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Neural encoding, or neural representation, is a field in neuroscience that focuses on characterizing how information is encoded in the spiking activity of neurons. Currently, little is known about how sensory neurons can preserve information from multiple stimuli given their broad receptive fields. Multiplexing is a neural encoding theory that posits that neurons temporally switch between encoding various stimuli in their receptive field. Here, we construct a statistically falsifiable single-neuron model for multiplexing using a competition-based framework. The spike train models are constructed using drift-diffusion models, implying an integrate-and-fire framework to model the temporal dynamics of the membrane potential of the neuron. In addition to a multiplexing-specific model, we develop alternative models that represent alternative encoding theories (normalization, winner-take-all, subadditivity, etc.) with some level of abstraction. Using information criteria, we perform model comparison to determine whether the data favor multiplexing over alternative theories of neural encoding. Analysis of spike trains from the inferior colliculus of two macaque monkeys provides tenable evidence of multiplexing and offers new insight into the timescales at which switching occurs.

A novel statistical method is proposed and investigated for estimating a heavy tailed density under mild smoothness assumptions. Statistical analyses of heavy-tailed distributions are susceptible to the problem of sparse information in the tail of the distribution getting washed away by unrelated features of a hefty bulk. The proposed Bayesian method avoids this problem by incorporating smoothness and tail regularization through a carefully specified semiparametric prior distribution, and is able to consistently estimate both the density function and its tail index at near minimax optimal rates of contraction. A joint, likelihood driven estimation of the bulk and the tail is shown to help improve uncertainty assessment in estimating the tail index parameter and offer more accurate and reliable estimates of the high tail quantiles compared to thresholding methods., Comment: Combined article with all technical details uploaded here to complement JASA publication

We recently reported the existence of fluctuations in neural signals that may permit neurons to code multiple simultaneous stimuli sequentially across time. This required deploying a novel statistical approach to permit investigation of neural activity at the scale of individual trials. Here we present tests using synthetic data to assess the sensitivity and specificity of this analysis. We fabricated datasets to match each of several potential response patterns derived from single-stimulus response distributions. In particular, we simulated dual stimulus trial spike counts that reflected fluctuating mixtures of the single stimulus spike counts, stable intermediate averages, single stimulus winner-take-all, or response distributions that were outside the range defined by the single stimulus responses (such as summation or suppression). We then assessed how well the analysis recovered the correct response pattern as a function of the number of simulated trials and the difference between the simulated responses to each "stimulus" alone. We found excellent recovery of the mixture, intermediate, and outside categories (>97% percent correct), and good recovery of the single/winner-take-all category (>90% correct) when the number of trials was >20 and the single-stimulus response rates were 50Hz and 20Hz respectively. Both larger numbers of trials and greater separation between the single stimulus firing rates improved categorization accuracy. These results provide a benchmark, and guidelines for data collection, for use of this method to investigate coding of multiple items at the individual-trial time scale., Comment: Accepted for publication in Neurons, Behavior, Data analysis, and Theory

Bayesian nonparametric regression under a rescaled Gaussian process prior offers smoothness-adaptive function estimation with near minimax-optimal error rates. Hierarchical extensions of this approach, equipped with stochastic variable selection, are known to also adapt to the unknown intrinsic dimension of a sparse true regression function. But it remains unclear if such extensions offer variable selection consistency, i.e., if the true subset of important variables could be consistently learned from the data. It is shown here that variable consistency may indeed be achieved with such models at least when the true regression function has finite smoothness to induce a polynomially larger penalty on inclusion of false positive predictors. Our result covers the high dimensional asymptotic setting where the predictor dimension is allowed to grow with the sample size. The proof utilizes Schwartz theory to establish that the posterior probability of wrong selection vanishes asymptotically. A necessary and challenging technical development involves providing sharp upper and lower bounds to small ball probabilities at all rescaling levels of the Gaussian process prior, a result that could be of independent interest.

Conventional analysis of neuroscience data involves computing average neural activity over a group of trials and/or a period of time. This approach may be particularly problematic when assessing the response patterns of neurons to more than one simultaneously presented stimulus. In such cases, the brain must represent each individual component of the stimuli bundle, but trial-and-time-pooled averaging methods are fundamentally unequipped to address the means by which multi-item representation occurs. We introduce and investigate a novel statistical analysis framework that relates the firing pattern of a single cell, exposed to a stimuli bundle, to the ensemble of its firing patterns under each constituent stimulus. Existing statistical tools focus on what may be called "first order stochasticity" in trial-to-trial variation in the form of unstructured noise around a fixed firing rate curve associated with a given stimulus. Our analysis is based upon the theoretical premise that exposure to a stimuli bundle induces additional stochasticity in the cell's response pattern, in the form of a stochastically varying recombination of its single stimulus firing rate curves. We discuss challenges to statistical estimation of such "second order stochasticity" and address them with a novel dynamic admixture Poisson process (DAPP) model. DAPP is a hierarchical point process model that decomposes second order stochasticity into a Gaussian stochastic process and a random vector of interpretable features, and, facilitates borrowing of information on the latter across repeated trials through latent clustering. We present empirical evidence of the utility of the DAPP analysis with synthetic and real neural recordings., Comment: 26 pages, 7 figures

