We consider the case of performing Bayesian inference for stochastic epidemic compartment models, using incomplete time course data consisting of incidence counts that are either the number of new infections or removals in time intervals of fixed length. We eschew the most natural Markov jump process representation for reasons of computational efficiency, and focus on a stochastic differential equation representation. This is further approximated to give a tractable Gaussian process, that is, the linear noise approximation (LNA). Unless the observation model linking the LNA to data is both linear and Gaussian, the observed data likelihood remains intractable. It is in this setting that we consider two approaches for marginalising over the latent process: a correlated pseudo-marginal method and analytic marginalisation via a Gaussian approximation of the observation model. We compare and contrast these approaches using synthetic data before applying the best performing method to real data consisting of removal incidence of oak processionary moth nests in Richmond Park, London. Our approach further allows comparison between various competing compartment models.
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Crystalline phase structure is essential for understanding the performance and properties of a material. Therefore, this study identified and quantified the crystalline phase structure of a sample based on the diffraction pattern observed when the crystalline sample was irradiated with electromagnetic waves such as X-rays. Conventional analysis necessitates experienced and knowledgeable researchers to shorten the list from many candidate crystalline phase structures. However, the Conventional diffraction pattern analysis is highly analyst-dependent and not objective. Additionally, there is no established method for discussing the confidence intervals of the analysis results. Thus, this study aimed to establish a method for automatically inferring crystalline phase structures from diffraction patterns using Bayesian inference. Our method successfully identified true crystalline phase structures with a high probability from 50 candidate crystalline phase structures. Further, the mixing ratios of selected crystalline phase structures were estimated with a high degree of accuracy. This study provided reasonable results for well-crystallized samples that clearly identified the crystalline phase structures.
Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks
We consider the problem of estimating the marginal independence structure of a Bayesian network from observational data in the form of an undirected graph called the unconditional dependence graph. We show that unconditional dependence graphs of Bayesian networks correspond to the graphs having equal independence and intersection numbers. Using this observation, a Gr\"obner basis for a toric ideal associated to unconditional dependence graphs of Bayesian networks is given and then extended by additional binomial relations to connect the space of all such graphs. An MCMC method, called GrUES (Gr\"obner-based Unconditional Equivalence Search), is implemented based on the resulting moves and applied to synthetic Gaussian data. GrUES recovers the true marginal independence structure via a penalized maximum likelihood or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior, for which the $20\%$ HPD credible sets include the true structure at a high rate for data-generating graphs with density at least $0.5$.
This note presents a refined local approximation for the logarithm of the ratio between the negative multinomial probability mass function and a multivariate normal density, both having the same mean-covariance structure. This approximation, which is derived using Stirling's formula and a meticulous treatment of Taylor expansions, yields an upper bound on the Hellinger distance between the jittered negative multinomial distribution and the corresponding multivariate normal distribution. Upper bounds on the Le Cam distance between negative multinomial and multivariate normal experiments ensue.
We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral equation when one surface is close to another or to obtain values at grid points. We replace the singular kernel with a regularized version having a length parameter $\delta$ in order to control discretization error. Analysis near the singularity leads to an expression for the error due to regularization which has terms with unknown coefficients multiplying known quantities. By computing the integral with three choices of $\delta$ we can solve for an extrapolated value that has regularization error reduced to $O(\delta^5)$. In examples with $\delta/h$ constant and moderate resolution we observe total error about $O(h^5)$. For convergence as $h \to 0$ we can choose $\delta$ proportional to $h^q$ with $q < 1$ to ensure the discretization error is dominated by the regularization error. With $q = 4/5$ we find errors about $O(h^4)$. For harmonic potentials we extend the approach to a version with $O(\delta^7)$ regularization; it typically has smaller errors but the order of accuracy is less predictable.
