Implicit Likelihood Estimator

Summary

The implicit-likelihood estimator ${\hat{\mu }}_{m+\ell }$ handles models whose likelihood $p(y\mid \theta ,d)$ can be sampled but not evaluated (e.g. random-effects / nuisance-variable models, where $p(y\mid \theta ,d)={\mathbb{E}}_{p(\psi \mid \theta )}[p(y\mid \theta ,\psi ,d)]$ is intractable). It learns two approximations — a marginal $q_m(y\mid d)$ and a likelihood $q_{\ell }(y\mid \theta ,d)$ — and plugs both into the EIG. Unlike the other three, it is not a bound on the EIG, but Lemma 2 bounds its error, so minimizing that bound trains it.

Overview

The posterior, marginal, and VNMC estimators all assume the likelihood $p(y\mid \theta ,d)$ can be evaluated pointwise. Many important models cannot: introduce nuisance latents $\psi$ (random effects, latent confounders) and the likelihood becomes an intractable integral, even though you can still simulate $y$ by sampling $\psi$ then $y$. The variational posterior ${\hat{\mu }}_{\text{post}}$ works here unchanged (it needs only samples), but ${\hat{\mu }}_{\text{marg}}$ does not — it contains $p(y\mid \theta ,d)$ explicitly. The fix is to approximate the likelihood too.

Main Content

Definition: Implicit-likelihood estimator (Foster 2019, Eq. 12)

Using a marginal approximation $q_m(y\mid d)$ and a likelihood approximation $q_{\ell }(y\mid \theta ,d)$:

$$\mathrm{EIG}(d)\approx {\mathcal{I}}_{m+\ell }(d):={\mathbb{E}}_{p(y,\theta \mid d)}\left[\log \frac{q_{\ell }(y\mid \theta ,d)}{q_m(y\mid d)}\right]\approx {\hat{\mu }}_{m+\ell }(d):=\frac{1}{N}\sum _{n=1}^N\log \frac{q_{\ell }(y_n\mid {\theta }_n,d)}{q_m(y_n\mid d)}.$$

This is not a bound on the EIG (unlike the other three estimators).

Lemma 2 — EIG estimation error bound (Foster 2019)

For any valid $q_m(y\mid d)$ and $q_{\ell }(y\mid \theta ,d)$, the EIG estimation error is bounded:

$$\left|{\mathcal{I}}_{m+\ell }(d)-\mathrm{EIG}(d)\right|\le -{\mathbb{E}}_{p(y,\theta \mid d)}\left[\log q_m(y\mid d)+\log q_{\ell }(y\mid \theta ,d)\right]+C,$$

where $C=-\mathrm{H}[p(y\mid d)]-{\mathbb{E}}_{p(\theta )}[\mathrm{H}[p(y\mid \theta ,d)]]$ does not depend on $q_m$ or $q_{\ell }$. The RHS is $0$ iff $q_m=p(y\mid d)$ and $q_{\ell }=p(y\mid \theta ,d)$ for almost all $y,\theta$.

Training implication

Lemma 2 says: learn $q_m$ and $q_{\ell }$ by maximizing ${\mathbb{E}}_{p(y,\theta \mid d)}[\log q_m(y\mid d)+\log q_{\ell }(y\mid \theta ,d)]$ via stochastic gradient ascent (two ordinary maximum-likelihood density-estimation problems), then substitute into Eq. 12. In general $q_m$ and $q_{\ell }$ are learned separately with no weight sharing; Foster 2019 §A.4 discusses the coupled case $q_m(y\mid d)={\mathbb{E}}_{p(\theta )}[q_{\ell }(y\mid \theta ,d)]$.

Where it sits among the four

From Foster 2019, Table 1: ${\hat{\mu }}_{m+\ell }$ is the only estimator marked implicit-likelihood ✓ besides ${\hat{\mu }}_{\text{post}}$, and it relies on approximating a distribution over $y$ (so prefer it, like ${\hat{\mu }}_{\text{marg}}$, when $\dim (y)\ll \dim (\theta )$). It gave the lowest empirical MSE on the mixed-effects benchmark (the implicit-likelihood problem) in Table 2.

Examples

Mixed-effects / item-response model (Foster 2019 §6.1, §6.3)

A psychology item-response model has common fixed effects $\theta$ (of interest) and per-participant random effects (nuisance $\psi$), making $p(y\mid \theta ,d)$ implicit. ${\hat{\mu }}_{m+\ell }$ gives the best EIG accuracy here and is used to drive the online adaptive face-perception experiment on Mechanical Turk, producing lower-entropy posteriors than random design. See Sequential and Adaptive BED.

Connections

• Generalizes [[Variational Marginal Estimator|${\hat{\mu }}_{\text{marg}}$]] (recover it when the likelihood is explicit, $q_{\ell }=p(y\mid \theta ,d)$).
• Anticipates likelihood-free ACE (Foster 2020, Theorem 2), which instead keeps the contrastive structure and replaces the likelihood with an unnormalized approximation while preserving a valid lower bound.
• Related to simulation-based / likelihood-free inference (LFIRE, ABC) for implicit models.

See Also

• Variational Marginal Estimator — the explicit-likelihood version
• Variational Posterior Estimator (Barber-Agakov) — the other implicit-capable estimator
• Likelihood-Free ACE and Gradient Estimation — Foster 2020’s bound-preserving implicit approach