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1. Connecting Hamilton-Jacobi Partial Differential Equations with Maximum a Posteriori and Posterior Mean Estimators for Some Non-convex Priors.

   https://doi.org/10.1007/978-3-030-03009-4_56-1  

   Darbon, J.; Langlois, G.P.; Meng, T. (May 2021, Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging. Springer,)

   null; null; null; null (Ed.)

   Many imaging problems can be formulated as inverse problems expressed as finite-dimensional optimization problems. These optimization problems generally consist of minimizing the sum of a data fidelity and regularization terms. In Darbon (SIAM J. Imag. Sci. 8:2268–2293, 2015), Darbon and Meng, (On decomposition models in imaging sciences and multi-time Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:1906.09502, 2019), connections between these optimization problems and (multi-time) Hamilton-Jacobi partial differential equations have been proposed under the convexity assumptions of both the data fidelity and regularization terms. In particular, under these convexity assumptions, some representation formulas for a minimizer can be obtained. From a Bayesian perspective, such a minimizer can be seen as a maximum a posteriori estimator. In this chapter, we consider a certain class of non-convex regularizations and show that similar representation formulas for the minimizer can also be obtained. This is achieved by leveraging min-plus algebra techniques that have been originally developed for solving certain Hamilton-Jacobi partial differential equations arising in optimal control. Note that connections between viscous Hamilton-Jacobi partial differential equations and Bayesian posterior mean estimators with Gaussian data fidelity terms and log-concave priors have been highlighted in Darbon and Langlois, (On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi partial differential equations, arXiv preprint arXiv:2003.05572, 2020). We also present similar results for certain Bayesian posterior mean estimators with Gaussian data fidelity and certain non-log-concave priors using an analogue of min-plus algebra techniques. 

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2. Graph-based prior and forward models for inverse problems on manifolds with boundaries

   https://doi.org/10.1088/1361-6420/ac3994  

   Harlim, John; Jiang, Shixiao W; Kim, Hwanwoo; Sanz-Alonso, Daniel (January 2022, Inverse Problems)

   Abstract This paper develops manifold learning techniques for the numerical solution of PDE-constrained Bayesian inverse problems on manifolds with boundaries. We introduce graphical Matérn-type Gaussian field priors that enable flexible modeling near the boundaries, representing boundary values by superposition of harmonic functions with appropriate Dirichlet boundary conditions. We also investigate the graph-based approximation of forward models from PDE parameters to observed quantities. In the construction of graph-based prior and forward models, we leverage the ghost point diffusion map algorithm to approximate second-order elliptic operators with classical boundary conditions. Numerical results validate our graph-based approach and demonstrate the need to design prior covariance models that account for boundary conditions. 

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3. HYPERDIFFERENTIAL SENSITIVITY ANALYSIS IN THE CONTEXT OF BAYESIAN INFERENCE APPLIED TO ICE-SHEET PROBLEMS

   https://doi.org/10.1615/Int.J.UncertaintyQuantification.2023047605  

   Reese, William; Hart, Joseph; Waanders, Bart_van Bloemen; Perego, Mauro; Jakeman, John D; Saibaba, Arvind K (January 2024, International Journal for Uncertainty Quantification)

   Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyperdifferential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice-sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate). 

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4. An Adaptive Empirical Bayesian Method for Sparse Deep Learning

   Wei Deng, Xiao Zhang (December 2019, Advances in neural information processing systems)

   We propose a novel adaptive empirical Bayesian (AEB) method for sparse deep learning, where the sparsity is ensured via a class of self-adaptive spike-and-slab priors. The proposed method works by alternatively sampling from an adaptive hierarchical posterior distribution using stochastic gradient Markov Chain Monte Carlo (MCMC) and smoothly optimizing the hyperparameters using stochastic approximation (SA). We further prove the convergence of the proposed method to the asymptotically correct distribution under mild conditions. Empirical applications of the proposed method lead to the state-of-the-art performance on MNIST and Fashion MNIST with shallow convolutional neural networks (CNN) and the state-of-the-art compression performance on CIFAR10 with Residual Networks. The proposed method also improves resistance to adversarial attacks. 

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5. Practical guidelines for Bayesian phylogenetic inference using Markov Chain Monte Carlo (MCMC)

   Barido-Sottani, J; Schwery, O; Warnock, RCM; Zhang, C; Wright, AM (June 2024, Open research Europe)

   Phylogenetic estimation is, and has always been, a complex endeavor. Estimating a phylogenetic tree involves evaluating many possible solutions and possible evolutionary histories that could explain a set of observed data, typically by using a model of evolution. Modern statistical methods involve not just the estimation of a tree, but also solutions to more complex models involving fossil record information and other data sources. Markov Chain Monte Carlo (MCMC) is a leading method for approximating the posterior distribution of parameters in a mathematical model. It is deployed in all Bayesian phylogenetic tree estimation software. While many researchers use MCMC in phylogenetic analyses, interpreting results and diagnosing problems with MCMC remain vexing issues to many biologists. In this manuscript, we will offer an overview of how MCMC is used in Bayesian phylogenetic inference, with a particular emphasis on complex hierarchical models, such as the fossilized birth-death (FBD) model. We will discuss strategies to diagnose common MCMC problems and troubleshoot difficult analyses, in particular convergence issues. We will show how the study design, the choice of models and priors, but also technical features of the inference tools themselves can all be adjusted to obtain the best results. Finally, we will also discuss the unique challenges created by the incorporation of fossil information in phylogenetic inference, and present tips to address them. 

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