ASPIRE: Accelerated Sequential Posterior Inference

• The paper demonstrates that reusing posterior computations in ASPIRE dramatically accelerates sequential Bayesian inference, reducing redundant evaluations.
• ASPIRE integrates iterative amortized refinement, recursive Bayesian reuse, and flow-based SMC bridging to enhance efficiency in high-dimensional inverse problems.
• Empirical results show up to 10^3× speedup with improved uncertainty calibration, enabling rapid updates in applications like ultrasound tomography and gravitational-wave analysis.

Accelerated Sequential Posterior Inference via Reuse (ASPIRE) denotes a class of algorithmic frameworks for Bayesian inference that exploit the reuse of posterior computations, amortized approximations, or functional density representations to dramatically accelerate sequential or repeated posterior evaluations. The unifying principle is to avoid redundant recomputation when new data arrive, different models are considered, or related posterior queries are required, by leveraging amortized inference, functional density fits, flow-based mappings, or hybrid offline–online procedures. ASPIRE methods are motivated by computational bottlenecks in high-dimensional inverse problems, sequential analysis, and model reanalysis settings.

1. Principles of ASPIRE: Foundational Concepts and Taxonomy

The core idea underlying ASPIRE approaches is to accelerate Bayesian posterior inference through the reuse and transformation of prior computations. Three major paradigms have emerged:

• Iterative Amortized Inference with Physics-Based Summaries: Combines amortized inference networks (e.g., conditional normalizing flows) with the iterative refinement of low-dimensional, physics-based summary statistics to bridge the gap between rapid, general amortization and high-fidelity, specialized inference (Orozco et al., 8 May 2024).
• Recursive Posterior Reuse for Sequential Bayesian Updating: Leverages recursive combinations of prior and proposal distributions (Prior- and Proposal-Recursive Bayes) to propagate posterior draws and likelihood computations across data blocks, reducing overall model evaluation costs while maintaining asymptotic correctness (Hooten et al., 2018).
• Posterior Transformation via Flow-Based Models and SMC Bridging: Constructs flexible normalizing flows on posterior samples from one model or dataset, then bridges to an alternative posterior via Sequential Monte Carlo (SMC), enabling rapid adaptation to new models or extended hypotheses without repeated full-data sampling (Williams, 6 Nov 2025).

ASPIRE methods contrast with direct sample reweighting and naive sequential importance approaches, which are generally prone to weight degeneracy and inefficiency in high-dimensional problems (Thijssen et al., 2017).

2. Methodological Formulations and Algorithmic Structure

(a) Iterative Amortized Posterior Refinement

Given unknown parameters $x \in \mathbb{R}^n$ and data $y \in \mathbb{R}^m$, with a prior $p(x)$ and likelihood $p(y|x)$, standard amortized variational inference (VI) seeks to train $q_\phi(x|y)$ by minimizing forward KL-divergence:

$$\min_\phi\, \mathbb{E}_{p(y)}[\,KL(p(x|y)\,\|\,q_\phi(x|y))\,] = \mathbb{E}_{p(x,y)}[-\log q_\phi(x|y)]$$

ASPIRE replaces the direct use of $y$ with an iteratively updated, lower-dimensional summary $s_j(y)$ informed by physical models. At each iteration $j$:

1. Compute score-based summaries at current fiducial points:

$$s_j^{(n)} = \nabla_x \log p(y^{(n)}|x) \big|_{x = x_j^{(n)}} = [\nabla F(x_j^{(n)})]^T(F(x_j^{(n)}) - y^{(n)})$$

where $F$ is the differentiable forward physics operator.

