Approximate Bayesian Computation (ABC) is a popular computational method for likelihood-free Bayesian inference. The term "likelihoodfree" refers to problems where the likelihood is intractable to compute or estimate directly, but where it is possible to generate simulated data X relatively easily given a candidate set of parameters θ simulated from a prior distribution. Parameters which generate simulated data within some tolerance δ of the observed data x * are regarded as plausible, and a collection of such θ is used to estimate the posterior distribution θ | X = x * . Suitable choice of δ is vital for ABC methods to return good approximations to θ in reasonable computational time.While ABC methods are widely used in practice, particularly in population genetics, rigorous study of the mathematical properties of ABC estimators lags behind practical developments of the method. We prove that ABC estimates converge to the exact solution under very weak assumptions and, under slightly stronger assumptions, quantify the rate of this convergence. In particular, we show that the bias of the ABC estimate is asymptotically proportional to δ 2 as δ ↓ 0. At the same time, the computational cost for generating one ABC sample increases like δ −q where q is the dimension of the observations. Rates of convergence are obtained by optimally balancing the mean squared error against the computational cost. Our results can be used to guide the choice of the tolerance parameter δ. keywords: Approximate Bayesian Computation, likelihood-free inference, Monte Carlo methods, convergence of estimators, rate of convergence MSC2010 classes: 62F12 (Asymptotic properties of estimators), 62F15 (Bayesian inference), 65C05 (Monte Carlo methods)

Amyotrophic lateral sclerosis (ALS) is an adult-onset neurodegenerative disease, characterized by progressive dysfunction and death of motor neurons. Although evidence for oxidative stress in ALS pathogenesis is well described, antioxidants have generally shown poor efficacy in animal models and human clinical trials. We have developed an in vitro screening cascade to identify antioxidant molecules capable of rescuing NSC34 motor neuron cells expressing an ALS-associated mutation of superoxide dismutase 1. We have tested known antioxidants and screened a library of 2000 small molecules. The library screen identified 164 antioxidant molecules, which were refined to the 9 most promising molecules in subsequent experiments. Analysis of the in silico properties of hit compounds and a review of published literature on their in vivo effectiveness have enabled us to systematically identify molecules with antioxidant activity combined with chemical properties necessary to penetrate the central nervous system. The top-performing molecules identified include caffeic acid phenethyl ester, esculetin, and resveratrol. These compounds were tested for their ability to rescue primary motor neuron cultures after trophic factor withdrawal, and the mechanisms of action of their antioxidant effects were investigated. Subsequent in vivo studies can be targeted using molecules with the greatest probability of success.

Wavelet shrinkage is an effective nonparametric regression technique, especially when the underlying curve has irregular features such as spikes or discontinuities. The basic idea is simple: take the discrete wavelet transform of data consisting of a signal corrupted by noise; shrink or remove the wavelet coefficients to remove the noise; then invert the discrete wavelet transform to form an estimate of the true underlying curve. Various researchers have proposed increasingly sophisticated methods of doing this by using real-valued wavelets. Complex-valued wavelets exist but are rarely used. We propose two new complex-valued wavelet shrinkage techniques: one based on multiwavelet style shrinkage and the other using Bayesian methods. Extensive simulations show that our methods almost always give significantly more accurate estimates than methods based on real-valued wavelets. Further, our multiwavelet style shrinkage method is both simpler and dramatically faster than its competitors. To understand the excellent performance of this method we present a new risk bound on its hard thresholded coefficients. Copyright 2004 Royal Statistical Society.

We extend the optimal symmetric group sequential tests of Eales & Jennison (1992) to the broader class of asymmetric designs. Two forms of asymmetry are considered; unequal type I and type II error rates and different emphases on expected sample sizes at the null and alternative hypotheses. We discuss the properties of our optimal designs and use them to assess the efficiency of the family of tests proposed by Pampallona & Tsiatis (1994) and two families of one-sided tests defined through error spending functions. We show that the error spending designs are highly efficient, while the easily implemented tests of Pampallona & Tsiatis are a little less efficient but still not far from optimal. Our results demonstrate that asymmetric designs can decrease the expected sample size under one hypothesis, but only at the expense of a significantly larger expected sample size under the other hypothesis.

There is a need for a reliable statistical test which is appropriate for assessing cospeciation of more than two phylogenies. We have developed an algorithm using a permutation method that can be used to test for and infer tri-trophic evolutionary relationships of organisms given both their phylogenies and pairwise interactions.An overall statistic has been developed based on the dominant eigenvalue of a covariance matrix, and compared to values of the statistic computed when tree labels are permuted. The resulting overall p-value is used to test for the presence or absence of cospeciation in a tri-trophic system. If cospeciation is detected, we propose new test statistics based on partial correlations to uncover more details about the relationships between multiple phylogenies.One of the strengths of our method is that it allows more parasites than hosts or more hosts than parasites, with multiple associations and more than one parasite attached to a host (or one parasite attached to multiple hosts). The new method does not require any parametric assumptions of the distribution of the data, and unlike the old methods, which utilise several pairwise steps, the overall statistic used is obtained in one step.We have applied our method to two published datasets where we obtained detailed information about the strength of associations among species with calculated partial p-values and one overall p-value from the dominant eigenvalue test statistic.Our permutation method produces reliable results with a clear procedure and statistics applied in an intuitive manner. Our algorithm is useful in testing evidence for three-way cospeciation in multiple phylogenies with tri-trophic associations and determining which phylogenies are involved in cospeciation.