Efficient non-conjugate Gaussian process factor models for spike count data using polynomial approximations
Princeton Neuroscience Institute（普林斯顿神经科学研究所）；Princeton University（普林斯顿大学）；University of California, Santa Barbara（加州大学圣巴巴拉分校）；Center for Visual Science（视觉科学中心）；University of Rochester（罗切斯特大学）
Stephen L. Keeley, David M. Zoltowski, Yiyi Yu, Jacob L. Yates, Spencer L. Smith, Jonathan W. Pillow
Gaussian Process Factor Analysis (GPFA) has been broadly applied to the problem of identifying smooth, low-dimensional temporal structure underlying large-scale neural recordings. However, spike trains are non-Gaussian, which motivates combining GPFA with discrete observation models for binned spike count data. The drawback to this approach is that GPFA priors are not conjugate to count model likelihoods, which makes inference challenging. Here we address this obstacle by introducing a fast, approximate inference method for non-conjugate GPFA models. Our approach uses orthogonal second-order polynomials to approximate the nonlinear terms in the non-conjugate log-likelihood, resulting in a method we refer to as \textit{polynomial approximate log-likelihood} (PAL) estimators. This approximation allows for accurate closed-form evaluation of marginal likelihoods and fast numerical optimization for parameters and hyperparameters. We derive PAL estimators for GPFA models with binomial, Poisson, and negative binomial observations and find the PAL estimation is highly accurate, and achieves faster convergence times compared to existing state-of-the-art inference methods. We also find that PAL hyperparameters can provide sensible initialization for black box variational inference (BBVI), which improves BBVI accuracy. We demonstrate that PAL estimators achieve fast and accurate extraction of latent structure from multi-neuron spike train data.
Unit-level models for survey data offer many advantages over their area-level counterparts, such as potential for more precise estimates and a natural benchmarking property. However two main challenges occur in this context: accounting for an informative survey design and handling non-Gaussian data types. The pseudo-likelihood approach is one solution to the former, and conjugate multivariate distribution theory offers a solution to the latter. By combining these approaches, we attain a unit-level model for count data that accounts for informative sampling designs and includes fully Bayesian m…
Department of Neurobiology（神经生物学系）；Department of Statistical Science（统计科学系）；Duke University（杜克大学）；Department of Electrical and Computer Engineering（电气与计算机工程系）
We characterize conjugate nonparametric Bayesian models as projective limits of conjugate, finite-dimensional Bayesian models. In particular, we identify a large class of nonparametric models representable as infinite-dimensional analogues of exponential family distributions and their canonical conjugate priors. This class contains most models studied in the literature, including Dirichlet processes and Gaussian process regression models. To derive these results, we introduce a representation of infinite-dimensional Bayesian models by projective limits of regular conditional probabilities. We …
We introduce a wavelet-domain functional analysis of variance (fANOVA) method based on a Bayesian hierarchical model. The factor effects are modeled through a spike-and-slab mixture at each location-scale combination along with a normal-inverse-Gamma (NIG) conjugate setup for the coefficients and errors. A graphical model called the Markov grove (MG) is designed to jointly model the spike-and-slab statuses at all location-scale combinations, which incorporates the clustering of each factor effect in the wavelet-domain thereby allowing borrowing of strength across location and scale. The poster…
In the density estimation model, the question of adaptive inference using Pólya tree-type prior distributions is considered. A class of prior densities having a tree structure, called spike-and-slab Pólya trees, is introduced. For this class, two types of results are obtained: first, the Bayesian posterior distribution is shown to converge at the minimax rate for the supremum norm in an adaptive way, for any Hölder regularity of the true density between $0$ and $1$, thereby providing adaptive counterparts to the results for classical Pólya trees in Castillo (2017). Second, the question of unce…
Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Data
Department of Mathematics ICSI and Department of Statistics（数学系ICSI和统计系）；University of California, San Diego, USA（美国加州大学圣地亚哥分校）；University of California, Berkeley, USA（美国加州大学伯克利分校）；Department of Mathematics ICSI, LBNL and Department of Statistics（数学系ICSI、劳伦斯伯克利国家实验室和统计系）
Gaussian random field (GRF) models are widely used in spatial statistics to capture spatially correlated error. We investigate the results of replacing Gaussian processes with Laplace moving averages (LMAs) in spatial generalized linear mixed models (SGLMMs). We demonstrate that LMAs offer improved predictive power when the data exhibits localized spikes in the response. SGLMMs with LMAs are shown to maintain analogous parameter inference and similar computing to Gaussian SGLMMs. We propose a novel discrete space LMA model for irregular lattices and construct conjugate samplers for LMAs with g…
Logistic regression models for binomial responses are routinely used in statistical practice. However, the maximum likelihood estimate may not exist due to data separability. We address this issue by considering a conjugate prior penalty which always produces finite estimates. Such a specification has a clear Bayesian interpretation and enjoys several invariance properties, making it an appealing prior choice. We show that the proposed method leads to an accurate approximation of the reduced-bias approach of Firth (1993), resulting in estimators with smaller asymptotic bias than the maximum-li…
We propose a voxel-wise general linear model with autoregressive noise and heteroscedastic noise innovations (GLMH) for analyzing functional magnetic resonance imaging (fMRI) data. The model is analyzed from a Bayesian perspective and has the benefit of automatically down-weighting time points close to motion spikes in a data-driven manner. We develop a highly efficient Markov Chain Monte Carlo (MCMC) algorithm that allows for Bayesian variable selection among the regressors to model both the mean (i.e., the design matrix) and variance. This makes it possible to include a broad range of explan…
Bayesian nonparametric hierarchical priors are highly effective in providing flexible models for latent data structures exhibiting sharing of information between and across groups. Most prominent is the Hierarchical Dirichlet Process (HDP), and its subsequent variants, which model latent clustering between and across groups. The HDP, may be viewed as a more flexible extension of Latent Dirichlet Allocation models (LDA), and has been applied to, for example, topic modelling, natural language processing, and datasets arising in health-care. We focus on analogous latent feature allocation models,…
Bayesian Variable Selection in Generalized Linear Models
University of Trento（特伦托大学）；Fondazione Bruno Kessler (FBK)（布鲁诺·孔塞雷基金会）；BCAM — Basque Center for Applied Mathematics（巴斯克应用数学中心）；Ikerbasque — Basque Foundation for Science（巴斯克科学基金会）
Accurate statistical models of neural spike responses can characterize the information carried by neural populations. But the limited samples of spike counts during recording usually result in model overfitting. Besides, current models assume spike counts to be Poisson-distributed, which ignores the fact that many neurons demonstrate over-dispersed spiking behaviour. Although the Negative Binomial Generalized Linear Model (NB-GLM) provides a powerful tool for modeling over-dispersed spike counts, the maximum likelihood-based standard NB-GLM leads to highly variable and inaccurate parameter est…
Neighbor Embedding for High-Dimensional Sparse Poisson Data
Department of Biomedical Engineering（生物医学工程系）；Center for Imaging Science（成像科学中心）；Kavli Neuroscience Discovery Institute（Kavli神经科学发现研究所）；Johns Hopkins University（约翰霍普金斯大学）
We study symmetric spiked matrix models with respect to a general class of noise distributions. Given a rank-1 deformation of a random noise matrix, whose entries are independently distributed with zero mean and unit variance, the goal is to estimate the rank-1 part. For the case of Gaussian noise, the top eigenvector of the given matrix is a widely-studied estimator known to achieve optimal statistical guarantees, e.g., in the sense of the celebrated BBP phase transition. However, this estimator can fail completely for heavy-tailed noise. In this work, we exhibit an estimator that works for h…
