15,240 research outputs found
Generalized structured additive regression based on Bayesian P-splines
Generalized additive models (GAM) for modelling nonlinear effects of continuous covariates are now well established tools for the applied statistician. In this paper we develop Bayesian GAM's and extensions to generalized structured additive regression based on one or two dimensional P-splines as the main building block. The approach extends previous work by Lang und Brezger (2003) for Gaussian responses. Inference relies on Markov chain Monte Carlo (MCMC) simulation techniques, and is either based on iteratively weighted least squares (IWLS) proposals or on latent utility representations of (multi)categorical regression models. Our approach covers the most common univariate response distributions, e.g. the Binomial, Poisson or Gamma distribution, as well as multicategorical responses. For the first time, we present Bayesian semiparametric inference for the widely used multinomial logit models. As we will demonstrate through two applications on the forest health status of trees and a space-time analysis of health insurance data, the approach allows realistic modelling of complex problems. We consider the enormous flexibility and extendability of our approach as a main advantage of Bayesian inference based on MCMC techniques compared to more traditional approaches. Software for the methodology presented in the paper is provided within the public domain package BayesX
Bayesian inference through encompassing priors and importance sampling for a class of marginal models for categorical data
We develop a Bayesian approach for selecting the model which is the most
supported by the data within a class of marginal models for categorical
variables formulated through equality and/or inequality constraints on
generalised logits (local, global, continuation or reverse continuation),
generalised log-odds ratios and similar higher-order interactions. For each
constrained model, the prior distribution of the model parameters is formulated
following the encompassing prior approach. Then, model selection is performed
by using Bayes factors which are estimated by an importance sampling method.
The approach is illustrated through three applications involving some datasets,
which also include explanatory variables. In connection with one of these
examples, a sensitivity analysis to the prior specification is also considered
Statistical coverage for supersymmetric parameter estimation: a case study with direct detection of dark matter
Models of weak-scale supersymmetry offer viable dark matter (DM) candidates.
Their parameter spaces are however rather large and complex, such that pinning
down the actual parameter values from experimental data can depend strongly on
the employed statistical framework and scanning algorithm. In frequentist
parameter estimation, a central requirement for properly constructed confidence
intervals is that they cover true parameter values, preferably at exactly the
stated confidence level when experiments are repeated infinitely many times.
Since most widely-used scanning techniques are optimised for Bayesian
statistics, one needs to assess their abilities in providing correct confidence
intervals in terms of the statistical coverage. Here we investigate this for
the Constrained Minimal Supersymmetric Standard Model (CMSSM) when only
constrained by data from direct searches for dark matter. We construct
confidence intervals from one-dimensional profile likelihoods and study the
coverage by generating several pseudo-experiments for a few benchmark sets of
pseudo-true parameters. We use nested sampling to scan the parameter space and
evaluate the coverage for the benchmarks when either flat or logarithmic priors
are imposed on gaugino and scalar mass parameters. The sampling algorithm has
been used in the configuration usually adopted for exploration of the Bayesian
posterior. We observe both under- and over-coverage, which in some cases vary
quite dramatically when benchmarks or priors are modified. We show how most of
the variation can be explained as the impact of explicit priors as well as
sampling effects, where the latter are indirectly imposed by physicality
conditions. For comparison, we also evaluate the coverage for Bayesian credible
intervals, and observe significant under-coverage in those cases.Comment: 30 pages, 5 figures; v2 includes major updates in response to
referee's comments; extra scans and tables added, discussion expanded, typos
corrected; matches published versio
Expectation Propagation for Poisson Data
The Poisson distribution arises naturally when dealing with data involving
counts, and it has found many applications in inverse problems and imaging. In
this work, we develop an approximate Bayesian inference technique based on
expectation propagation for approximating the posterior distribution formed
from the Poisson likelihood function and a Laplace type prior distribution,
e.g., the anisotropic total variation prior. The approach iteratively yields a
Gaussian approximation, and at each iteration, it updates the Gaussian
approximation to one factor of the posterior distribution by moment matching.
We derive explicit update formulas in terms of one-dimensional integrals, and
also discuss stable and efficient quadrature rules for evaluating these
integrals. The method is showcased on two-dimensional PET images.Comment: 25 pages, to be published at Inverse Problem
Quantum collapse rules from the maximum relative entropy principle
We show that the von Neumann--Lueders collapse rules in quantum mechanics
always select the unique state that maximises the quantum relative entropy with
respect to the premeasurement state, subject to the constraint that the
postmeasurement state has to be compatible with the knowledge gained in the
measurement. This way we provide an information theoretic characterisation of
quantum collapse rules by means of the maximum relative entropy principle.Comment: v2: some references added, improved presentation, result generalised
to cover nonfaithful states; v3: cross-ref to arXiv:1408.3502 added, v4: some
small corrections plus reference to a published version adde
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