Out of the participants in a randomized experiment with anticipated heterogeneous treatment effects, is it possible to identify which subjects have a positive treatment effect? While subgroup analysis has received attention, claims about individual participants are much more challenging. We frame the problem in terms of multiple hypothesis testing: each individual has a null hypothesis (stating that the potential outcomes are equal, for example) and we aim to identify those for whom the null is false (the treatment potential outcome stochastically dominates the control one, for example). We develop a novel algorithm that identifies such a subset, with nonasymptotic control of the false discovery rate (FDR). Our algorithm allows for interaction -- a human data scientist (or a computer program) may adaptively guide the algorithm in a data-dependent manner to gain power. We show how to extend the methods to observational settings and achieve a type of doubly-robust FDR control. We also propose several extensions: (a) relaxing the null to nonpositive effects, (b) moving from unpaired to paired samples, and (c) subgroup identification. We demonstrate via numerical experiments and theoretical analysis that the proposed method has valid FDR control in finite samples and reasonably high identification power.
Multi-level modeling is an important approach for analyzing complex survey data using multi-stage sampling. However, estimation of multi-level models can be challenging when we combine several datasets with distinct hierarchies with sampling weights. This paper presents a method for combining multiple datasets with different hierarchical structures due to distinct informative sampling designs for the same survey. To develop an approach with complete generality, we propose to define a pseudo-cluster, a cluster containing only a singleton observation, to unify the data structure and thereby enable estimation of multi-level models incorporating sampling weights across the combined sample. We justify incorporating sampling weights at each level of the hierarchical model and in doing-so define a pseudo-likelihood estimation procedure. Simulation studies are used to illustrate the effect of incorporating sampling designs in this challenging multi-level modeling scenario. We demonstrate in the simulation studies that considering a linear mixed model with sampling weights provides unbiased estimates of model parameters and enhances the estimation of the variance components of the random effects. The proposed method is illustrated through a novel application from the National Survey of Healthcare Organizations and Systems that sought to determine which organizational characteristics or traits, as measured in the surveys, have the strongest average relationship to the percentage of depression and anxiety diagnoses in physician practices in the US.
In ptychographic imaging, the trade-off between the number of acquisitions and the resultant imaging quality presents a complex optimization problem. Increasing the number of acquisitions typically yields reconstructions with higher spatial resolution and finer details. Conversely, a reduction in measurement frequency often compromises the quality of the reconstructed images, manifesting as increased noise and coarser details. To address this challenge, we employ sparsity priors to reformulate the ptychographic reconstruction task as a total variation regularized optimization problem. We introduce a new computational framework, termed the ptychographic proximal total-variation (PPTV) solver, designed to integrate into existing ptychography settings without necessitating hardware modifications. Through comprehensive numerical simulations, we validate that PPTV-driven coded ptychography is capable of producing highly accurate reconstructions with a minimal set of eight intensity measurements. Convergence analysis further substantiates the robustness, stability, and computational feasibility of the proposed PPTV algorithm. Experimental results obtained from optical setups unequivocally demonstrate that the PPTV algorithm facilitates high-throughput, high-resolution imaging while significantly reducing the measurement burden. These findings indicate that the PPTV algorithm has the potential to substantially mitigate the resource-intensive requirements traditionally associated with high-quality ptychographic imaging, thereby offering a pathway toward the development of more compact and efficient ptychographic microscopy systems.
We investigate the combinatorics of max-pooling layers, which are functions that downsample input arrays by taking the maximum over shifted windows of input coordinates, and which are commonly used in convolutional neural networks. We obtain results on the number of linearity regions of these functions by equivalently counting the number of vertices of certain Minkowski sums of simplices. We characterize the faces of such polytopes and obtain generating functions and closed formulas for the number of vertices and facets in a 1D max-pooling layer depending on the size of the pooling windows and stride, and for the number of vertices in a special case of 2D max-pooling.
Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. These systems generically involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the magnitude of the steady-state probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry and the strength of nonequilibrium transport of measure. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework that estimates the score of this density. We show that the score, together with the microscopic equations of motion, gives direct access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles, spatial regions, and degrees of freedom. To represent the score, we introduce a novel, spatially-local transformer-based network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of interacting active particles undergoing motility-induced phase separation (MIPS). We show that a single instance of our network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.
We propose an approach to compute inner and outer-approximations of the sets of values satisfying constraints expressed as arbitrarily quantified formulas. Such formulas arise for instance when specifying important problems in control such as robustness, motion planning or controllers comparison. We propose an interval-based method which allows for tractable but tight approximations. We demonstrate its applicability through a series of examples and benchmarks using a prototype implementation.
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