' what $z$ likely gave rise to $x$ by computing $p_\theta(z, x)$. The usual Bayesian method is to estimate the integral $p_\theta(x) = \int p_\theta(x, z)p(z)dz$, then compute by Bayes rule $p_\theta(z, x) = \frac$. This is expensive to perform in general, but if we can simply find a good approximation $q_\phi(z, x) \approx p_\theta(z, x)$ for most $x, z$, then we can infer $z$ from $x$ cheaply. Thus, the search for a good $q_\phi$ is also called amortized inference.

All in all, we have found a problem of variational Bayesian inference.

Deriving the ELBO

A basic result in variational inference is that minimizing the Kullback–Leibler divergence (KL-divergence) is equivalent to maximizing the log-likelihood:\mathbb_ ln p_\theta (x)= -H(p^*) - D_(p^*(x) \, p_\theta(x))where H(p^*) = -\mathbb \mathbb E_ ln p^*(x)/math> is the entropy of the true distribution. So if we can maximize \mathbb_ ln p_\theta (x)/math>, we can minimize $D_(p^*(x) \, p_\theta(x))$, and consequently find an accurate approximation $p_\theta \approx p^*$.

To maximize \mathbb_ ln p_\theta (x)/math>, we simply sample many $x_i\sim p^*(x)$, i.e. use Importance samplingN\max_\theta \mathbb_ ln p_\theta (x)approx \max_\theta \sum_i \ln p_\theta (x_i)

In order to maximize $\sum_i \ln p_\theta (x_i)$, it's necessary to find $\ln p_\theta(x)$:$\ln p_\theta(x) = \ln \int p_\theta(x, z) p(z)dz$This usually has no closed form and must be estimated. The usual way to estimate integrals is Monte Carlo integration with importance sampling:\int p_\theta(x, z) p(z)dz = \mathbb E_\left frac\right/math>where $q_\phi(z, x)$ is a sampling distribution over $z$ that we use to perform the Monte Carlo integration.

So we see that if we sample $z\sim q_\phi(\cdot, x)$, then $\frac$ is an unbiased estimator of $p_\theta(x)$. Unfortunately, this does not give us an unbiased estimator of $\ln p_\theta(x)$, because $\ln$ is nonlinear. Indeed, we have by Jensen's inequality, \ln p_\theta(x)= \ln \mathbb E_\left frac\right\geq \mathbb E_\left ln\frac\right/math>In fact, all the obvious estimators of $\ln p_\theta(x)$ are biased downwards, because no matter how many samples of $z_i\sim q_\phi(\cdot , x)$ we take, we have by Jensen's inequality:\mathbb E_\left \ln \left(\frac 1N \sum_i \frac\right)
\right\leq \ln \mathbb E_\left \frac 1N \sum_i \frac
\right= \ln p_\theta(x) Subtracting the right side, we see that the problem comes down to a biased estimator of zero:\mathbb E_\left \ln \left(\frac 1N \sum_i \frac\right)
\right\leq 0By the delta method, we have\mathbb E_\left \ln \left(\frac 1N \sum_i \frac\right)
\right\approx -\frac \mathbb V_\left frac\right= O(N^)If we continue with this, we would obtain the importance-weighted autoencoder. But we return to the simplest case with $N=1$:\ln p_\theta(x)= \ln \mathbb E_\left frac\right\geq \mathbb E_\left ln\frac\right/math>The tightness of the inequality has a closed form:\ln p_\theta(x)- \mathbb E_\left ln\frac\right= D_(q_\phi(\cdot , x)\, p_\theta(\cdot , x))\geq 0We have thus obtained the ELBO function:$L(\phi, \theta; x) := \ln p_\theta(x) - D_(q_\phi(\cdot , x)\, p_\theta(\cdot , x))$

Maximizing the ELBO

For fixed $x$, the optimization $\max_ L(\phi, \theta; x)$ simultaneously attempts to maximize $\ln p_\theta(x)$ and minimize $D_(q_\phi(\cdot , x)\, p_\theta(\cdot , x))$. If the parametrization for $p_\theta$ and $q_\phi$ are flexible enough, we would obtain some $\hat\phi, \hat \theta$, such that we have simultaneously

