Selecting and averaging relaxed clock models in Bayesian tip dating of Mesozoic birds

Relaxed clock models are fundamental in Bayesian clock dating, but a single distribution characterizing the clock variation is typically selected. Hence, I developed a new reversible-jump Markov chain Monte Carlo (rjMCMC) algorithm for drawing posterior samples between the independent lognormal (ILN) and independent gamma rates (IGR) clock models. The ability of the rjMCMC algorithm to infer the true model was verified through simulations. I then applied the algorithm to the Mesozoic bird data previously analyzed under the white noise (WN) clock model. In comparison, averaging over the ILN and IGR models provided more reliable estimates of the divergence times and evolutionary rates. The ILN model showed slightly better fit than the IGR model and much better fit than the autocorrelated lognormal (ALN) clock model. When the data were partitioned, different partitions showed heterogeneous model fit for ILN and IGR clocks. The implementation provides a general framework for selecting and averaging relaxed clock models in Bayesian dating analyses.

Bayesian estimation of a phenotype network structure using reversible jump Markov chain Monte Carlo

<p>In this thesis we aim to estimate the unknown phenotype network structure existing among multiple interacting quantitative traits, assuming the genetic architecture is known. We begin by taking a frequentist approach and implement a score-based greedy hill-climbing search strategy using AICc to estimate an unknown phenotype network structure. This approach was inconsistent and overfitting was common, so we then propose a Bayesian approach that extends on the reversible jump Markov chain Monte Carlo algorithm. Our approach makes use of maximum likelihood estimates in the chain, so we have an efficient sampler using well-tuned proposal distributions. The common approach is to assume uniform priors over all network structures; however, we introduce a prior on the number of edges in the phenotype network structure, which prefers simple models with fewer directed edges. We determine that the relationship between the prior penalty and the joint posterior probability of the true model is not monotonic, there is some interplay between the two. Simulation studies were carried out and our approach is also applied to a published data set. It is determined that larger trait-to-trait effects are required to recover the phenotype network structure; however, mixing is generally slow, a common occurrence with reversible jump Markov chain Monte Carlo methods. We propose the use of a double step to combine two steps that alter the phenotype network structure. This proposes larger steps than the traditional birth and death move types, possibly changing the dimension of the model by more than one. This double step helped the sampler move between different phenotype network structures in simulated data sets.</p>

Bayesian estimation of a phenotype network structure using reversible jump Markov chain Monte Carlo

<p>In this thesis we aim to estimate the unknown phenotype network structure existing among multiple interacting quantitative traits, assuming the genetic architecture is known. We begin by taking a frequentist approach and implement a score-based greedy hill-climbing search strategy using AICc to estimate an unknown phenotype network structure. This approach was inconsistent and overfitting was common, so we then propose a Bayesian approach that extends on the reversible jump Markov chain Monte Carlo algorithm. Our approach makes use of maximum likelihood estimates in the chain, so we have an efficient sampler using well-tuned proposal distributions. The common approach is to assume uniform priors over all network structures; however, we introduce a prior on the number of edges in the phenotype network structure, which prefers simple models with fewer directed edges. We determine that the relationship between the prior penalty and the joint posterior probability of the true model is not monotonic, there is some interplay between the two. Simulation studies were carried out and our approach is also applied to a published data set. It is determined that larger trait-to-trait effects are required to recover the phenotype network structure; however, mixing is generally slow, a common occurrence with reversible jump Markov chain Monte Carlo methods. We propose the use of a double step to combine two steps that alter the phenotype network structure. This proposes larger steps than the traditional birth and death move types, possibly changing the dimension of the model by more than one. This double step helped the sampler move between different phenotype network structures in simulated data sets.</p>

Forecasting Software Using Laplacian AR Model based on Bootstrap-Reversible Jump MCMC: Application on Stock Price Data

