Bridging Inference in Bayesian Models

• Bridging inference is a hybrid approach that integrates MCMC transitions into variational approximations to enhance posterior fidelity.
• It employs auxiliary variables and a reverse model to construct an augmented ELBO that tightens as the inverse matching improves.
• This method is applied in deep generative models and Bayesian learning to achieve a tunable trade-off between computational efficiency and inference accuracy.

Bridging inference refers to the theoretical and algorithmic synthesis of distinct inference paradigms—most frequently, Markov chain Monte Carlo (MCMC) and variational inference (VI)—into a unified framework that leverages the complementary advantages of both. By constructing hybrid inference algorithms that embed MCMC transitions within variational approximations and explicitly optimize for an enhanced evidence lower bound (ELBO), bridging inference aims to balance fast, objective-driven optimization with asymptotic accuracy for approximating complex posteriors. This approach provides a principled methodology for interpolating between deterministic and stochastic inference, enabling practitioners to flexibly trade computational expenditure for posterior fidelity.

1. Formulation of Bridging Inference

Bridging inference is fundamentally characterized by augmenting the standard variational family with auxiliary random variables that represent trajectories of a Markov chain. In classical VI, the goal is to approximate the posterior $p(z|x)$ by a tractable family $q(z|x)$, optimizing the ELBO: $$\log p(x) \geq \mathbb{E}_{q(z|x)}[\log p(x, z) - \log q(z|x)] \equiv \mathcal{L}.$$ Bridging inference generalizes this approximation by considering not just $z$, but auxiliary variables $y = (z_0, z_1, ..., z_{T-1})$ encoding the sequence of intermediate MCMC states: $$q(y, z_T | x) = q(z_0 | x) \prod_{t=1}^T q_t(z_t | x, z_{t-1}).$$ The overall variational approximation over $z_T$ now emerges from “flowing” the initial variational sample through $T$ MCMC-like transitions.

To control for the effect of these transitions, a reverse/inverse model $r(y|x, z_T)$ is introduced, yielding an augmented variational lower bound: $$\mathcal{L}_{\text{aux}} = \mathbb{E}_{q(y, z_T|x)}[\log p(x, z_T) + \log r(y|x, z_T) - \log q(y, z_T|x)].$$ This reduces to the standard ELBO minus an expected KL-divergence term between the pathwise distributions under the variational forward and reverse models. If the inverse model $r(y|x, z_T)$ perfectly matches $q(y|x, z_T)$, the bound becomes tight.

2. Markov Chain Variational Inference (MCVI) Framework

The Markov Chain Variational Inference (MCVI) framework operationalizes bridging inference by making the following transitions explicit:

• Forward Markov Chain: Start with $z_0 \sim q(z_0|x)$, then for $t=1,\dots,T$ sample $z_t \sim q_t(z_t|x,z_{t-1})$.
• Reverse Model: Assume $r(y | x, z_T) = \prod_{t=1}^T r_t(z_{t-1}|x, z_t)$.
• Augmented Lower Bound: The resulting bound is

  $$\log p(x) \geq \mathbb{E}_{q} \left[ \log\frac{p(x, z_T)}{q(z_0|x)} + \sum_{t=1}^T \log\frac{r_t(z_{t-1}|x, z_t)}{q_t(z_t|x, z_{t-1})} \right].$$

  Each MCMC transition thus contributes a log-ratio “correction” to the ELBO.

MCVI Algorithm

1. Draw $z_0 \sim q(z_0|x)$; initialize $L \leftarrow \log p(x, z_0) - \log q(z_0|x)$.
2. For $t = 1,\dots,T$:
   1. Sample $z_t \sim q_t(z_t|x, z_{t-1})$,
   2. Compute $$\alpha_t = \frac{p(x, z_t) r_t(z_{t-1}|x, z_t)}{p(x, z_{t-1}) q_t(z_t|x, z_{t-1})}$$,
   3. Update $L \leftarrow L + \log \alpha_t$.

3. Optimize the variational and reverse model parameters via stochastic gradients, with reparameterization as appropriate.

This strategy admits unbiased Monte Carlo estimation of the bound and supports backpropagation through the entire chain for end-to-end optimization.

