Estimating Distributions of Parameters in Nonlinear State Space Models with Replica Exchange Particle Marginal Metropolis–Hastings Method

Extracting latent nonlinear dynamics from observed time-series data is important for understanding a dynamic system against the background of the observed data. A state space model is a probabilistic graphical model for time-series data, which describes the probabilistic dependence between latent variables at subsequent times and between latent variables and observations. Since, in many situations, the values of the parameters in the state space model are unknown, estimating the parameters from observations is an important task. The particle marginal Metropolis–Hastings (PMMH) method is a method for estimating the marginal posterior distribution of parameters obtained by marginalization over the distribution of latent variables in the state space model. Although, in principle, we can estimate the marginal posterior distribution of parameters by iterating this method infinitely, the estimated result depends on the initial values for a finite number of times in practice. In this paper, we propose a replica exchange particle marginal Metropolis–Hastings (REPMMH) method as a method to improve this problem by combining the PMMH method with the replica exchange method. By using the proposed method, we simultaneously realize a global search at a high temperature and a local fine search at a low temperature. We evaluate the proposed method using simulated data obtained from the Izhikevich neuron model and Lévy-driven stochastic volatility model, and we show that the proposed REPMMH method improves the problem of the initial value dependence in the PMMH method, and realizes efficient sampling of parameters in the state space models compared with existing methods.

Effective aggregation of gappy replicated time series using INLA

<p>Soil CO2 efflux data from automated chambers provide an important constraint for ecosystem and soil respiration. Usually, half-hourly time series of several replicated chambers have to be aggregated to plot-level while gaps in the time series have to be accommodated. Gaps cause jumps and other problems in aggregation of replicated measurement in each half-hour, therefore, lookup tables and machine learning approaches are used to fill gaps beforehand.</p><p>Here, we present an alternative fully Bayesian approach for the combined gap-filling and aggregation based on Integrated Nested Laplace Approximation (INLA). This method integrates all information from every measurement across replicates and across time and therefore efficiently estimates the correlation structure among all observations. It provides the full marginal posterior distribution of the aggregated time series at the plot level across the time span of the time series. We compare several aggregation approaches using four years of data from 16 automatic chambers at the eddy-covariance site in Majadas de Tietar in Spain (ES-LM1, ES-LMa).</p><p>This approach is applicable for other replicated time series as well. We further explore its usage for analysing time-varying effects across treatments and habitats and its usage for gap-filling net ecosystem exchange (NEE) data based on the full correlation structure in a data-cube of time and environmental conditions.</p>

A Note on the Bayesian Modeling of the Stratigraphic Chronology of Canímar Abajo, Cuba

We implement a Bayesian statistical analysis of the chronology of Canímar Abajo in Cuba in order to estimate two episodes of burial activity and the period of time corresponding to the hiatus between them. We show that by using simple Bayesian modeling, conclusions can easily be reached by the analysis of the marginal posterior distribution of each parameter of the model. However, we also suggest and describe new statistical tools that exploit the joint posterior distribution of collections of dates. These new tools give complementary information regarding the chronology of human activity.

A Bayesian Adaptive Ensemble Kalman Filter for Sequential State and Parameter Estimation

This paper proposes new methodology for sequential state and parameter estimation within the ensemble Kalman filter. The method is fully Bayesian and propagates the joint posterior distribution of states and parameters over time. To implement the method, the authors consider three representations of the marginal posterior distribution of the parameters: a grid-based approach, a Gaussian approximation, and a sequential importance sampling (SIR) approach with kernel resampling. In contrast to existing online parameter estimation algorithms, the new method explicitly accounts for parameter uncertainty and provides a formal way to combine information about the parameters from data at different time periods. The method is illustrated and compared to existing approaches using simulated and real data.

