Inferring the parameters of models describing biological systems is an important problem in the reverse engineering of the mechanisms underlying these systems. Much work has focussed on parameter inference of stochastic and ordinary differential equation models using Approximate Bayesian Computation (ABC). While there is some recent work on inference in spatial models, this remains an open problem. Simultaneously, advances in topological data analysis (TDA), a field of computational mathematics, has enabled spatial patterns in data to be characterised. Here, we focus on recent work using topological data analysis to study different regimes of parameter space of a well-studied model of angiogenesis. We propose a method for combining TDA with ABC for inferring parameters in the Anderson-Chaplain model of angiogenesis. We demonstrate that this topological approach outperforms simpler statistics based on spatial features of the data. This is a first step towards a larger framework of spatial parameter inference for biological systems, for which there may be a variety of filtrations, vectorisations, and summary statistics to be considered.
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Network lasso is a method for solving a multi-task learning problem through the regularized maximum likelihood method. A characteristic of network lasso is setting a different model for each sample. The relationships among the models are represented by relational coefficients. A crucial issue in network lasso is to provide appropriate values for these relational coefficients. In this paper, we propose a Bayesian approach to solve multi-task learning problems by network lasso. This approach allows us to objectively determine the relational coefficients by Bayesian estimation. The effectiveness of the proposed method is shown in a simulation study and a real data analysis.
We develop a methodology for proving central limit theorems in network models with strategic interactions and homophilous agents. Since data often consists of observations on a single large network, we consider an asymptotic framework in which the network size tends to infinity. In the presence of strategic interactions, network moments are generally complex functions of components, where a node's component consists of all alters to which it is directly or indirectly connected. We find that a modification of "exponential stabilization" conditions from the stochastic geometry literature provides a useful formulation of weak dependence for moments of this type. We establish a CLT for a network moments satisfying stabilization and provide a methodology for deriving primitive sufficient conditions for stabilization using results in branching process theory. We apply the methodology to static and dynamic models of network formation.
The ability for a robot to navigate with only the use of vision is appealing due to its simplicity. Traditional vision-based navigation approaches required a prior map-building step that was arduous and prone to failure, or could only exactly follow previously executed trajectories. Newer learning-based visual navigation techniques reduce the reliance on a map and instead directly learn policies from image inputs for navigation. There are currently two prevalent paradigms: end-to-end approaches forego the explicit map representation entirely, and topological approaches which still preserve some loose connectivity of the space. However, while end-to-end methods tend to struggle in long-distance navigation tasks, topological map-based solutions are prone to failure due to spurious edges in the graph. In this work, we propose a learning-based topological visual navigation method with graph update strategies that improve lifelong navigation performance over time. We take inspiration from sampling-based planning algorithms to build image-based topological graphs, resulting in sparser graphs yet with higher navigation performance compared to baseline methods. Also, unlike controllers that learn from fixed training environments, we show that our model can be finetuned using a relatively small dataset from the real-world environment where the robot is deployed. We further assess performance of our system in real-world deployments.
In recent years, the transformer has established itself as a workhorse in many applications ranging from natural language processing to reinforcement learning. Similarly, Bayesian deep learning has become the gold-standard for uncertainty estimation in safety-critical applications, where robustness and calibration are crucial. Surprisingly, no successful attempts to improve transformer models in terms of predictive uncertainty using Bayesian inference exist. In this work, we study this curiously underpopulated area of Bayesian transformers. We find that weight-space inference in transformers does not work well, regardless of the approximate posterior. We also find that the prior is at least partially at fault, but that it is very hard to find well-specified weight priors for these models. We hypothesize that these problems stem from the complexity of obtaining a meaningful mapping from weight-space to function-space distributions in the transformer. Therefore, moving closer to function-space, we propose a novel method based on the implicit reparameterization of the Dirichlet distribution to apply variational inference directly to the attention weights. We find that this proposed method performs competitively with our baselines.
We present symbolic and numerical methods for computing Poisson brackets on the spaces of measures with positive densities of the plane, the 2-torus, and the 2-sphere. We apply our methods to compute symplectic areas of finite regions for the case of the 2-sphere, including an explicit example for Gaussian measures with positive densities.