1. Train $q_{\theta_j}(x|s)$ (typically a normalizing flow) over $\{(x^{(n)}, s_j^{(n)})\}$.
2. Update fiducials:

$$x_{j+1}^{(n)} \approx \mathbb{E}_{x\sim q_{\theta_j}(\cdot|s_j^{(n)})}[x]$$

After $J$ refinements, the resulting $q_{\theta_J}(x|s_{J})$ yields approximate posterior samples for new $y_{\text{obs}}$ via a cheap online evaluation, requiring only low-rank summary updates and flow evaluations (Orozco et al., 8 May 2024).

(b) Recursive Bayesian Reuse

The ASPIRE approach in (Hooten et al., 2018) combines Prior-Recursive and Proposal-Recursive Bayes. Sequential data blocks $\{y_j\}_{j=1}^J$ are processed as follows:

1. Fit $p(\theta|y_1)$ via MCMC.
2. For each subsequent block $j$, reuse posterior draws as proposals:
   • Precompute likelihoods $L_{j,k} = \log p(y_j|\theta_{1}^{(k)}, y_{1:j-1})$.
   • Run a Metropolis-Hastings update on $\theta$ using these proposals, with acceptance ratio based only on available precomputed likelihoods.
3. Final draws from $p(\theta|y_{1:J})$ are built by recombining earlier computations.

This reduces total complexity from $O(L(n))$ per full inference to $O(J \cdot L(n/J))$ with $J$ partitions and yields substantial speedup (Hooten et al., 2018).

(c) Flow-Based Posterior Reanalysis with SMC Bridging

For settings where new models (priors or likelihoods) are considered for fixed data, ASPIRE uses the following steps:

1. Fit a normalizing flow $q_{\phi}$ to samples $\{\theta^{(i)}\}$ from the existing posterior.
2. Initialize particles $\theta_0^{(i)} \sim q_{\phi}$.
3. Define a path of bridging distributions:

$$p_t(\theta) \propto q_{\phi}(\theta)^{1-\beta_t} [L_2(\theta)\,p(\theta|M_2)]^{\beta_t} \quad \text{for } \beta_t \in [0,1]$$

1. Run SMC: at each step, update particle weights for $\Delta \beta = \beta_t - \beta_{t-1}$, resample if necessary, and apply short MCMC moves to maintain diversity.

Resulting samples and evidence estimates for $M_2$ match those from full reanalysis, at 4–10× lower cost (Williams, 6 Nov 2025).

3. Computational Complexity and Efficiency

A summary of computational costs and characteristics across the main ASPIRE instantiations is as follows:

ASPIRE Variant	Offline/Precompute Cost	Online/Update Cost	Online Speedup
Iterative Amortized (physics summary)	$O(NJ\,{\rm cost}_{\text{grad}})$	$O(J\,{\rm cost}_{\text{grad}})$	$10^2$–$10^3\times$ vs non-amortized
Recursive Reuse (prior/proposal)	$O(J\,L(n/J))$	$O(1)$ per MCMC draw	$4\times$ (GP model), $59\times$ (state-space)
Flow-based SMC bridging	$O(N{\rm flow\ fit})$	$O({\rm SMC}{\rm steps})$	$4$–$10\times$

In instance, ASPIRE in (Orozco et al., 8 May 2024) for transcranial ultrasound computed tomography achieves posterior uncertainty calibration, reducing Uncertainty Coverage Error (UCE) from $1.6 \to 0.5$ as a function of iteration (see empirical Sections 6.1-6.3).

Benchmarks in (Williams, 6 Nov 2025) demonstrate reduced likelihood evaluations (up to 10×) for posterior adaptation in gravitational-wave analyses, reproducing reference posterior and evidence within the credible tolerance.