3. Hamiltonian Variational Inference (HVI) and Advanced Instances

Hamiltonian Variational Inference (HVI) is a specific instance where the Markov transitions $q_t$ are parameterized by Hamiltonian Monte Carlo (HMC) dynamics. Here, the Markov steps involve the introduction of auxiliary momentum variables, leapfrog integration, and deterministic, volume-preserving mappings, enabling efficient exploration of complex posteriors.

In HVI, the forward and reverse transitions are typically deterministic given the momenta, and the reverse model $r_t$ is constructed to mirror the Hamiltonian flow’s time-reversal. Because HMC preserves volume, the variational family is substantially richer than standard Gaussian posteriors, capturing posterior correlations, skewness, and multi-modality.

4. Theoretical Properties and Trade-Offs

Bridging inference inherits several favorable theoretical characteristics:

• Tightness of the Bound: With optimal inverse models $r_t$, the ELBO corrections (log-ratios) are nonnegative, ensuring that each Markov step can only tighten the lower bound, or leave it unchanged if the approximation is already exact.
• Speed–Accuracy Trade-Off: The number of MCMC steps $T$ serves as a tunable hyperparameter governing computational cost and approximation accuracy. Few steps yield rapid, coarse approximations (as in vanilla VI), while additional steps incrementally bridge toward MCMC-like accuracy at added computational expense.
• Differentiable Inference: If all Markov transitions are differentiable (by reparameterization, e.g., $z_t = g_\theta(u_t, x)$ with $u_t \sim$ fixed noise), backpropagation enables joint optimization of all variational and inverse parameters.

5. Empirical Performance and Applications

Empirical studies in the original work confirm several practical benefits:

• Gaussian Toy Models: For a bivariate Gaussian, optimizing over-relaxation parameters within the bridging framework accelerates convergence compared to classical Gibbs sampling.
• Beta-Binomial Overdispersion: Inclusion of even a single HMC step dramatically reduces bias in the variational posterior.
• Deep Generative Models: In VAEs trained on MNIST, augmenting inference networks with a small (1–5) number of HVI steps improves the marginal log-likelihood, achieving better generative performance and posterior accuracy compared to standard amortized VI.

These benefits are attributed to the enriched variational family enabled by the MCVI/HVI hybridization and improved exploration of the posterior landscape.

6. Implementation Considerations and Limitations

• Inverse Model Design: Practical success depends on the expressiveness of the reverse model $r_t$; intractability can limit the tightness of the bound, and careful design or parameterization may be required.
• Computational Overhead: Each additional MCMC step increases forward and backward-pass runtime, potentially offsetting gains in approximation quality. However, reparameterized transitions and parallelized computation can mitigate some of this overhead.
• Choice of MCMC Kernel: The transition kernel $q_t$ must be efficiently sampleable and differentiable. For HMC, leapfrog step size, trajectory length, and momentum proposals must be tuned for stability.
• Deployment Strategy: In settings requiring amortized inference (e.g., inference networks in VAEs), the framework is well-suited, as it supports SGD-based updates and batch-wise sampling. For “classic” non-neural models, care must be taken in parameterization and computation of reverse densities.

7. Impact and Future Directions

Bridging inference, as formalized by the MCVI/HVI paradigm, provides a unified perspective for incorporating the benefits of both variational methods (explicit objectives, efficiency) and MCMC (asymptotic accuracy, flexibility). It is extensible to more complex MCMC schemes, auxiliary variable constructions, and multi-level variational approximations. The methodology has broad implications for fields requiring fast and accurate approximate inference, including deep generative modeling, Bayesian deep learning, and online amortized inference scenarios.

Further research may explore designing adaptive schemes for the number and type of bridging steps, improved inverse model learning, and generalizations beyond canonical VI/MCMC pairings (e.g., flows, SMC, or particle-based methods).

---

Table: Comparison of Inference Strategies

Method	Posterior Family	Objective	Asymptotic Exactness	Speed
Variational Inference (VI)	Parametric (e.g., Gaussian)	ELBO	No	High
Markov Chain Monte Carlo (MCMC)	Markov chain stationary distributions	None	Yes	Low
Bridging Inference (MCVI/HVI)	Parametric + MCMC-augmented	Tightened ELBO	Approximate/Interpolated	Tunable

Bridging inference thus fills the continuum between pure VI and MCMC, presenting a principled avenue for optimizing, calibrating, and deploying Bayesian models in contemporary applied settings (Salimans et al., 2014).