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The amortization gap measures how much evidence lower bound is lost by using the distribution returned by amortized inference instead of the best distribution available for that observation in the same variational family. Hold the generative parameters $\theta$ fixed. Write ${\mathcal{L}}_x(q)$ for the evidence lower bound and suppose the optimum $q_x^{\ast }={\mathrm{argmax}}_{q\in \mathcal{Q}}{\mathcal{L}}_x(q)$ is attained. For the shared rule $q_{\varphi }(\cdot \mid x)\in \mathcal{Q}$,The remaining variational approximation gap is $G_{\mathrm{approx}}(x)=\log p_{\theta }(x)-{\mathcal{L}}_x(q_x^{\ast })$, so the total loss isThus a restrictive recognition network or incomplete training can leave an amortization gap even when the variational family can represent the exact posterior distribution.^[ref]

For an original example, stipulate $p(z\mid x)=N(2x,1)$ for equally weighted observations $x=-1$ and $x=1$. The variational family $\mathcal{Q}=\{N(m,1):m\in \mathbb{R}\}$ represents both posterior distributions exactly when fitted separately, so $G_{\mathrm{approx}}=0$. Restrict the shared rule to $q_b(z\mid x)=N(x+b,1)$: it can adjust the intercept but its slope is fixed at one. The Kullback-Leibler divergence between these unit-variance normal distributions is $(b-x{)}^2/2$. Averaging the amortization gap over the two observations givesThe best shared intercept $b=0$ still loses $1/2$ nat on average. A current rule with $b=1$ loses one nat: half a nat can be removed by training this rule better, while the remaining half requires changing its fixed-slope restriction. This separates the variational family's expressive power from the shared mapping's capacity and current training state.

In a numerical comparison, a locally improved per-observation distribution need not be the global optimum $q_x^{\ast }$. Its improvement supplies a lower bound on the true amortization gap; failing to find an improvement does not establish a zero gap. Comparisons must keep both $\theta$ and $\mathcal{Q}$ fixed.

Amortized inference learns one shared rule that maps observations to approximate posterior distributions, spreading the work of fitting that rule across many inputs. Instead of separately optimizing variational parameters for every observation $x$, a recognition network returns the parameters of $q_{\varphi }(z\mid x)$. The shared parameters $\varphi$ are learned across data; evaluating the rule for a new input is then a different operation from fitting it.^[ref]

A scalar Gaussian model gives an exact example. Let $z\sim N(0,1)$ and $x\mid z\sim N(z,1)$. Completing the square givesThe shared rule ${\mu }_{\varphi }(x)=ax+b$, with $a=1/2$, $b=0$ and fixed variance $1/2$, represents the exact posterior for every real $x$. Evaluating $x/2$ replaces a separate optimization for each observation in this stipulated model. It does not imply that every nonlinear latent-variable model has an exact rule this simple.

Sharing can introduce an additional restriction even when the chosen distribution family is adequate. For observations $x=-2$ and $x=2$, the exact posterior means are $-1$ and $1$. The Gaussian family $N(m,1/2)$ can represent each exactly when $m$ is fitted separately. If the shared rule is constrained to output one constant mean for both inputs, its best equal-weight choice is $m=0$. Each case then hasThis original example isolates an amortization gap: the loss from restricting how observation-specific variational parameters are produced, rather than an inability of the per-observation Gaussian family to express the target. The shared mapping's capacity and its training therefore matter alongside the family of probability distributions.

David Duvenaud coauthored the 2018 study of inference suboptimality with Chris Cremer and Xuechen Li. Their analysis distinguishes distribution-family limitations from a shared inference rule's shortcomings.^[ref]

An original diagnostic example makes that distinction numerical. Fix the generative model and one observation. Suppose its exact log evidence is $-5$, the globally best evidence lower bound within a chosen family is $-5.2$, and a recognition network returns a distribution with bound $-5.9$. The variational approximation gap is $0.2$ nat; the amortization gap is $0.7$ nat. They sum to the total $0.9$-nat inference deficit.

These stipulated exact values isolate the diagnostic, rather than reproducing experimental measurements. Improving only the shared rule could recover at most $0.7$ nat within that family. Comparing a different generative model would change the target posterior distribution and would no longer isolate the same gap. This is a contribution of the joint paper; its byline does not specify individual responsibility for each derivation.

The variational approximation gap is the part of the evidence lower bound deficit that remains even after optimizing separately for an observation within a specified variational family. With the generative model fixed and finite $\log p_{\theta }(x)$, defineThe equality uses the usual evidence lower bound identity for variational inference. A supremum allows for a best value that is not attained. The amortization gap separately measures failure of the shared inference rule to reach this value.^[ref]

An original example shows what independence can cost. Stipulate a centred two-dimensional multivariate normal distribution as the posterior distribution, with covariance matrixRestrict the variational family to centred independent normal distributions, $q=N(0,\mathrm{diag}(v_1,v_2))$ with $v_1,v_2>0$. This is a simple mean-field variational inference restriction. Writing $s=1-{\rho }^2$, the Kullback-Leibler divergence between normal distributions becomesEach variance contributes a term $v_i/s-\log v_i$, minimized at $v_i=s$. ConsequentlyFor $\rho =0.6$, the best independent variances are both $0.64$, and the unavoidable deficit is about $0.2231$ nat. Matching both marginal variances to one instead would give about $0.3394$ nat. The reverse Kullback-Leibler divergence therefore chooses narrower independent marginals here, while still failing to represent the correlation.

Enlarging the family to include the full covariance matrix permits $q=p$ and reduces this example's gap to zero. More generally, inclusion ${\mathcal{Q}}_1\subseteq {\mathcal{Q}}_2$ cannot increase the optimum approximation gap at a fixed model. Merely training the same shared rule longer cannot remove a strictly positive family restriction. Numerical improvements are not proofs that the infimum has been reached.

Xuechen Li coauthored Inference Suboptimality in Variational Autoencoders by Chris Cremer and others with Chris Cremer and David Duvenaud. Their numerical method refits variational parameters per observation and uses the Monte Carlo method with 100 samples to estimate the evidence lower bound during that optimization.^[ref]

An original sampling calculation explains why sample count matters. Suppose independent contributions to a bound estimate have standard deviation two nats. A sample mean from $N$ such contributions has standard error $2/\sqrt{N}$ nats: $0.4$ at $N=25$ and $0.2$ at $N=100$. Quadrupling the sample count halves this sampling error under the independence assumption. These are assumed variances, not measured performance figures from the paper.

Reducing this error helps compare trial distributions, but neither a stable noisy bound nor a numerical stopping rule certifies the globally best member of the family. The mathematical variational approximation gap and amortization gap use that optimum; a measured local improvement can underestimate the latter. This distinction connects the joint experimental method to uncertainty in the numerical diagnostic.