4. Statistical and Practical Properties

ASPIRE’s statistical guarantees depend on the paradigm:

• Iterative refinement: Each refinement step tightens the amortization gap by specializing the inference network to improved local summaries, enabling posterior means and covariances to approach their true values within a small number of iterations ($J \sim 3$–$4$ typically suffices) (Orozco et al., 8 May 2024).
• Recursive Bayesian reuse: Draws are asymptotically correct for the full posterior provided the stage-1 chain mixes and proposal resampling is unbiased; effective sample size (ESS) typically increases compared to naive MCMC, due to better proposal matching (Hooten et al., 2018).
• Flow–SMC bridging: Provided the normalizing flow covers the support of the new posterior, the SMC yields unbiased samples and evidence under the alternative model; Jensen–Shannon divergences between ASPIRE and baseline posteriors are $<1.5$ mnats and evidence matches within $\pm0.2$ log-units (Williams, 6 Nov 2025).

A key practical caveat is that, if the new posterior is much more concentrated or otherwise not covered by the prior or earlier posterior samples, importance reweighting or flow fitting may fail (weight degeneracy, poor tail coverage) (Thijssen et al., 2017). Sufficient overlap between reused and target distributions is essential for robust sequential inference.

5. Applications and Empirical Results

Inverse Problems and Imaging

• Transcranial Ultrasound Computed Tomography (TUCT): ASPIRE yields substantial improvements in root-mean-square error (RMSE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM) across iterations. It succeeds where end-to-end amortized flows fail to resolve tissue interfaces.
• Diffusion Models for Posterior Sampling: By incorporating transition models (e.g., ViViT) to re-initialize denoising trajectories, ASPIRE achieves 20–25× faster inference in real-time ultrasound video reconstruction with no loss in fidelity, enabling frame rates $>100$ Hz and up to 8% PSNR gains in high-motion settings (Stevens et al., 9 Sep 2024).

Scientific Model Updating and Sequential Data Analysis

• Sea-Surface Temperature (SST) Geostatistics: ASPIRE enables matching of full-data posterior means and credible intervals at 4× faster overall cost for Matérn models with large spatial data (Hooten et al., 2018).
• State-space Models (e.g., ecological count data): Reuse-based ASPIRE achieves $59\times$ faster posterior updates for time-series models of animal counts, while matching joint-inference posteriors.
• Gravitational-Wave Data Analysis: Posterior reanalysis for alternative waveform models or injected physical effects is achieved with $4$–$10\times$ fewer likelihood evaluations and robust recovery of model evidence (Williams, 6 Nov 2025).

6. Limitations, Recommendations, and Extensions

Limitations include:

• The necessity for sufficient overlap between existing and target posterior/model support to avoid degeneracy in importance, flow, or copula approaches.
• Diminishing returns or failure when new data contradict earlier posterior mass or when the desired posterior is highly multimodal and distinct (Thijssen et al., 2017).
• Iterative amortized methods require access to physics models for summary construction and the adjoint for gradient computation, which may not always be available (Orozco et al., 8 May 2024).

Best-practice recommendations:

• Partition data such that each block is large enough to yield stable posteriors (typically $J=2$–$5$).
• Use flow-based or copula-based functional approximations for densities in moderate-to-high dimension; Gaussian Process regression excels under $D \leq 4$ and moderate $N$ (Thijssen et al., 2017).
• Precompute and store blockwise likelihoods or embeddings in recursive implementations for maximal efficiency.
• Employ MCMC rejuvenation and diagnostic checks (e.g., Gelman–Rubin $\hat{R}$, ESS) to monitor convergence and mixing.

Potential extensions include the use of gradient-based SMC proposals, persistent particle populations across reanalyses, and adaptation to GPU-accelerated evaluation pipelines. Generalization to domains such as cosmology or epidemiological modeling requires only the ability to construct or approximate mappings from earlier posterior draws, and to evaluate new likelihood and prior terms (Williams, 6 Nov 2025).

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In summary, ASPIRE denotes a broad suite of technically rigorous approaches that accelerate Bayesian posterior inference by judicious reuse and transformation of previously computed densities, samples, or summary information. These frameworks address critical computational barriers in sequential analysis, model reanalysis, and inverse problems, achieving significant efficiency gains without sacrificing statistical calibration or flexibility.