W wielu modelach statystyki bayesowskiej kluczowym problemem jest obliczanie całek względem rozkładów a posteriori, które są skomplikowane i możliwe jest jedynie wyznaczenie ich gęstości z dokładnością do stałej normującej. W tej sytuacji najczęściej stosowanym i bardzo skutecznym narzędziem są markowowskie metody Monte Carlo (Markov Chain Monte Carlo, MCMC). Jest to rodzina algorytmów, które p...

This article introduces a Markov chain Monte Carlo (MCMC) method for sampling the parameters of a multinomial logit model from their posterior distribution. Let yi ∈ {0, . . . ,M} denote the categorical response of subject i with covariates xi = (xi1, . . . , xip) T . Let X = (x1, . . . ,xn) T denote the design matrix, and let y = (y1, . . . , yn) T . Multinomial logit models relate yi to xi th...

In this paper we propose a Bayesian method for estimating hyperbolic diffusion models. The approach is based on the Markov Chain Monte Carlo (MCMC) method after discretization via the Milstein scheme. Our simulation study shows that the hyperbolic diffusion exhibits many of the stylized facts about asset returns documented in the financial econometrics literature, such as a slowly declining aut...

In this article, we propose new Monte Carlo methods for computing a single marginal likelihood or several marginal likelihoods for the purpose of Bayesian model comparisons. The methods are motivated by Bayesian variable selection, in which the marginal likelihoods for all subset variable models are required to compute. The proposed estimates use only a single Markov chain Monte Carlo (MCMC) ou...

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most flexible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. A...

The following is a simple example to show two important properties of a Markov chain Monte Carlo (MCMC) sampler and to illustrate the basic functionality of the method and issues relating to it’s usage. A Markov chain is a series where the realisation of the next element in the series, Y , is dependent only on the current state, X, and occurs with probability, P (Y |X). So the even number serie...

Usually in multivariate regression problem it is assumed that residuals of outputs are independent of each other. In many applications a more realistic model would allow dependencies between the outputs. In this paper we show how a Bayesian treatment using Markov Chain Monte Carlo (MCMC) method can allow for a full covariance matrix with Multi Layer Perceptron (MLP) neural networks.

We motivate the use of convergence diagnostic techniques for Markov Chain Monte Carlo algorithms and review various methods proposed in the MCMC literature. A common notation is established and each method is discussed with particular emphasis on implementational issues and possible extensions. The methods are compared in terms of their interpretability and applicability and recommendations are...

Decision tree learning is a popular approach for classification and regression in machine learning and statistics, and Bayesian formulations— which introduce a prior distribution over decision trees, and formulate learning as posterior inference given data—have been shown to produce competitive performance. Unlike classic decision tree learning algorithms like ID3, C4.5 and CART, which work in ...

Subset Simulation is an adaptive simulation method that efficiently solves structural reliability problems with many random variables. The method requires sampling from conditional distributions, which is achieved through Markov Chain Monte Carlo (MCMC) algorithms. This paper discusses different MCMC algorithms proposed for Subset Simulation and introduces a novel approach for MCMC sampling in ...