This paper considers Bayesian parameter estimation of dynamic systems using a Markov Chain Monte Carlo (MCMC) approach. The Metroplis-Hastings (MH) algorithm is employed, and the main contribution of the paper is to examine and illustrate the efficacy of a particular proposal density based on energy preserving Hamiltonian dynamics, which results in what is known in the statistics literature as ``Hamiltonian Monte--Carlo'' (HMC). The very significant utility of this approach is that, as will be illustrated, it greatly reduces (almost to the point of elimination) the typically very high correlation in the Metropolis--Hastings chain which has been observed by several authors to restrict the application of the MH approach to only very low dimension model structures. The paper illustrates how the HMC approach may be applied to both significant dimension linear and nonlinear model structures, even when the system order is unknown, and using both simulated and real data.
Implicit Processes (IPs) are flexible priors that can describe models such as Bayesian neural networks, neural samplers and data generators. IPs allow for approximate inference in function-space. This avoids some degenerate problems of parameter-space approximate inference due to the high number of parameters and strong dependencies. For this, an extra IP is often used to approximate the posterior of the prior IP. However, simultaneously adjusting the parameters of the prior IP and the approximate posterior IP is a challenging task. Existing methods that can tune the prior IP result in a Gaussian predictive distribution, which fails to capture important data patterns. By contrast, methods producing flexible predictive distributions by using another IP to approximate the posterior process cannot fit the prior IP to the observed data. We propose here a method that can carry out both tasks. For this, we rely on an inducing-point representation of the prior IP, as often done in the context of sparse Gaussian processes. The result is a scalable method for approximate inference with IPs that can tune the prior IP parameters to the data, and that provides accurate non-Gaussian predictive distributions.
The nature of the Fermi gamma-ray Galactic Center Excess (GCE) has remained a persistent mystery for over a decade. Although the excess is broadly compatible with emission expected due to dark matter annihilation, an explanation in terms of a population of unresolved astrophysical point sources e.g., millisecond pulsars, remains viable. The effort to uncover the origin of the GCE is hampered in particular by an incomplete understanding of diffuse emission of Galactic origin. This can lead to spurious features that make it difficult to robustly differentiate smooth emission, as expected for a dark matter origin, from more "clumpy" emission expected for a population of relatively bright, unresolved point sources. We use recent advancements in the field of simulation-based inference, in particular density estimation techniques using normalizing flows, in order to characterize the contribution of modeled components, including unresolved point source populations, to the GCE. Compared to traditional techniques based on the statistical distribution of photon counts, our machine learning-based method is able to utilize more of the information contained in a given model of the Galactic Center emission, and in particular can perform posterior parameter estimation while accounting for pixel-to-pixel spatial correlations in the gamma-ray map. This makes the method demonstrably more resilient to certain forms of model misspecification. On application to Fermi data, the method generically attributes a smaller fraction of the GCE flux to unresolved point sources when compared to traditional approaches. We nevertheless infer such a contribution to make up a non-negligible fraction of the GCE across all analysis variations considered, with at least $38^{+9}_{-19}\%$ of the excess attributed to unresolved points sources in our baseline analysis.
Active inference, a corollary of the free energy principle, is a formal way of describing the behavior of certain kinds of random dynamical systems that have the appearance of sentience. In this chapter, we describe how active inference combines Bayesian decision theory and optimal Bayesian design principles under a single imperative to minimize expected free energy. It is this aspect of active inference that allows for the natural emergence of information-seeking behavior. When removing prior outcomes preferences from expected free energy, active inference reduces to optimal Bayesian design, i.e., information gain maximization. Conversely, active inference reduces to Bayesian decision theory in the absence of ambiguity and relative risk, i.e., expected utility maximization. Using these limiting cases, we illustrate how behaviors differ when agents select actions that optimize expected utility, expected information gain, and expected free energy. Our T-maze simulations show optimizing expected free energy produces goal-directed information-seeking behavior while optimizing expected utility induces purely exploitive behavior and maximizing information gain engenders intrinsically motivated behavior.
This research addresses a new tool for data analysis known as Topological Data Analysis TDA It underlies an area of Mathematics known as Combinatorial Algebra or more recently Algebraic Topology which through making strong use of Computation Statistics Probability and Topology among other concepts extracts mathematical characteristics from a set of data that allow us associate create and infer general and quality information about them
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