Since its introduction in 2001, MrBayes has grown in popularity as a software package for Bayesian phylogenetic inference using Markov chain Monte Carlo (MCMC) methods. With this note, we announce the release of version 3.2, a major upgrade to the latest official release presented in 2003. The new version provides convergence diagnostics and allows multiple analyses to be run in parallel with convergence progress monitored on the fly. The introduction of new proposals and automatic optimization of tuning parameters has improved convergence for many problems. The new version also sports significantly faster likelihood calculations through streaming single-instruction-multiple-data extensions (SSE) and support of the BEAGLE library, allowing likelihood calculations to be delegated to graphics processing units (GPUs) on compatible hardware. Speedup factors range from around 2 with SSE code to more than 50 with BEAGLE for codon problems. Checkpointing across all models allows long runs to be completed even when an analysis is prematurely terminated. New models include relaxed clocks, dating, model averaging across time-reversible substitution models, and support for hard, negative, and partial (backbone) tree constraints. Inference of species trees from gene trees is supported by full incorporation of the Bayesian estimation of species trees (BEST) algorithms. Marginal model likelihoods for Bayes factor tests can be estimated accurately across the entire model space using the stepping stone method. The new version provides more output options than previously, including samples of ancestral states, site rates, site d(N)/d(S) rations, branch rates, and node dates. A wide range of statistics on tree parameters can also be output for visualization in FigTree and compatible software.

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MrBayes 3.2: Efficient Bayesian Phylogenetic Inference and Model Choice across a Large Model Space

Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

Meyn & Tweedie is back! The bible on Markov chains in general state spaces has been brought up to date to reflect developments in the field since 1996 - many of them sparked by publication of the first edition. The pursuit of more efficient simulation algorithms for complex Markovian models, or algorithms for computation of optimal policies for controlled Markov models, has opened new directions for research on Markov chains. As a result, new applications have emerged across a wide range of topics including optimisation, statistics, and economics. New commentary and an epilogue by Sean Meyn summarise recent developments and references have been fully updated. This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background.

Markov Chains and Stochastic Stability

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

We generalize the method proposed by Gelman and Rubin (1992a) for monitoring the convergence of iterative simulations by comparing between and within variances of multiple chains, in order to obtain a family of tests for convergence. We review methods of inference from simulations in order to develop convergence-monitoring summaries that are relevant for the purposes for which the simulations are used. We recommend applying a battery of tests for mixing based on the comparison of inferences from individual sequences and from the mixture of sequences. Finally, we discuss multivariate analogues, for assessing convergence of several parameters simultaneously.

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General methods for monitoring convergence of iterative simulations

We review and extend results related to optimal scaling of Metropolis–Hastings algorithms. We present various theoretical results for the high-dimensional limit. We also present simulation studies which confirm the theoretical results in finite-dimensional contexts.

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Optimal scaling for various Metropolis-Hastings algorithms

We investigate the use of adaptive MCMC algorithms to automatically tune the Markov chain parameters during a run. Examples include the Adaptive Metropolis (AM) multivariate algorithm of Haario, Saksman, and Tamminen (2001), Metropolis-within-Gibbs algorithms for nonconjugate hierarchical models, regionally adjusted Metropolis algorithms, and logarithmic scalings. Computer simulations indicate that the algorithms perform very well compared to nonadaptive algorithms, even in high dimension.

https://www.jstor.org/stable/pdf/25651249.pdf

Examples of Adaptive MCMC

This paper surveys various results about Markov chains on gen- eral (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sucient conditions for geomet- ric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift con- ditions. Necessary and sucient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equa- tion or direct regeneration constructions. Finally, optimal scaling and weak convergence results for Metropolis-Hastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.

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General state space Markov chains and MCMC algorithms

We review recent work concerning optimal proposal scalings for Metropolis-Hastings MCMC algorithms, and adaptive MCMC algorithms for trying to improve the algorithm on the y.

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Optimal Proposal Distributions and Adaptive MCMC

We consider the optimal scaling problem for proposal distributions in Hastings–Metropolis algorithms derived from Langevin diffusions. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterized by its overall acceptance rate, independently of the target distribution. The asymptotically optimal acceptance rate is 0.574. We show that, as a function of dimension n, the complexity of the algorithm is O(n1/3), which compares favourably with the O(n) complexity of random walk Metropolis algorithms. We illustrate this comparison with some example simulations.

Jeffrey S. Rosenthal

Papers

Optimal scaling for various Metropolis-Hastings algorithms

Examples of Adaptive MCMC

General state space Markov chains and MCMC algorithms

Optimal Proposal Distributions and Adaptive MCMC

Optimal scaling of discrete approximations to Langevin diffusions