On the curved exponential family in the Stochatic Approximation Expectation Maximization Algorithm

The Expectation-Maximization Algorithm (EM) is a widely used method allowing to estimate the maximum likelihood of models involving latent variables. When the Expectation step cannot be computed easily, one can use stochastic versions of the EM such as the Stochastic Approximation EM. This algorithm, however, has the drawback to require the joint likelihood to belong to the curved exponential family. To overcome this problem, \cite{kuhn2005maximum} introduced a rewriting of the model which ``exponentializes'' it by considering the parameter as an additional latent variable following a Normal distribution centered on the newly defined parameters and with fixed variance. The likelihood of this new exponentialized model now belongs to the curved exponential family. Although often used, there is no guarantee that the estimated mean is close to the maximum likelihood estimate of the initial model. In this paper, we quantify the error done in this estimation while considering the exponentialized model instead of the initial one. By verifying those results on an example, we see that a trade-off must be made between the speed of convergence and the tolerated error. Finally, we propose a new algorithm allowing a better estimation of the parameter in a reasonable computation time to reduce the bias.

Improving estimation of species distribution from citizen-science records using data-integration models

Mapping species distributions is a crucial but challenging requirement of wildlife management. The frequent need to sample vast expanses of potential habitat increases the cost of planned surveys and rewards accumulation of opportunistic observations. In this paper, we integrate planned survey data from roost counts with opportunistic samples from eBird, WikiAves and Xeno-canto citizen-science platforms to map the geographic range of the endangered Vinaceous-breasted Parrot. We demonstrate the estimation and mapping of species occurrence based on data integration while accounting for specifics of each data set, including observation technique and uncertainty about the observations. Our analysis illustrates 1) the incorporation of sampling effort, spatial autocorrelation, and site covariates in a joint-likelihood, hierarchical, data-integration model; 2) the evaluation of the contribution of each data set, as well as the contribution of effort covariates, spatial autocorrelation, and site covariates to the predictive ability of fitted models using a cross-validation approach; and 3) how spatial representation of the latent occupancy state (i.e. realized occupancy) helps identify areas with high uncertainty that should be prioritized in future field work. Our results reveal a Vinaceous-breasted Parrot geographic range of 434,670 square kilometers, which is three times larger than the Extant area previously reported in the IUCN Red List. The exclusion of one data set at a time from the analyses always resulted in worse predictions by the models of truncated data than by the full model, which included all data sets. Likewise, exclusion of spatial autocorrelation, site covariates, or sampling effort resulted in worse predictions. The integration of different data sets into one joint-likelihood model produced a more reliable representation of the species range than any individual data set taken on its own improving the use of citizen science data in combination with planned survey results.

The Theory and Implementation of Bayesian Networks Structural Learning using Tabu Search Algorithm

This paper portrays the hypothesis and execution of Bayesian systems basic getting the hang of utilizing unthinkable pursuit calculation. Bayesian systems give an extremely broad but powerful graphical language for calculating joint likelihood disseminations. Finding the ideal structure of Bayesian systems from information has been demonstrated to be NP-hard. In this paper, unthinkable hunt has been created to give progressively proficient structure. We actualized auxiliary learning in Bayesian systems with regards to information characterization. With the end goal of correlation, we considered order task and applied general Bayesian systems alongside this classifier to certain databases. Our trial results show that the Tabu pursuit can locate the great structure with the less time multifaceted nature. The reenactment results affirmed that utilizing Tabu hunt so as to discover Bayesian systems structure improves the grouping exactness.

Evaluation of a pair-based, joint-likelihood association approach for regional infrasound event identification

SUMMARY A Bayesian framework for the association of infrasonic detections is presented and evaluated for analysis at regional propagation scales. A pair-based, joint-likelihood association approach is developed that identifies events by computing the probability that individual detection pairs are attributable to a hypothetical common source and applying hierarchical clustering to identify events from the pair-based analysis. The framework is based on a Bayesian formulation introduced for infrasonic source localization and utilizes the propagation models developed for that application with modifications to improve the numerical efficiency of the analysis. Clustering analysis is completed using hierarchical analysis via weighted linkage for a non-Euclidean distance matrix defined by the negative log-joint-likelihood values. The method is evaluated using regional synthetic data with propagation distances of hundreds of kilometres in order to study the sensitivity of the method to uncertainties and errors in backazimuth and time of arrival. The method is found to be robust and stable for typical uncertainties, able to effectively distinguish noise detections within the data set from those in events, and can be made numerically efficient due to its ease of parallelization.

