Temporal Posterior Inference (TPI)

• Temporal Posterior Inference (TPI) is a Bayesian framework that updates posterior distributions of time-indexed latent variables using sequential observations.
• It integrates methods from time-series analysis, probabilistic programming, and structured sparsity to efficiently track evolving latent processes.
• Algorithmic implementations like expectation propagation and variational Bayes enable scalable, robust inference across high-dimensional spatio-temporal applications.

Temporal Posterior Inference (TPI) refers to a family of Bayesian inference frameworks that update and condition probabilistic beliefs over time-dependent latent variables, signals, or execution traces, leveraging the temporal evolution and structure of both data and latent processes. TPI unifies classical and contemporary approaches in time-series analysis, probabilistic programming, sequential variational inference, and structured sparsity, anchored by efficient algorithmic procedures for tracking the full temporal posterior—sometimes over path or trace space—given sequential or temporally-structured observations.

1. Fundamental Concepts and Mathematical Formulation

Temporal Posterior Inference constructs and updates the posterior distribution of latent state trajectories, model parameters, or execution traces, conditioned on a sequence of temporal observations. Given data arriving over time—possibly in blocks or as streams—and a generative or probabilistic program model, TPI computes

$p(\theta_{1:T} \mid y_{1:T})$

where $\theta_{1:T}$ encodes time-indexed latent quantities and $y_{1:T}$ are corresponding observations.

Key mathematical instances include:

• Sequential Bayes' rule for parameter filtering:

$$p(\theta \mid y_{1:T_{n+1}}) \propto p(\theta \mid y_{1:T_n}) p(y_{T_n+1:T_{n+1}} \mid \theta)$$

• Path-level posterior in probabilistic programming:

$$P(\varphi\mid \psi) = \frac{P(\pi \in \mathcal{L}(\varphi)\cap \mathcal{L}(\psi))}{P(\pi \in \mathcal{L}(\psi))}$$

where $\varphi, \psi$ are temporal logic specifications and $\pi$ ranges over execution traces (Wang et al., 25 Dec 2025).

• Full posterior over run-length and state for latent change-point models (Prat-Carrabin et al., 2021):

$p_{t+1}(s,\tau) \propto g(x_{t+1}\mid s)\left[ \mathbbm{1}_{\tau=0}\sum_{\tau'}q(\tau')\int a(s|s') p_t(s',\tau') ds' + \mathbbm{1}_{\tau>0} (1-q(\tau-1)) p_t(s,\tau-1) \right]$

2. TPI in Structured Spatio-Temporal Models

In high-dimensional spatio-temporal settings, TPI is instantiated by hierarchical sparsity-inducing models, e.g., spatio-temporal spike-and-slab priors which encode joint spatial and temporal dependency for latent activations. The model defined in (Andersen et al., 2015) includes:

• Latent signals $x_t$, binary supports $z_t$, and inclusion control fields $\gamma_t$ over time.
• Gaussian linear or probit likelihoods for each $y_t$.
• Spike-and-slab prior:

$$p(X\mid Z) = \prod_{t=1}^T\prod_{i=1}^D [(1-z_{i,t})\delta(x_{i,t}) + z_{i,t}\mathcal{N}(x_{i,t}\mid\rho,\tau)]$$

• Hierarchical Gaussian-process prior on inclusion field $\Gamma$, with Kronecker-structured covariance encoding both spatial and temporal correlations.

Posterior inference in this framework involves computation over $2^{DT}$ binary spike configurations and joint Gaussian latent fields—tractable only via variational approximations or message passing (Andersen et al., 2015).

3. Algorithmic Realizations

Three canonical classes of algorithms operationalize TPI in contemporary research:

3.1 Expectation Propagation for Spatio-Temporal Models

Expectation propagation (EP) approximates the full intractable posterior by iteratively refining marginal (site) approximations through moment-matching. For TPI under spatio-temporal spike-and-slab priors:

• The joint posterior is approximated with structured products of Gaussian and Bernoulli factors over $(X,Z,\Gamma)$.
• EP updates comprise forming cavity distributions, computing moments of the tilted distribution (incorporating the current site and cavity), and re-matching site parameters to fit these moments.
• Computational complexity is mitigated using three approximations: low-rank (LR-EP), common-precision (CP-EP), and group-wise (G-EP) schemes, enabling practical TPI for large dimensional ($D,T$) problems (Andersen et al., 2015).

3.2 Variational Bayes and Its Sequential Recursive Updates

Updating Variational Bayes (UVB) is a recursive variational method for streaming-data Bayesian inference (Tomasetti et al., 2019):

• At each increment, the previous variational posterior $q_{\lambda_n^*}(\theta|y_{1:T_n})$ is used as a pseudo-prior for the data increment.
• The next variational posterior minimizes the KL divergence to the updated pseudo-posterior, targeting only the data accrued since the last update.
• Importance-sampled UVB (UVB-IS) further enhances speed by reusing draws from the previous variational posterior and adjusting gradients via importance weighting.