Linear quantile regression is a powerful tool to investigate how predictors may affect a response heterogeneously across different quantile levels. Unfortunately, existing approaches find it extremely difficult to adjust for any dependency between observation units, largely because such methods are not based upon a fully generative model of the data. For analyzing spatially indexed data, we address this difficulty by generalizing the joint quantile regression model of Yang and Tokdar (2017) and characterizing spatial dependence via a Gaussian or $t$ copula process on the underlying quantile levels of the observation units. A Bayesian semiparametric approach is introduced to perform inference of model parameters and carry out spatial quantile smoothing. An effective model comparison criteria is provided, particularly for selecting between different model specifications of tail heaviness and tail dependence. Extensive simulation studies and an application to particulate matter concentration in northeast US are presented to illustrate substantial gains in inference quality, accuracy and uncertainty quantification over existing alternatives., Comment: 30 pages, 10 figures

In genomic applications, there is often interest in identifying genes whose time-course expression trajectories exhibit periodic oscillations with a period of approximately 24 hours. Such genes are usually referred to as circadian, and their identification is a crucial step toward discovering physiological processes that are clock-controlled. It is natural to expect that the expression of gene i at time j might depend to some degree on the expression of the other genes measured at the same time. However, widely-used rhythmicity detection techniques do not accommodate for the potential dependence across genes. We develop a Bayesian approach for periodicity identification that explicitly takes into account the complex dependence structure across time-course trajectories in gene expressions. We employ a latent factor representation to accommodate dependence, while representing the true trajectories in the Fourier domain allows for inference on period, phase, and amplitude of the signal. Identification of circadian genes is allowed through a carefully chosen variable selection prior on the Fourier basis coefficients. The methodology is applied to a novel mouse liver circadian dataset. Although motivated by time-course gene expression array data, the proposed approach is applicable to the analysis of dependent functional data at broad.

Variable selection is a key issue when analyzing high-dimensional data. The explosion of data with large sample sizes and dimensionality brings new challenges to this problem in both inference accuracy and computational complexity. To alleviate these problems, we propose a new scalable Markov chain Monte Carlo (MCMC) sampling algorithm for "large $p$ small $n$" scenarios by generalizing multiple-try Metropolis to discrete model spaces and further incorporating neighborhood-based stochastic search. The proof of reversibility of the proposed MCMC algorithm is provided. Extensive simulation studies are performed to examine the efficiency of the new algorithm compared with existing methods. A real data example is provided to illustrate the prediction performances of the new algorithm., Comment: 28 pages; 5 figures; 5 tables

In dynamic topic modeling, the proportional contribution of a topic to a document depends on the temporal dynamics of that topic's overall prevalence in the corpus. We extend the Dynamic Topic Model of Blei and Lafferty (2006) by explicitly modeling document level topic proportions with covariates and dynamic structure that includes polynomial trends and periodicity. A Markov Chain Monte Carlo (MCMC) algorithm that utilizes Polya-Gamma data augmentation is developed for posterior inference. Conditional independencies in the model and sampling are made explicit, and our MCMC algorithm is parallelized where possible to allow for inference in large corpora. To address computational bottlenecks associated with Polya-Gamma sampling, we appeal to the Central Limit Theorem to develop a Gaussian approximation to the Polya-Gamma random variable. This approximation is fast and reliable for parameter values relevant in the text mining domain. Our model and inference algorithm are validated with multiple simulation examples, and we consider the application of modeling trends in PubMed abstracts. We demonstrate that sharing information across documents is critical for accurately estimating document-specific topic proportions. We also show that explicitly modeling polynomial and periodic behavior improves our ability to predict topic prevalence at future time points.

