Accelerating Parallel Tempering: Quantile Tempering Algorithm (QuanTA)

Using MCMC to sample from a target distribution, $\pi(x)$ on a $d$-dimensional state space can be a difficult and computationally expensive problem. Particularly when the target exhibits multimodality, then the traditional methods can fail to explore the entire state space and this results in a bias sample output. Methods to overcome this issue include the parallel tempering algorithm which utilises an augmented state space approach to help the Markov chain traverse regions of low probability density and reach other modes. This method suffers from the curse of dimensionality which dramatically slows the transfer of mixing information from the auxiliary targets to the target of interest as $d \rightarrow \infty$. This paper introduces a novel prototype algorithm, QuanTA, that uses a Gaussian motivated transformation in an attempt to accelerate the mixing through the temperature schedule of a parallel tempering algorithm. This new algorithm is accompanied by a comprehensive theoretical analysis quantifying the improved efficiency and scalability of the approach; concluding that under weak regularity conditions the new approach gives accelerated mixing through the temperature schedule. Empirical evidence of the effectiveness of this new algorithm is illustrated on canonical examples

Weight-Preserving Simulated Tempering

Simulated tempering is popular method of allowing MCMC algorithms to move between modes of a multimodal target density {\pi}. One problem with simulated tempering for multimodal targets is that the weights of the various modes change for different inverse-temperature values, sometimes dramatically so. In this paper, we provide a fix to overcome this problem, by adjusting the mode weights to be preserved (i.e., constant) over different inverse-temperature settings. We then apply simulated tempering algorithms to multimodal targets using our mode weight correction. We present simulations in which our weight-preserving algorithm mixes between modes much more successfully than traditional tempering algorithms. We also prove a diffusion limit for an version of our algorithm, which shows that under appropriate assumptions, our algorithm mixes in time O(d [log d]^2)

A utility based approach to information for stochastic differential equations

AbstractA Bayesian perspective is taken to quantify the amount of information learned from observing a stochastic process, Xt, on the interval [0, T] which satisfies the stochastic differential equation, dXt = S(θ, t, Xt)dt+σ(t, Xt)dBt. Information is defined as a change in expected utility when the experimenter is faced with the decision problem of reporting beliefs about the parameter of interest θ. For locally asymptotic mixed normal families we establish an asymptotic relationship between the Shannon information of the posterior and Fisher's information of the process. In particular we compute this measure for the linear case (S(θ, t, Xt) = θS(t, Xt)), Brownian motion with drift, the Ornstein-Uhlenbeck process and the Bessel process

Time-resolved velocity map imaging of methyl elimination from photoexcited anisole

To date, H-atom elimination from heteroaromatic molecules following UV excitation has been extensively studied, with the focus on key biological molecules such as chromophores of DNA bases and amino acids. Extending these studies to look at elimination of other non-hydride photoproducts is essential in creating a more complete picture of the photochemistry of these biomolecules in the gas-phase. To this effect, CH3 elimination in anisole has been studied using time resolved velocity map imaging (TR-VMI) for the first time, providing both time and energy information on the dynamics following photoexcitation at 200 nm. The extra dimension of energy afforded by these measurements has enabled us to address the role of πσ* states in the excited state dynamics of anisole as compared to the hydride counterpart (phenol), providing strong evidence to suggest that only CH3 fragments eliminated with high kinetic energy are due to direct dissociation involving a 1πσ* state. These measurements also suggest that indirect mechanisms such as statistical unimolecular decay could be contributing to the dynamics at much longer times

Exact Bayesian inference for diffusion driven Cox processes

In this paper we present a novel methodology to perform Bayesian inference for Cox processes in which the intensity function is driven by a diffusion process. The novelty lies on the fact that no discretisation error is involved, despite the non-tractability of both the likelihood function and the transition density of the diffusion. The methodology is based on an MCMC algorithm and its exactness is built on retrospective sampling techniques. The efficiency of the methodology is investigated in some simulated examples and its applicability is illustrated in some real data analyses

Major‐element composition of sediments in terms of weathering and provenance: Implications for crustal recycling

The elemental composition of a sediment is set by the composition of its protolith and modified by weathering, sorting, and diagenesis. An important problem is deconvolving these contributions to a sediment's composition to arrive at information about processes that operate on the Earth's surface. We approach this problem by developing a predictive and invertible model of sedimentary major‐element composition. We compile a dataset of sedimentary rock, river sediment, soil, and igneous rock compositions. Principal component analysis of the dataset shows that most variation can be simplified to a small number of variables. We thus show that any sediment's composition can be described with just two vectors of igneous evolution and weathering. We hence define a model for sedimentary composition as a combination of these processes. A 1:1 correspondence is observed between predictions and independent data. The log‐ratios ln(K2O/MgO) and ln(Al2O3/Na2O) are found to be simple proxies for, respectively, the model's protolith and weathering indices. Significant deviations from the model can be explained by sodium‐calcium exchange. Using this approach, we show that the major‐element composition of the upper continental crust has been modified by weathering and we calculate the amount of each element that it must have lost to modify it to its present composition. By extrapolating modern weathering rates over the age of the crust we conclude that it has not retained a significant amount of the necessarily produced weathering restite. This restite has likely been subducted into the mantle, indicating a crust‐to‐mantle recycling rate of 1.33 ± 0.89×1013 kg yr‐1

CLTs and asymptotic variance of time-sampled Markov chains

For a Markov transition kernel P and a probability distribution μ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel Pμ = Σkμ(k)Pk. In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker's and Metropolis algorithms in terms of asymptotic variance