The application of the Bootstrap-Metropolis-Hastings algorithm is limited to fixed dimension models. In various fields, data often has a variable dimension model. The Laplacian autoregressive (AR) model includes a variable dimension model so that the Bootstrap-Metropolis-Hasting algorithm cannot be applied. This article aims to develop a Bootstrap reversible jump Markov Chain Monte Carlo (MCMC) algorithm to estimate the Laplacian AR model. The parameters of the Laplacian AR model were estimated using a Bayesian approach. The posterior distribution has a complex structure so that the Bayesian estimator cannot be calculated analytically. The Bootstrap-reversible jump MCMC algorithm was applied to calculate the Bayes estimator. This study provides a procedure for estimating the parameters of the Laplacian AR model. Algorithm performance was tested using simulation studies. Furthermore, the algorithm is applied to the finance sector to predict stock price on the stock market. In general, this study can be useful for decision makers in predicting future events. The novelty of this study is the triangulation between the bootstrap algorithm and the reversible jump MCMC algorithm. The Bootstrap-reversible jump MCMC algorithm is useful especially when the data is large and the data has a variable dimension model. The study can be extended to the Laplacian Autoregressive Moving Average (ARMA) model.

On the Reversible Jump Markov Chain Monte Carlo (RJMCMC) Algorithm for Extreme Value Mixture Distribution as a Location-Scale Transformation of the Weibull Distribution

Data with a multimodal pattern can be analyzed using a mixture model. In a mixture model, the most important step is the determination of the number of mixture components, because finding the correct number of mixture components will reduce the error of the resulting model. In a Bayesian analysis, one method that can be used to determine the number of mixture components is the reversible jump Markov chain Monte Carlo (RJMCMC). The RJMCMC is used for distributions that have location and scale parameters or location-scale distribution, such as the Gaussian distribution family. In this research, we added an important step before beginning to use the RJMCMC method, namely the modification of the analyzed distribution into location-scale distribution. We called this the non-Gaussian RJMCMC (NG-RJMCMC) algorithm. The following steps are the same as for the RJMCMC. In this study, we applied it to the Weibull distribution. This will help many researchers in the field of survival analysis since most of the survival time distribution is Weibull. We transformed the Weibull distribution into a location-scale distribution, which is the extreme value (EV) type 1 (Gumbel-type for minima) distribution. Thus, for the mixture analysis, we call this EV-I mixture distribution. Based on the simulation results, we can conclude that the accuracy level is at minimum 95%. We also applied the EV-I mixture distribution and compared it with the Gaussian mixture distribution for enzyme, acidity, and galaxy datasets. Based on the Kullback–Leibler divergence (KLD) and visual observation, the EV-I mixture distribution has higher coverage than the Gaussian mixture distribution. We also applied it to our dengue hemorrhagic fever (DHF) data from eastern Surabaya, East Java, Indonesia. The estimation results show that the number of mixture components in the data is four; we also obtained the estimation results of the other parameters and labels for each observation. Based on the Kullback–Leibler divergence (KLD) and visual observation, for our data, the EV-I mixture distribution offers better coverage than the Gaussian mixture distribution.

Reversible Jump MCMC for Deghosting in MSPSR Systems

This paper deals with bistatic track association and deghosting in the classical frequency modulation (FM)-based multi-static primary surveillance radar (MSPSR). The main contribution of this paper is a novel algorithm for bistatic track association and deghosting. The proposed algorithm is based on a hierarchical model which uses the Indian buffet process (IBP) as the prior probability distribution for the association matrix. The inference of the association matrix is then performed using the classical reversible jump Markov chain Monte Carlo (RJMCMC) algorithm with the usage of a custom set of the moves proposed by the sampler. A detailed description of the moves together with the underlying theory and the whole model is provided. Using the simulated data, the algorithm is compared with the two alternative ones and the results show the significantly better performance of the proposed algorithm in such a simulated setup. The simulated data are also used for the analysis of the properties of Markov chains produced by the sampler, such as the convergence or the posterior distribution. At the end of the paper, further research on the proposed method is outlined.