Rates of contraction of posterior distributions based on Gaussian process priors

Biological data objects often have both of the following features: (i) they are functions rather than single numbers or vectors, and (ii) they are correlated owing to phylogenetic relationships. In this paper, we give a flexible statistical model for such data, by combining assumptions from phylogenetics with Gaussian processes. We describe its use as a non-parametric Bayesian prior distribution, both for prediction (placing posterior distributions on ancestral functions) and model selection (comparing rates of evolution across a phylogeny, or identifying the most likely phylogenies consistent with the observed data). Our work is integrative, extending the popular phylogenetic Brownian motion and Ornstein–Uhlenbeck models to functional data and Bayesian inference, and extending Gaussian process regression to phylogenies. We provide a brief illustration of the application of our method.

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Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

Inverse Problems ◽

10.1088/1361-6420/ac4839 ◽

2022 ◽

Author(s):

Hanne Kekkonen

Keyword(s):

Inverse Problem ◽

Convergence Rate ◽

Gaussian Process ◽

Reconstruction Error ◽

Posterior Distributions ◽

Smooth Functions ◽

True Parameter ◽

Linear Inverse Problem ◽

Point Evaluations ◽

Truncated Gaussian

Abstract We consider the statistical non-linear inverse problem of recovering the absorption term f>0 in the heat equation with given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

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A General Nonstationary and Time-Varying Mixed Signal Blind Source Separation Method Based on Online Gaussian Process

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s021800142058015x ◽

2020 ◽

Vol 34 (11) ◽

pp. 2058015

Author(s):

Pengju He ◽

Mi Qi ◽

Wenhui Li ◽

Mengyang Tang ◽

Ziwei Zhao

Keyword(s):

Gaussian Process ◽

Blind Source Separation ◽

Likelihood Function ◽

Source Separation ◽

Microphone Arrays ◽

Time Varying ◽

Posterior Distributions ◽

Source Signal ◽

Mixing Matrix ◽

Nonstationary Source

Most nonstationary and time-varying mixed source separation algorithms are based on the model of instantaneous mixtures. However, the observation signal is a convolutional mixed source in reverberation environment, such as mobile voice received by indoor microphone arrays. In this paper, a time-varying convolution blind source separation (BSS) algorithm for nonstationary signals is proposed, which can separate both time-varying instantaneous mixtures and time-varying convolution mixtures. We employ the variational Bayesian (VB) inference method with Gaussian process (GP) prior for separating the nonstationary source frame by frame from the time-varying convolution signal, in which the prior information of the mixing matrix and the source signal are obtained by the Gaussian autoregressive method, and the posterior distributions of parameters (source signal and mixing matrix) are obtained by the VB learning. In the learning process, the learned parameters and hyperparameters are propagated to the next frame for VB inference as the prior which is combined with the likelihood function to get the posterior distribution. The experimental results show that the proposed algorithm is effective for separating time-varying mixed speech signals.

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The Variational Gaussian Approximation Revisited

Neural Computation ◽

10.1162/neco.2008.08-07-592 ◽

2009 ◽

Vol 21 (3) ◽

pp. 786-792 ◽

Cited By ~ 52

Author(s):

Manfred Opper ◽

Cédric Archambeau

Keyword(s):

Machine Learning ◽

Gaussian Process ◽

Learning Community ◽

Gaussian Approximation ◽

Good Reason ◽

Random Variables ◽

Gaussian Process Regression ◽

Variational Approximation ◽

Posterior Distributions ◽

The Relationship

The variational approximation of posterior distributions by multivariate gaussians has been much less popular in the machine learning community compared to the corresponding approximation by factorizing distributions. This is for a good reason: the gaussian approximation is in general plagued by an [Formula: see text] number of variational parameters to be optimized, N being the number of random variables. In this letter, we discuss the relationship between the Laplace and the variational approximation, and we show that for models with gaussian priors and factorizing likelihoods, the number of variational parameters is actually [Formula: see text]. The approach is applied to gaussian process regression with nongaussian likelihoods.

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Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

Journal of Applied Probability ◽

10.1017/s0021900200003041 ◽

2007 ◽

Vol 44 (02) ◽

pp. 393-408 ◽

Cited By ~ 1

Author(s):

Allan Sly

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

White Noise ◽

Gaussian Process ◽

Discrete Data ◽

Hölder Exponent ◽

Scaling Properties ◽

Multifractional Brownian Motion ◽

Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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A Stochastic Model for the Inheritance of the Cancer Proneness Phenotype

Methods of Information in Medicine ◽

10.1055/s-0038-1636689 ◽

1987 ◽

Vol 26 (03) ◽

pp. 117-123

Author(s):

P. Tautu ◽

G. Wagner

Keyword(s):

Stochastic Model ◽

Gaussian Process ◽

Covariance Function ◽

Neoplastic Disease ◽

Disease Phenotype ◽

Probabilistic Representation ◽

Temporal Behaviour ◽

Continuous Trait ◽

Continuous Parameter ◽

Level Zero

SummaryA continuous parameter, stationary Gaussian process is introduced as a first approach to the probabilistic representation of the phenotype inheritance process. With some specific assumptions about the components of the covariance function, it may describe the temporal behaviour of the “cancer-proneness phenotype” (CPF) as a quantitative continuous trait. Upcrossing a fixed level (“threshold”) u and reaching level zero are the extremes of the Gaussian process considered; it is assumed that they might be interpreted as the transformation of CPF into a “neoplastic disease phenotype” or as the non-proneness to cancer, respectively.

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