$\ln p_(x) \approx \max_\theta \ln p_\theta(x); \quad q_(\cdot , x)\approx p_(\cdot , x)$Since\mathbb_ ln p_\theta (x)= -H(p^*) - D_(p^*(x) \, p_\theta(x))we have$\ln p_(x) \approx \max_\theta -H(p^*) - D_(p^*(x) \, p_\theta(x))$and so$\hat\theta \approx \arg\min D_(p^*(x) \, p_\theta(x))$In other words, maximizing the ELBO would simultaneously allow us to obtain an accurate generative model $p_ \approx p^*$ and an accurate discriminative model $q_(\cdot , x)\approx p_(\cdot , x)$.

Main forms

The ELBO has many possible expressions, each with some different emphasis.\mathbb_\left ln\frac\right= \int q_\phi(z, x)\ln\fracdzThis form shows that if we sample $z\sim q_\phi(\cdot , x)$, then $\ln\frac$ is an unbiased estimator of the ELBO.$\ln p_\theta(x) - D_(q_\phi(\cdot , x) \;\, \; p_\theta(\cdot , x))$This form shows that the ELBO is a lower bound on the evidence $\ln p_\theta(x)$, and that maximizing the ELBO with respect to $\phi$ is equivalent to minimizing the KL-divergence from $p_\theta(\cdot , x)$ to $q_\phi(\cdot , x)$.\mathbb_ z)- D_(q_\phi(\cdot , x) \;\, \; p)This form shows that maximizing the ELBO simultaneously attempts to keep $q_\phi(\cdot , x)$ close to $p$ and concentrate $q_\phi(\cdot , x)$ on those $z$ that maximizes $\ln p_\theta (x, z)$. That is, the approximate posterior $q_\phi(\cdot , x)$ balances between staying close to the prior $p$ and moving towards the maximum likelihood $\arg\max_z \ln p_\theta (x, z)$.H(q_\phi(\cdot , x)) + \mathbb_ x)+ \ln p_\theta(x)This form shows that maximizing the ELBO simultaneously attempts to keep the entropy of $q_\phi(\cdot , x)$ high, and concentrate $q_\phi(\cdot , x)$ on those $z$ that maximizes $\ln p_\theta (z, x)$. That is, the approximate posterior $q_\phi(\cdot , x)$ balances between being a uniform distribution and moving towards the maximum a posteriori $\arg\max_z \ln p_\theta (z, x)$.

Data-processing inequality

Suppose we take $N$ independent samples from $p^*$, and collect them in the dataset $D = \$, then we have empirical distribution $q_D(x) = \frac 1N \sum_i \delta_$.

Fitting $p_\theta(x)$ to $q_D(x)$ can be done, as usual, by maximizing the loglikelihood $\ln p_\theta(D)$:$D_(q_D(x) \, p_\theta(x)) = -\frac 1N \sum_i \ln p_\theta(x_i) - H(q_D)= -\frac 1N \ln p_\theta(D) + H(q_D)$Now, by the ELBO inequality, we can bound $\ln p_\theta(D)$, and thus$D_(q_D(x) \, p_\theta(x)) \leq -\frac 1N L(\phi, \theta; D) - H(q_D)$The right-hand-side simplifies to a KL-divergence, and so we get:$D_(q_D(x) \, p_\theta(x)) \leq -\frac 1N \sum_i L(\phi, \theta; x_i) - H(q_D)= D_(q_(x, z); p_\theta(x, z))$This result can be interpreted as a special case of the data processing inequality.

In this interpretation, maximizing $L(\phi, \theta; D)= \sum_i L(\phi, \theta; x_i)$ is minimizing $D_(q_(x, z); p_\theta(x, z))$, which upper-bounds the real quantity of interest $D_(q_(x); p_\theta(x))$ via the data-processing inequality. That is, we append a latent space to the observable space, paying the price of a weaker inequality for the sake of more computationally efficient minimization of the KL-divergence.

References

Notes

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Theory of probability distributions