Continuous-Time Evidence Lower Bound

• Continuous-Time ELBO is a variational lower bound for continuous-time SDE models that integrates data likelihoods with stochastic optimal control for irregular time series.
• It employs Doob’s h-transform and neural control parameterizations to derive tractable approximations via a stochastic optimal control framework.
• The approach enables efficient learning and simulation-free inference using piecewise linear drift approximations and modern network architectures.

The continuous-time Evidence Lower Bound (ELBO) is a variational lower bound formulated for probabilistic models that evolve according to continuous-time latent state-space dynamics, notably those driven by stochastic differential equations (SDEs). It provides a foundation for scalable inference and learning in irregularly observed time series, enabling the integration of data likelihoods and pathwise regularization. The continuous-time ELBO considered here arises from a stochastic optimal control (SOC) perspective, establishing a rigorous connection between Doob’s $h$-transform, Feynman–Kac path measures, and amortized variational inference with neural control parameterizations (Park et al., 2024).

1. Feynman–Kac Path Measures and the Posterior in State-space SDEs

Let $(X_t)_{t\in[0,T]}$ denote a $\mathbb{R}^d$-valued diffusion evolving under an SDE

$dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$

where $b$ is the drift and $W_t$ standard Brownian motion. Observations $\{Y_{t_i}\}_{i=1}^k$ are made at irregular time-stamps $0=t_0<t_1<\cdots<t_k=T$, each with likelihood $g_i(y_{t_i}|X_{t_i})$. The joint path-observation posterior, or the Feynman–Kac model, is

$$\mathbb{P}^*\bigl(dX_{0:T}\mid Y_{t_1}=y_{t_1},\dots,Y_{t_k}=y_{t_k}\bigr) = \frac1{Z(y_{1:k})}\,\prod_{i=1}^k g_i\bigl(y_{t_i}\mid X_{t_i}\bigr)\;\mathbb{P}(dX_{0:T}),$$

with $(X_t)_{t\in[0,T]}$0 the marginal likelihood. Normalized potentials are defined as $(X_t)_{t\in[0,T]}$1, $(X_t)_{t\in[0,T]}$2, with the property $(X_t)_{t\in[0,T]}$3.

A multi-marginal Doob's $(X_t)_{t\in[0,T]}$4-transform yields the posterior dynamics as

$(X_t)_{t\in[0,T]}$5

where $(X_t)_{t\in[0,T]}$6 is a “backward survival” function propagating posterior information. This SDE generates exactly the posterior law $(X_t)_{t\in[0,T]}$7 with the correct initial condition $(X_t)_{t\in[0,T]}$8.

2. Variational Family, Amortization, and Auxiliary Variables

The intractability of $(X_t)_{t\in[0,T]}$9 motivates a tractable variational family: controlled SDEs parameterized by neural controls,

$\mathbb{R}^d$0

with induced path law $\mathbb{R}^d$1. In amortized inference, per-observation latent variables $\mathbb{R}^d$2 are encoded via $\mathbb{R}^d$3 and decoded with $\mathbb{R}^d$4. The control $\mathbb{R}^d$5 is parameterized to depend on latent histories or the full latent collection $\mathbb{R}^d$6, typically via a transformer or RNN.

3. Stochastic Optimal Control Formulation and Dynamic Programming

SOC theory provides a variational foundation for continuous-time inference. Define the cost functional

$\mathbb{R}^d$7

The value function $\mathbb{R}^d$8 satisfies, on subintervals $\mathbb{R}^d$9, the Hamilton–Jacobi–Bellman (HJB) PDE

$dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$0

where $dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$1 is the infinitesimal generator of the prior SDE. The minimizer is $dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$2. The Hopf–Cole transform $dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$3 linearizes the HJB, relating $dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$4 to the solution of backward Kolmogorov equations and recovering the Doob control.

4. Variational Bound and Continuous-Time ELBO Construction

Using Girsanov’s theorem,

$dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$5

the KL divergence between the variational path law and the path posterior is

$dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$6

excluding a vanishing initial-law term at optimum. Setting

$dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$7

yields a tight variational characterization: $dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$8, so that $dX_t = b(t,X_t)\,dt + dW_t,\quad X_0\sim\mu_0,$9. Minimizing $b$0 corresponds to optimal control and tightens the ELBO.

Combining the path-space bound with the standard VAE objective over latent variables, the negative ELBO is given by

$b$1

where $b$2 depends on encoded latents $b$3. The key object is

$b$4

This structure enables end-to-end training by maximizing $b$5 across all variational and generative parameters.

5. Assumptions, Practical Strategies, and Simulation-free ELBO

All drifts $b$6 and controls $b$7 are assumed Lipschitz with linear growth to guarantee strong solution existence for SDEs. The Hopf–Cole transform and use of Girsanov require Novikov-type moment conditions for validity. In practical implementation, the optimal drift $b$8 is replaced by a parameteric neural control $b$9 optimized via the ELBO.

Costly simulation of pathwise SDEs and backpropagation through continuous-time integrators can be circumvented by a piecewise locally linear drift ansatz: $W_t$0 for $W_t$1, so that state marginals evolve as Gaussian processes with closed-form updates. This approximation enables efficient, simulation-free parallel ELBO computation. Amortized control construction is performed by modern attention-based networks (e.g., transformers) operating over $W_t$2.

6. Summary and Practical Implementation

The continuous-time ELBO, as realized in the "Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series," synthesizes stochastic optimal control, Feynman–Kac representations, and deep amortized inference, resulting in an end-to-end objective: $W_t$3 All nested expectations are tractable using 1. Sampling of $W_t$4 from the encoder, 2. Neural construction of $W_t$5 via sequence models, 3. Either numerical SDE simulation or closed-form marginal propagations in the piecewise linear case, 4. Likelihood decoding via $W_t$6. The formulation provides a theoretically grounded and computationally practical route to sequential data assimilation in continuous time, particularly for irregular time series (Park et al., 2024).