1,661 research outputs found
Bayesian Credibility for GLMs
We revisit the classical credibility results of Jewell and B\"uhlmann to
obtain credibility premiums for a GLM using a modern Bayesian approach. Here
the prior distributions can be chosen without restrictions to be conjugate to
the response distribution. It can even come from out-of-sample information if
the actuary prefers.
Then we use the relative entropy between the "true" and the estimated models
as a loss function, without restricting credibility premiums to be linear. A
numerical illustration on real data shows the feasibility of the approach, now
that computing power is cheap, and simulations software readily available
Entropic effects in large-scale Monte Carlo simulations
The efficiency of Monte Carlo samplers is dictated not only by energetic
effects, such as large barriers, but also by entropic effects that are due to
the sheer volume that is sampled. The latter effects appear in the form of an
entropic mismatch or divergence between the direct and reverse trial moves. We
provide lower and upper bounds for the average acceptance probability in terms
of the Renyi divergence of order 1/2. We show that the asymptotic finitude of
the entropic divergence is the necessary and sufficient condition for
non-vanishing acceptance probabilities in the limit of large dimensions.
Furthermore, we demonstrate that the upper bound is reasonably tight by showing
that the exponent is asymptotically exact for systems made up of a large number
of independent and identically distributed subsystems. For the last statement,
we provide an alternative proof that relies on the reformulation of the
acceptance probability as a large deviation problem. The reformulation also
leads to a class of low-variance estimators for strongly asymmetric
distributions. We show that the entropy divergence causes a decay in the
average displacements with the number of dimensions n that are simultaneously
updated. For systems that have a well-defined thermodynamic limit, the decay is
demonstrated to be n^{-1/2} for random-walk Monte Carlo and n^{-1/6} for Smart
Monte Carlo (SMC). Numerical simulations of the LJ_38 cluster show that SMC is
virtually as efficient as the Markov chain implementation of the Gibbs sampler,
which is normally utilized for Lennard-Jones clusters. An application of the
entropic inequalities to the parallel tempering method demonstrates that the
number of replicas increases as the square root of the heat capacity of the
system.Comment: minor corrections; the best compromise for the value of the epsilon
parameter in Eq. A9 is now shown to be log(2); 13 pages, 4 figures, to appear
in PR
Bayesian interpretation of Generalized empirical likelihood by maximum entropy
We study a parametric estimation problem related to moment condition models.
As an alternative to the generalized empirical likelihood (GEL) and the
generalized method of moments (GMM), a Bayesian approach to the problem can be
adopted, extending the MEM procedure to parametric moment conditions. We show
in particular that a large number of GEL estimators can be interpreted as a
maximum entropy solution. Moreover, we provide a more general field of
applications by proving the method to be robust to approximate moment
conditions
Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization
We address the inverse problem of cosmic large-scale structure reconstruction
from a Bayesian perspective. For a linear data model, a number of known and
novel reconstruction schemes, which differ in terms of the underlying signal
prior, data likelihood, and numerical inverse extra-regularization schemes are
derived and classified. The Bayesian methodology presented in this paper tries
to unify and extend the following methods: Wiener-filtering, Tikhonov
regularization, Ridge regression, Maximum Entropy, and inverse regularization
techniques. The inverse techniques considered here are the asymptotic
regularization, the Jacobi, Steepest Descent, Newton-Raphson,
Landweber-Fridman, and both linear and non-linear Krylov methods based on
Fletcher-Reeves, Polak-Ribiere, and Hestenes-Stiefel Conjugate Gradients. The
structures of the up-to-date highest-performing algorithms are presented, based
on an operator scheme, which permits one to exploit the power of fast Fourier
transforms. Using such an implementation of the generalized Wiener-filter in
the novel ARGO-software package, the different numerical schemes are
benchmarked with 1-, 2-, and 3-dimensional problems including structured white
and Poissonian noise, data windowing and blurring effects. A novel numerical
Krylov scheme is shown to be superior in terms of performance and fidelity.
These fast inverse methods ultimately will enable the application of sampling
techniques to explore complex joint posterior distributions. We outline how the
space of the dark-matter density field, the peculiar velocity field, and the
power spectrum can jointly be investigated by a Gibbs-sampling process. Such a
method can be applied for the redshift distortions correction of the observed
galaxies and for time-reversal reconstructions of the initial density field.Comment: 40 pages, 11 figure
Efficient computation of the first passage time distribution of the generalized master equation by steady-state relaxation
The generalized master equation or the equivalent continuous time random walk
equations can be used to compute the macroscopic first passage time
distribution (FPTD) of a complex stochastic system from short-term microscopic
simulation data. The computation of the mean first passage time and additional
low-order FPTD moments can be simplified by directly relating the FPTD moment
generating function to the moments of the local FPTD matrix. This relationship
can be physically interpreted in terms of steady-state relaxation, an extension
of steady-state flow. Moreover, it is amenable to a statistical error analysis
that can be used to significantly increase computational efficiency. The
efficiency improvement can be extended to the FPTD itself by modelling it using
a Gamma distribution or rational function approximation to its Laplace
transform
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