Bayesian Inverse Planning Model

• Bayesian Inverse Planning Model is a probabilistic framework that infers latent causes and parameters from noisy observations using Bayes’ theorem.
• It employs adaptive surrogates that iteratively refine forward model approximations to focus on high-information regions, drastically reducing computational costs.
• Its applications in scientific computing, robotics, and engineering design enable more reliable uncertainty quantification and efficient parameter recovery compared to traditional methods.

A Bayesian Inverse Planning Model is a probabilistic framework for inferring latent causes, goals, preferences, or parameters from observed outcomes—often formulated as inferring the most plausible underlying “plan” or “intent” given partial or noisy observations. In the context of computational inverse problems governed by complex physical systems, these models enable efficient and accurate recovery of unknown parameters, leveraging Bayes’ rule to unify prior knowledge with observational evidence. The approach has profound implications for fields such as scientific computing, engineering design, robotics, and uncertainty quantification, especially where forward model evaluations are computationally intensive (1309.5524).

1. The Bayesian Inverse Problem Formulation

In a canonical Bayesian inverse problem, a forward map $G$ maps the parameter vector $y \in \mathbb{R}^{n_y}$ to a predicted data vector $d \in \mathbb{R}^{n_d}$. The prior density $\pi(y)$ encodes assumptions or information about the parameters before observing data, while the likelihood function $\pi(d|y)$ describes the plausibility of observing $d$ for a given $y$—commonly incorporating additive noise models to account for measurement or model error. Bayes’ theorem then yields the posterior: $$\pi(y|d) = \frac{\pi(d|y) \pi(y)}{I}, \quad I = \int \pi(d|y)\pi(y) \, dy.$$ For problems where $G$ is a complex, nonlinear mapping (e.g., a PDE model), direct application of this rule involves repeated, computationally expensive forward solves, motivating the development of surrogate models and adaptive sampling strategies.

2. Surrogate Modeling and Adaptive Construction

Traditional surrogate modeling for Bayesian inverse planning relies on constructing approximations to the forward model $G$. A typical approach is the prior-based polynomial chaos expansion (PC), where $G$ is represented as

$$\tilde{G}_N(y) = \sum_{|\mathbf{i}|\leq N} a_{\mathbf{i}} \Psi_{\mathbf{i}}(y),$$

with orthogonal basis functions $\Psi_{\mathbf{i}}$ defined with respect to the prior $\pi(y)$. While this yields a global L$^2$ approximation under the prior, it is often unnecessarily expensive: in practice, the posterior distribution is typically concentrated in a small, data-dependent region of parameter space.

To address this, the adaptive surrogate construction iteratively sharpens accuracy where the posterior is concentrated. This process involves:

• Defining a parameterized family of biasing distributions $p(y; v)$ (e.g., Gaussians).
• Iteratively optimizing the bias toward the posterior by minimizing the Kullback–Leibler divergence $\mathrm{KL}(\pi^* \| p)$, equivalently maximizing

$D(v) = \int L(G(y)) \ln p(y; v)\, \pi(y)\, dy,$

using stochastic optimization.

• At each iteration $k$:
  • Construct a local surrogate $\tilde{G}_k(y)$ (e.g., sparse PC expansion) focused on $p(y; v_k)$.
  • Use importance sampling to estimate objective functions and update $v_{k+1}$.
  • Employ a tempering parameter $\lambda$ in the likelihood (raising $L$ to $1/\lambda$ power), decreasing $\lambda$ to gradually focus surrogacy from prior-like regions to the posterior.
• Repeat until $\lambda = 1$, so the surrogate targets the region where the posterior mass is located (1309.5524).

3. Efficiency and Posterior-Focused Accuracy

The adaptive surrogate approach achieves dramatic reductions in the number of expensive forward model calls while enhancing accuracy in the posterior region. For example:

• In source inversion for contaminant transport, a prior-weighted surrogate required 417 forward evaluations for a specified polynomial order, whereas the adaptive method achieved equivalent or better posterior approximation using only 36 targeted evaluations.
• In an inverse heat conduction problem, the Kullback–Leibler divergence between the true and approximate posteriors was reduced from over $124$ (prior-based) to as low as $0.0032$ (adaptive surrogate) for relevant parameter coefficients.
• The final biasing distribution from the adaptive process can be used as the proposal in MCMC to greatly improve mixing rates, as shown by lower autocorrelation in posterior samples relative to classic samplers.

These gains are rooted in localizing model accuracy to regions where the posterior is non-negligible, underlining the importance of posterior-focused surrogate construction in high-dimensional and nonlinear settings (1309.5524).

4. Algorithmic Workflow and Applications

The adaptive surrogate-based Bayesian inverse planning model operates through a well-defined workflow:

1. Start with a prior over parameters and initial biasing distribution.
2. Iteratively:
   • Construct a local polynomial chaos expansion (surrogate) for the forward map using samples from the current biasing distribution.
   • Evaluate the K-L divergence objective via importance sampling.
   • Update the biasing parameters, possibly analytically for certain distribution families, and update the surrogate accordingly.
   • Adjust the likelihood tempering to increasingly “zoom” onto the posterior.
3. Terminate when tempering is complete and the surrogate is focused on the posterior.
4. Use the final biasing distribution as an efficient proposal for independence-type MCMC.

Applications demonstrated with this approach include:

• Source inversion in contaminant transport: Inferring source location from sensor measurements, with the forward model a diffusion PDE.
• Inverse nonlinear heat conduction: Recovering unknown boundary heat flux represented via spectral (Fourier) coefficients. In both cases, posterior mean, variance, skewness, and autocovariance of recovered parameters are accurately reconstructed with greatly reduced computational costs (1309.5524).

5. Limitations and Methodological Considerations

The adaptive Bayesian inverse planning model exhibits certain limitations and requirements:

• Performance improvements depend on the posterior being markedly more concentrated than the prior; for diffuse posteriors, conventional global surrogates may be required.
• Choice of biasing distribution family impacts success—simple Gaussians may not suffice for highly multimodal posteriors, though extensions to mixtures are possible.
• Surrogate accuracy is contingent on the forward model’s smoothness and the chosen polynomial basis; strong local nonlinearities or discontinuities may still present challenges.
• The approach does not require derivative information from the forward model, but in cases where such information is accessible, derivative-informed optimization could accelerate convergence. These nuances underscore the need to match surrogate construction and adaptive stochastic optimization to the properties of the forward model and the targeted inverse problem (1309.5524).

6. Broader Implications for Computational Inverse Planning

The adaptive surrogate construction represents a significant advancement for scalable uncertainty quantification in computationally demanding Bayesian inverse planning problems. Its contributions include:

• Reducing the computational burden in high-dimensional, non-linear inverse problems by focusing approximation resources where data is most informative.
• Providing a principled stochastic optimization framework to iteratively tune surrogate accuracy to the posterior.
• Demonstrating orders-of-magnitude improvements in MCMC efficiency and posterior accuracy, with direct applications to scenarios where “forward model calls” are the dominant computational cost.
• Facilitating more reliable and informative uncertainty estimates in applications such as sensor placement, parameter recovery in physical systems, and engineering design.

Overall, this methodological innovation lays the groundwork for principled, efficient Bayesian inverse planning across a variety of data-limited, complex systems, rendering previously intractable inverse problems accessible to practical solution and enabling advanced uncertainty quantification in real-world computational science (1309.5524).