Additive-interactive regression has recently been shown to offer attractive minimax error rates over traditional nonparametric multivariate regression in a wide variety of settings, including cases where the predictor count is much larger than the sample size and many of the predictors have important effects on the response, potentially through complex interactions. We present a Bayesian implementation of additive-interactive regression using an additive Gaussian process (AGP) prior and develop an efficient Markov chain sampler that extends stochastic search variable selection in this setting. Careful prior and hyper-parameter specification are developed in light of performance and computational considerations, and key innovations address difficulties in exploring a joint posterior distribution over multiple subsets of high dimensional predictor inclusion vectors. The method offers state-of-the-art support and interaction recovery while improving dramatically over competitors in terms of prediction accuracy on a diverse set of simulated and real data. Results from real data studies provide strong evidence that the additive-interactive framework is an attractive modeling platform for high-dimensional nonparametric regression., Comment: 28 pages; 9 figures; 5 tables

Minimax $L_2$ risks for high-dimensional nonparametric regression are derived under two sparsity assumptions: (1) the true regression surface is a sparse function that depends only on $d=O(\log n)$ important predictors among a list of $p$ predictors, with $\log p=o(n)$; (2) the true regression surface depends on $O(n)$ predictors but is an additive function where each additive component is sparse but may contain two or more interacting predictors and may have a smoothness level different from other components. For either modeling assumption, a practicable extension of the widely used Bayesian Gaussian process regression method is shown to adaptively attain the optimal minimax rate (up to $\log n$ terms) asymptotically as both $n,p\to\infty$ with $\log p=o(n)$., Comment: Published at http://dx.doi.org/10.1214/14-AOS1289 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Gaussian process (GP) models are widely used to emulate propagation uncertainty in computer experiments. GP emulation sits comfortably within an analytically tractable Bayesian framework. Apart from propagating uncertainty of the input variables, a GP emulator trained on finitely many runs of the experiment also offers error bars for response surface estimates at unseen input values. This helps select future input values where the experiment should be run to minimize the uncertainty in the response surface estimation. However, traditional GP emulators use stationary covariance functions, which perform poorly and lead to sub-optimal selection of future input points when the response surface has sharp local features, such as a jump discontinuity or an isolated tall peak. We propose an easily implemented non-stationary GP emulator, based on two stationary GPs, one nested into the other, and demonstrate its superior ability in handling local features and selecting future input points from the boundaries of such features.

It is now known that an extended Gaussian process model equipped with rescaling can adapt to different smoothness levels of a function valued parameter in many nonparametric Bayesian analyses, offering a posterior convergence rate that is optimal (up to logarithmic factors) for the smoothness class the true function belongs to. This optimal rate also depends on the dimension of the function's domain and one could potentially obtain a faster rate of convergence by casting the analysis in a lower dimensional subspace that does not amount to any loss of information about the true function. In general such a subspace is not known a priori but can be explored by equipping the model with variable selection or linear projection. We demonstrate that for nonparametric regression, classification, density estimation and density regression, a rescaled Gaussian process model equipped with variable selection or linear projection offers a posterior convergence rate that is optimal (up to logarithmic factors) for the lowest dimension in which the analysis could be cast without any loss of information about the true function. Theoretical exploration of such dimension reduction features appears novel for Bayesian nonparametric models with or without Gaussian processes., Comment: 14 pages

It is shown that a simple Dirichlet process mixture of multivariate normals offers Bayesian density estimation with adaptive posterior convergence rates. Toward this, a novel sieve for non-parametric mixture densities is explored, and its rate adaptability to various smoothness classes of densities in arbitrary dimension is demonstrated. This sieve construction is expected to offer a substantial technical advancement in studying Bayesian non-parametric mixture models based on stick-breaking priors., Comment: 12 pages

We show that rate-adaptive multivariate density estimation can be performed using Bayesian methods based on Dirichlet mixtures of normal kernels with a prior distribution on the kernel's covariance matrix parameter. We derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is optimal minimax for the smoothness class to which the true density belongs. No prior knowledge of smoothness is assumed. The sufficient conditions are shown to hold for the Dirichlet location mixture of normals prior with a Gaussian base measure and an inverse-Wishart prior on the covariance matrix parameter. Locally H\"older smoothness classes and their anisotropic extensions are considered. Our study involves several technical novelties, including sharp approximation of finitely differentiable multivariate densities by normal mixtures and a new sieve on the space of such densities., Comment: 29 pages

We propose a Bayesian test of normality for univariate or multivariate data against alternative nonparametric models characterized by Dirichlet process mixture distributions. The alternative models are based on the principles of embedding and predictive matching. They can be interpreted to offer random granulation of a normal distribution into a mixture of normals with mixture components occupying a smaller volume the farther they are from the distribution center. A scalar parametrization based on latent clustering is used to cover an entire spectrum of separation between the normal distributions and the alternative models. An efficient sequential importance sampler is developed to calculate Bayes factors. Simulations indicate the proposed test can detect non-normality without favoring the nonparametric alternative when normality holds., Comment: 24 pages, 5 figures, 1 table