Analysis of a genetically structured variance heterogeneity model using the Box–Cox transformation

SummaryOver recent years, statistical support for the presence of genetic factors operating at the level of the environmental variance has come from fitting a genetically structured heterogeneous variance model to field or experimental data in various species. Misleading results may arise due to skewness of the marginal distribution of the data. To investigate how the scale of measurement affects inferences, the genetically structured heterogeneous variance model is extended to accommodate the family of Box–Cox transformations. Litter size data in rabbits and pigs that had previously been analysed in the untransformed scale were reanalysed in a scale equal to the mode of the marginal posterior distribution of the Box–Cox parameter. In the rabbit data, the statistical evidence for a genetic component at the level of the environmental variance is considerably weaker than that resulting from an analysis in the original metric. In the pig data, the statistical evidence is stronger, but the coefficient of correlation between additive genetic effects affecting mean and variance changes sign, compared to the results in the untransformed scale. The study confirms that inferences on variances can be strongly affected by the presence of asymmetry in the distribution of data. We recommend that to avoid one important source of spurious inferences, future work seeking support for a genetic component acting on environmental variation using a parametric approach based on normality assumptions confirms that these are met.

Inferring the trajectory of genetic variance in the course of artificial selection

A method is proposed to infer genetic parameters within a cohort, using data from all individuals in an experiment. An application is the study of changes in additive genetic variance over generations, employing data from all generations. Inferences about the genetic variance in a given generation are based on its marginal posterior distribution, estimated via Markov chain Monte Carlo methods. As defined, the additive genetic variance within the group is directly related to the amount of selection response to be expected if parents are chosen within the group. Results from a simulated selection experiment are used to illustrate properties of the method. Four sets of data are analysed: directional selection with and without environmental trend, and random selection, with and without environmental trend. In all cases, posterior credibility intervals of size 95% assign relatively high density to values of the additive genetic variance and heritability in the neighbourhood of the true values. Properties and generalizations of the method are discussed.

Bayesian Inference of Genetic Parameters and Selection Response for Litter Size Components in Pigs

Three contemporary lines were formed from the progeny of 50 French Large White sows. In the first line, gilts were selected for ovulation rate at puberty. In the second line, they were selected for prenatal survival of the first two parities, corrected for ovulation rate. The control constituted the third line. Ovulation rate at puberty was analyzed using an animal model with a batch effect. Prenatal survival was analyzed with a repeatability animal model that included batch and parity effects. Flat priors were used to represent vague previous knowledge about parity and batch effects. Additive and residual effects were represented assuming that they were a priori normally distributed. Variance components were assumed to follow either uniform or inverted chi-square distributions, a priori. The use of different priors did not affect the results substantially. Heritabilities for ovulation rate ranged from 0.32 to 0.39, and from 0.11 to 0.16 for prenatal survival, depending on the prior used. The mean of the marginal posterior distribution of response to four generations of selection ranged from 0.38 to 0.40 ova per generation, and from 1.1 to 1.3% of the mean survival rate for average survival per generation.

Application of Bayesian inference in the comparison of lactation curves of Merino ewes

Bayesian theory is applied to compare the characteristics of the estimated lactation curves of two groups of 5-year-old Merino ewes. The diets of the two groups were supplemented respectively by DL-methionine and maleyl-DLmethionine. The purpose is to illustrate the Bayesian approach when analysing for the effect of supplement on the lactation pattern of the sheep. Using Wood's model, the posterior distributions of the model parameters are determined for the two groups. This is achieved by assuming a hierarchical Bayes model and applying the Gibbs sampler, a sampling based computer intensive algorithm that is very efficient in obtaining marginal distributions of functions of parameters. The Gibbs sampler enables us to obtain marginal posterior distribution of characteristics of the lactation curve such as peak yield, time of peak yield, persistency and total milk yield. The results are notable differences in the marginal posterior distributions of mean peak milk yield and mean total yield. The posterior probability that the mean peak milk yield of the group supplemented by maleyl-DL-methionine is higher than that of the group with DL-methionine supplement is 0·98, while the same probability for mean total yield is 0·83.