Design of Hamiltonian Monte Carlo for perfect simulation of general continuous distributions

Hamiltonian Monte Carlo (HMC) is an efficient method of simulating smooth distributions and has motivated the widely used No-U-turn Sampler (NUTS) and software Stan. We build on NUTS and the technique of “unbiased sampling” to design HMC algorithms that produce perfect simulation of general continuous distributions that are amenable to HMC. Our methods enable separation of Markov chain Monte Carlo convergence error from experimental error, and thereby provide much more powerful MCMC convergence diagnostics than current state-of-the-art summary statistics which confound these two errors. Objective comparison of different MCMC algorithms is provided by the number of derivative evaluations per perfect sample point. We demonstrate the methodology with applications to normal, t and normal mixture distributions up to 100 dimensions, and a 12-dimensional Bayesian Lasso regression. HMC runs effectively with a goal of 20 to 30 points per trajectory. Numbers of derivative evaluations per perfect sample point range from 390 for a univariate normal distribution to 12,000 for a 100-dimensional mixture of two normal distributions with modes separated by six standard deviations, and 22,000 for a 100-dimensional t-distribution with four degrees of freedom

Understanding environmental and fisheries factors causing fluctuations in mud crab and blue swimmer crab fisheries in Northern Australia to inform harvest strategies

The current project investigated relationships between environmental factors and harvests of crabs in the Gulf of Carpentaria (GoC), northern Australia. This was in response to industry and managerial concerns about consistent declines in harvests of GoC Giant Mud Crab (Scylla serrata). In the orthern Territory (NT), declines occurred between 2009 and 2016, whilst in Queensland (Qld), declines occurred between 2013 and 2016. The declines occurred despite different management arrangements (e.g. NT harvests females, whereas Qld does not), suggesting common environmental factors were involved

Perfect simulation from unbiased simulation

We show that any application of the technique of unbiased simulation becomes perfect simulation when coalescence of the two coupled Markov chains can be practically assured in advance. This happens when a fixed number of iterations is high enough that the probability of needing any more to achieve coalescence is negligible; we suggest a value of $10^{-20}$. This finding enormously increases the range of problems for which perfect simulation, which exactly follows the target distribution, can be implemented. We design a new algorithm to make practical use of the high number of iterations by producing extra perfect sample points with little extra computational effort, at a cost of a small, controllable amount of serial correlation within sample sets of about 20 points. Different sample sets remain completely independent. The algorithm includes maximal coupling for continuous processes, to bring together chains that are already close. We illustrate the methodology on a simple, two-state Markov chain and on standard normal distributions up to 20 dimensions. Our technical formulation involves a nonzero probability, which can be made arbitrarily small, that a single perfect sample point may have its place taken by a "string" of many points which are assigned weights, each equal to $\pm 1$, that sum to~$1$. A point with a weight of $-1$ is a "hole", which is an object that can be cancelled by an equivalent point that has the same value but opposite weight $+1$.Comment: 17 pages, 4 figures; for associated R scripts, see https://github.com/George-Leigh/PerfectSimulatio

Not All Kinds of Revegetation Are Created Equal: Revegetation Type Influences Bird Assemblages in Threatened Australian Woodland Ecosystems

The value for biodiversity of large intact areas of native vegetation is well established. The biodiversity value of regrowth vegetation is also increasingly recognised worldwide. However, there can be different kinds of revegetation that have different origins. Are there differences in the richness and composition of biotic communities in different kinds of revegetation? The answer remains unknown or poorly known in many ecosystems. We examined the conservation value of different kinds of revegetation through a comparative study of birds in 193 sites surveyed over ten years in four growth types located in semi-cleared agricultural areas of south-eastern Australia. These growth types were resprout regrowth, seedling regrowth, plantings, and old growth

Average counts for growth types in a particular year and season.

<p>Average counts for growth types in a particular year and season.</p

Correspondence analysis biplots of bird species and growth type.

<p>The diagrams are: (left) first versus third dimensions from correspondence analysis and (right) second and third dimensions from correspondence analysis. Distances between species approximate the chi-squared distance between species distributions (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034527#pone.0034527-Lep1" target="_blank">[59]</a> for details of the approach used in data analyses).</p

The South-west Slopes study region of southern New South Wales, south-eastern Australia.

<p>The South-west Slopes study region of southern New South Wales, south-eastern Australia.</p

Biplot of the first two canonical axes showing species and growth types.

<p>Distances between species approximate the chi-squared distance between species distributions (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0034527#pone.0034527-Lep1" target="_blank">[59]</a> for details of the approach used in data analyses).</p