We address the issue of knots selection for Gaussian predictive process methodology. Predictive process approximation provides an effective solution to the cubic order computational complexity of Gaussian process models. This approximation crucially depends on a set of points, called knots, at which the original process is retained, while the rest is approximated via a deterministic extrapolation. Knots should be few in number to keep the computational complexity low, but provide a good coverage of the process domain to limit approximation error. We present theoretical calculations to show that coverage must be judged by the canonical metric of the Gaussian process. This necessitates having in place a knots selection algorithm that automatically adapts to the changes in the canonical metric affected by changes in the parameter values controlling the Gaussian process covariance function. We present an algorithm toward this by employing an incomplete Cholesky factorization with pivoting and dynamic stopping. Although these concepts already exist in the literature, our contribution lies in unifying them into a fast algorithm and in using computable error bounds to finesse implementation of the predictive process approximation. The resulting adaptive predictive process offers a substantial automatization of Guassian process model fitting, especially for Bayesian applications where thousands of values of the covariance parameters are to be explored., Comment: 20 pages, 5 figures

We propose a flexible and identifiable version of the two-groups model, motivated by hierarchical Bayes considerations, that features an empirical null and a semiparametric mixture model for the non-null cases. We use a computationally efficient predictive recursion marginal likelihood procedure to estimate the model parameters, even the nonparametric mixing distribution. This leads to a nonparametric empirical Bayes testing procedure, which we call PRtest, based on thresholding the estimated local false discovery rates. Simulations and real-data examples demonstrate that, compared to existing approaches, PRtest's careful handling of the non-null density can give a much better fit in the tails of the mixture distribution which, in turn, can lead to more realistic conclusions., Comment: 18 pages, 4 figures, 3 tables

Predictive recursion is an accurate and computationally efficient algorithm for nonparametric estimation of mixing densities in mixture models. In semiparametric mixture models, however, the algorithm fails to account for any uncertainty in the additional unknown structural parameter. As an alternative to existing profile likelihood methods, we treat predictive recursion as a filter approximation to fitting a fully Bayes model, whereby an approximate marginal likelihood of the structural parameter emerges and can be used for inference. We call this the predictive recursion marginal likelihood. Convergence properties of predictive recursion under model mis-specification also lead to an attractive construction of this new procedure. We show pointwise convergence of a normalized version of this marginal likelihood function. Simulations compare the performance of this new marginal likelihood approach that of existing profile likelihood methods as well as Dirichlet process mixtures in density estimation. Mixed-effects models and an empirical Bayes multiple testing application in time series analysis are also considered.

Mixture models have received considerable attention recently and Newton [Sankhy\={a} Ser. A 64 (2002) 306--322] proposed a fast recursive algorithm for estimating a mixing distribution. We prove almost sure consistency of this recursive estimate in the weak topology under mild conditions on the family of densities being mixed. This recursive estimate depends on the data ordering and a permutation-invariant modification is proposed, which is an average of the original over permutations of the data sequence. A Rao--Blackwell argument is used to prove consistency in probability of this alternative estimate. Several simulations are presented, comparing the finite-sample performance of the recursive estimate and a Monte Carlo approximation to the permutation-invariant alternative along with that of the nonparametric maximum likelihood estimate and a nonparametric Bayes estimate., Comment: Published in at http://dx.doi.org/10.1214/08-AOS639 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In the spirit of modeling inference for microarrays as multiple testing for sparse mixtures, we present a similar approach to a simplified version of quantitative trait loci (QTL) mapping. Unlike in case of microarrays, where the number of tests usually reaches tens of thousands, the number of tests performed in scans for QTL usually does not exceed several hundreds. However, in typical cases, the sparsity $p$ of significant alternatives for QTL mapping is in the same range as for microarrays. For methodological interest, as well as some related applications, we also consider non-sparse mixtures. Using simulations as well as theoretical observations we study false discovery rate (FDR), power and misclassification probability for the Benjamini-Hochberg (BH) procedure and its modifications, as well as for various parametric and nonparametric Bayes and Parametric Empirical Bayes procedures. Our results confirm the observation of Genovese and Wasserman (2002) that for small p the misclassification error of BH is close to optimal in the sense of attaining the Bayes oracle. This property is shared by some of the considered Bayes testing rules, which in general perform better than BH for large or moderate $p$'s., Comment: Published in at http://dx.doi.org/10.1214/193940307000000158 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)