Exact joint likelihood of pseudo-Cℓ estimates from correlated Gaussian cosmological fields

ABSTRACT We present the exact joint likelihood of pseudo-Cℓ power spectrum estimates measured from an arbitrary number of Gaussian cosmological fields. Our method is applicable to both spin-0 fields and spin-2 fields, including a mixture of the two, and is relevant to cosmic microwave background (CMB), weak lensing, and galaxy clustering analyses. We show that Gaussian cosmological fields are mixed by a mask in such a way that retains their Gaussianity and derive exact expressions for the covariance of the cut-sky spherical harmonic coefficients, the pseudo-aℓms, without making any assumptions about the mask geometry. We then show that each auto or cross-pseudo-Cℓ estimator can be written as a quadratic form, and apply the known joint distribution of quadratic forms to obtain the exact joint likelihood of a set of pseudo-Cℓ estimates in the presence of an arbitrary mask. We show that the same formalism can be applied to obtain the exact joint likelihood of quadratic maximum likelihood power spectrum estimates. Considering the polarization of the CMB as an example, we show using simulations that our likelihood recovers the full, exact multivariate distribution of EE, BB, and EB pseudo-Cℓ power spectra. Our method provides a route to robust cosmological constraints from future CMB and large-scale structure surveys in an era of ever-increasing statistical precision.

The Portrait Problem: Bayesian Inference with Joint Likelihood

Chapter 7 discusses the “Portrait Problem,” which concerns the dispute about whether a portrait frequently associated with Thomas Bayes (and used, in fact, as the cover of this book!) is actually a picture of him. In doing so, the chapter highlights the fact that multiple pieces of information can be used in a Bayesian analysis. A key concept in this chapter is that multiple sources of data can be combined in a Bayesian inference framework. The main take-home point is that Bayesian analysis can be very, very flexible. A Bayesian analysis is possible as long as the likelihood of observing the data under each hypothesis can be computed. The chapter also discusses the concepts of joint likelihood and independence.

Analysis of Error Structure for Additive Biomass Equations on the Use of Multivariate Likelihood Function

Research Highlights: this study developed additive biomass equations respectively from nonlinear regression (NLR) on original data and linear regression (LR) on a log-transformed scale by nonlinear seemingly unrelated regression (NSUR). To choose appropriate regression form, the error structures (additive vs. multiplicative) of compatible biomass equations were determined on the use of the multivariate likelihood function which extended the method of likelihood analysis to the general occasion of a contemporaneously correlated set of equations. Background and Objectives: both NLR and LR could yield the expected predictions for allometric scaling relationship. In recent studies, there are vigorous debates on which regression (NLR or LR) should apply. The main aim of this paper is to analyze the error structure of a compatible system of biomass equations to choose more appropriate regression. Materials and Methods: based on biomass data of 270 trees for three tree species, additive biomass equations were developed respectively for NLR and LR by NSUR. Multivariate likelihood functions were computed to determine the error structure based on the multivariate probability density function. The anti-log correction factor which kept the additive property was obtained separately using the arithmetic and weighted average of basic correction factors from each equation to assess two model specifications on the comparably original scale. Results: the assumption of additive error structure was well favored for an additive system of three species based on the joint likelihood function. However, the error structure of each component equation calculated from the conditional likelihood function for compatible equations might be different. The performance of additive equations corrected by a weighted average of basic correction factor from each component equation performed better than that of the arithmetic average and held good property of compatibility after corrected. Conclusions: NLR provided a better fit for additive biomass equations of three tree species. Additive equations which confirmed the responding assumption of error structure performed better. The joint likelihood function on the use of the multivariate likelihood function could be used to analyze the error structure of the additive system which was a result of a tradeoff for each component equation. Based on the average of correction factors from each component equation to correct the bias of additive equations was feasible for the hold of additive property, which might lead to a poor correction effect for some component equation.