UVB/UVB-IS achieve amortized computational scaling in dynamic settings and robustness provided the variational family remains sufficiently expressive across updates (Tomasetti et al., 2019).

3.3 Temporal Posterior Inference over Execution Traces and Temporal Logic

For probabilistic programs and temporal logic specifications, TPI computes rigorous posterior probabilities over omega-regular properties along execution traces (Wang et al., 25 Dec 2025):

• Execution traces are evaluated with respect to automata (typically deterministic Rabin automata) encoding temporal properties.
• Posterior satisfaction probabilities given temporal observations are formalized as the ratios of path measures.
• Rigorous soundness is provided via stochastic barrier certificates—supermartingale or submartingale witnesses—to bound satisfaction probabilities, synthesized using semidefinite or linear programming techniques.

This approach is implemented in the tool TPInfer, yielding certified posterior bounds for nontrivial temporal queries in infinite-state or unbounded-loop PPLs (Wang et al., 25 Dec 2025).

4. Computational and Theoretical Properties

TPI algorithms are assessed along several axes:

• Scalability: Factorizations and low-rank approximations reduce the cubic complexity associated with naive posterior updates in spatio-temporal models (Andersen et al., 2015).
• Error Control: In recursive sequential methods (UVB/UVB-IS) two key sources of error are the deviation of variational approximations from the true posterior and the accumulation of bias when using successive pseudo-priors. Well-behaved models and modest data increments yield subadditive error accumulation (Tomasetti et al., 2019).
• Convergence Guarantees: For stochastic gradient-based variational inference, convergence to stationary points is ensured under classic Robbins-Monro step-size decay conditions (Tomasetti et al., 2019).
• Rigorous Bounds: For omega-regular TPI, convergence of computed upper/lower bounds to the true posterior probability is established as polynomial barrier degrees and counting thresholds increase, provided feasible certificates exist (Wang et al., 25 Dec 2025).

5. Empirical Domains and Applications

TPI is applied across several scientific and engineering domains:

• Compressed Sensing and Source Localization: Enhanced recovery of structured, time-varying signals matrixes using spatio-temporal spike-and-slab TPI, demonstrated on synthetic, EEG, and classification data (Andersen et al., 2015).
• Time-Series Forecasting and Clustering: UVB and UVB-IS outperform batch variational Bayes (SVB) in real-time autoregressive forecasting, mixture clustering, and streaming hierarchical models, with substantial speed and competitive accuracy (Tomasetti et al., 2019).
• Probabilistic Program Verification: TPI for omega-regular properties yields certified probabilistic bounds over safety and liveness conditions in probabilistic programs with complex temporal logic (Wang et al., 25 Dec 2025).
• Human Sequential Inference: TPI describes mechanisms for sample-based human Bayesian inference in dynamic environments with temporal statistics, capturing both adaptation to statistical structure and "lawful" behavioral variability (Prat-Carrabin et al., 2021).

6. Extensions, Practical Considerations, and Toolchains

Advanced TPI practice involves:

• Hyperparameter Learning: Type-II maximum likelihood, Central Composite Design (CCD), or Bayesian optimization for kernel scales and variances are efficient when integrated with the EP TPI pipeline (Andersen et al., 2015).
• Model Richness and Diagnostics: Sequential VB methods monitor ELBO, effective sample size (ESS) in importance sampling, and sliding-window estimates to control drift and overfitting (Tomasetti et al., 2019).
• Automated Certificate Synthesis: For temporal logic-based inference, polynomial templates for barrier certificates are synthesized and optimized using SOSTOOLS, CPLEX, and tools like Spot for automata conversion (Wang et al., 25 Dec 2025).
• Sample-Based Approximations: Particle filter TPI variants scale to real-time inference in environments where memory and computation are constrained (e.g., simulating human decision processes) (Prat-Carrabin et al., 2021).

7. Evaluation and Benchmarks

Empirically, TPI has been shown to deliver quantitative improvements and rigorous guarantees:

• Recovery accuracy and computational efficiency outpace classical (batch) inference in high-dimensional and streaming settings (Andersen et al., 2015, Tomasetti et al., 2019).
• Certified upper/lower bounds for temporal logic properties are robust to solver and template parametrization, enclosing simulation estimates and converging under degree/bound increases (Wang et al., 25 Dec 2025).
• Particle-based TPI matches observed adaptive and variable patterns of human inference in sequential, volatile environments, supporting sample-based cognitive models (Prat-Carrabin et al., 2021).

The frameworks and algorithms associated with Temporal Posterior Inference constitute a general methodology for tractable, interpretable, and rigorous Bayesian inference in temporally structured stochastic models, with growing impact across probabilistic machine learning, program verification, and cognitive science.