61 research outputs found
Multi-dimensional Density of States by Multicanonical Monte Carlo
Full text linkMulti-dimensional density of states provides a useful description of complex
frustrated systems. Recent advances in Monte Carlo methods enable efficient
calculation of the density of states and related quantities, which renew the
interest in them. Here we calculate density of states on the plane (energy,
magnetization) for an Ising Model with three-spin interactions on a random
sparse network, which is a system of current interest both in physics of glassy
systems and in the theory of error-correcting codes. Multicanonical Monte Carlo
algorithm is successfully applied, and the shape of densities and its
dependence on the degree of frustration is revealed. Efficiency of
multicanonical Monte Carlo is also discussed with the shape of a projection of
the distribution simulated by the algorithm.Comment: Presented at SPDSA 2004, Hayama, Japa
W-kernel and essential subspace for frequencist's evaluation of Bayesian estimators
Full text linkThe posterior covariance matrix W defined by the log-likelihood of each
observation plays important roles both in the sensitivity analysis and
frequencist's evaluation of the Bayesian estimators. This study focused on the
matrix W and its principal space; we term the latter as an essential subspace.
First, it is shown that they appear in various statistical settings, such as
the evaluation of the posterior sensitivity, assessment of the frequencist's
uncertainty from posterior samples, and stochastic expansion of the loss; a key
tool to treat frequencist's properties is the recently proposed Bayesian
infinitesimal jackknife approximation (Giordano and Broderick (2023)). In the
following part, we show that the matrix W can be interpreted as a reproducing
kernel; it is named as W-kernel. Using the W-kernel, the essential subspace is
expressed as a principal space given by the kernel PCA. A relation to the
Fisher kernel and neural tangent kernel is established, which elucidates the
connection to the classical asymptotic theory; it also leads to a sort of
Bayesian-frequencist's duality. Finally, two applications, selection of a
representative set of observations and dimensional reduction in the approximate
bootstrap, are discussed. In the former, incomplete Cholesky decomposition is
introduced as an efficient method to compute the essential subspace. In the
latter, different implementations of the approximate bootstrap for posterior
means are compared.Comment: 48 pages, 10 figures. Revised and enlarged version of ISM Research
Memorandum No.122
Posterior Covariance Information Criterion
Full text linkWe introduce an information criterion, PCIC, for predictive evaluation based
on quasi-posterior distributions. It is regarded as a natural generalisation of
the widely applicable information criterion (WAIC) and can be computed via a
single Markov chain Monte Carlo run. PCIC is useful in a variety of predictive
settings that are not well dealt with in WAIC, including weighted likelihood
inference and quasi-Bayesian predictio
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