Estimating discrete Markov models from various incomplete data schemes

Abstract

The parameters of a discrete stationary Markov model are transition probabilities between states. Traditionally, data consist in sequences of observed states for a given number of individuals over the whole observation period. In such a case, the estimation of transition probabilities is straightforwardly made by counting one-step moves from a given state to another. In many real-life problems, however, the inference is much more difficult as state sequences are not fully observed, namely the state of each individual is known only for some given values of the time variable. A review of the problem is given, focusing on Monte Carlo Markov Chain (MCMC) algorithms to perform Bayesian inference and evaluate posterior distributions of the transition probabilities in this missing-data framework. Leaning on the dependence between the rows of the transition matrix, an adaptive MCMC mechanism accelerating the classical Metropolis-Hastings algorithm is then proposed and empirically studied.

Data-intensive modeling of forest dynamics

“…We provide in the following a brief presentation of Gibbs sampling. Additional implementation details are in Appendix, and we refer to Robert and Casella (2004) for the general principles underlying MCMC algorithms and to Pasanisi et al (2012) for an extended description of Gibbs sampling to infer transition probabilities in temporal sequences. In addition to the full explanation below, we also provide a pseudocode of the procedure (Box 1).…”

“…with Dir is the Dirichlet distribution, g are biasing factors set here uniformly to 1 as we include no prior knowledge on the shape of the transition matrix (Pasanisi et al, 2012). w i,j are the sufficient statistics reflecting the transitions in the augmented data Z [hÀ1] , formally defined as…”

“…To overcome the issue of irregular samplings in time specific to forest inventory data, we develop a Gibbs sampling procedure for augmenting the data and infer the transition probabilities. Our particular use of Gibbs sampling (Pasanisi et al, 2012) has a substantial scientific novelty, as this is the first application of this statistical machinery to overcome the problems of irregularities in the forest inventory sample design. In this paper, we demonstrate the power of this statistical methodology in our application, and deliver it as readyto-go tool for other applications by explaining every step, providing pseudocode, and original R code.…”

Data-intensive modeling of forest dynamics

“…We provide in the following a brief presentation of Gibbs sampling. Additional 237 implementation details are in Appendix 1.2, and we refer to Robert and Casella (2004) for 238 the general principles underlying MCMC algorithms and to Pasanisi et al (2012) for an ex-239 tended description of Gibbs sampling to infer transition probabilities in temporal sequences.…”

An appraisal of the classic forest succession paradigm with the shade tolerance index

“…To overcome this difficulty, we employed here the original methodology developed in Lienard et al (2014). Specifically, we applied Gibbs sampling, a Monte Carlo Markov Chain implementation (Robert and Casella, 2004) to estimate the transition probabilities, with the specific guidelines provided by Pasanisi et al (2012). We provide in the following a brief description of the algorithm; please refer to Lienard et al (2014) for an extended description. First, we constructed a temporal sequence S p of the shade tolerance index for each plot p , by inserting the discretized value of shade tolerance index s ( p,i ) measured in the i -th year, at position i of S p .…”