Abstract:
Introduction Vitamin D deficiency has been associated with increased risk of colorectal cancer (CRC), but causal relationship has not yet been confirmed. We investigate the direction of causation between vitamin D and CRC by extending the conventional approaches to allow pleiotropic relationships and by explicitly modelling unmeasured confounders. Methods Plasma 25-hydroxyvitamin D (25-OHD), genetic variants associated with 25-OHD and CRC, and other relevant information was available for 2645 individuals (1057 CRC cases and 1588 controls) and included in the model. We investigate whether 25-OHD is likely to be causally associated with CRC, or vice versa, by selecting the best modelling hypothesis according to Bayesian predictive scores. We examine consistency for a range of prior assumptions. Results Model comparison showed preference for the causal association between low 25-OHD and CRC over the reverse causal hypothesis. This was confirmed for posterior mean deviances obtained for both models (11.5 natural log units in favour of the causal model), and also for deviance information criteria (DIC) computed for a range of prior distributions. Overall, models ignoring hidden confounding or pleiotropy had significantly poorer DIC scores. Conclusion Results suggest causal association between 25-OHD and colorectal cancer, and support the need for randomised clinical trials for further confirmations.

Abstract:
In the Bayesian approach, the a priori knowledge about the input of a mathematical model is described via a probability measure. The joint distribution of the unknown input and the data is then conditioned, using Bayes' formula, giving rise to the posterior distribution on the unknown input. In this setting we prove posterior consistency for nonlinear inverse problems: a sequence of data is considered, with diminishing fluctuations around a single truth and it is then of interest to show that the resulting sequence of posterior measures arising from this sequence of data concentrates around the truth used to generate the data. Posterior consistency justifies the use of the Bayesian approach very much in the same way as error bounds and convergence results for regularisation techniques do. As a guiding example, we consider the inverse problem of reconstructing the diffusion coefficient from noisy observations of the solution to an elliptic PDE in divergence form. This problem is approached by splitting the forward operator into the underlying continuum model and a simpler observation operator based on the output of the model. In general, these splittings allow us to conclude posterior consistency provided a deterministic stability result for the underlying inverse problem and a posterior consistency result for the Bayesian regression problem with the push-forward prior. Moreover, we prove posterior consistency for the Bayesian regression problem based on the regularity, the tail behaviour and the small ball probabilities of the prior.

Abstract:
Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. We show that posterior inconsistency in tail index is surprisingly common for both parametric and nonparametric mixture models. We then present a set of sufficient conditions under which posterior consistency in tail index can be achieved, and verify these conditions for Pareto mixture models under general mixing priors.

Abstract:
We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.

Abstract:
In this paper, we consider the so-called Shape Invariant Model which stands for the estimation of a function $f^0$ submitted to a random translation of law $g^0$ in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process $\PP_{f^0,g^0}$ as well as $f^0$ and $g^0$. In this perspective, we adopt a Bayesian point of view and find prior on $f$ and $g$ such that the posterior distribution concentrates around $\PP_{f^0,g^0}$ at a polynomial rate when $n$ goes to $+\infty$. We obtain a logarithmic posterior contraction rate for the shape $f^0$ and the distribution $g^0$. We also derive logarithmic lower bounds for the estimation of $f^0$ and $g^0$ in a frequentist paradigm.

Abstract:
Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The measurement model for an unknown Gaussian random variable $U(x,\omega)$ is \begin{eqnarray*} M(y,\omega) = A(U(x,\omega) )+ \delta\hspace{.2mm}\mathcal{E}(y,\omega), \end{eqnarray*} where $A$ is a finitely many times smoothing linear hypoelliptic operator and $\delta>0$ is the noise magnitude. The covariance operator $C_U$ of $U$ is $2r$ times smoothing, self-adjoint, injective and elliptic pseudodifferential operator. If $\mathcal{E}$ was taking values in $L^2$ then in Gaussian case solving the conditional mean (and maximum a posteriori) estimate is linked to solving the minimisation problem \begin{eqnarray*} T_\delta(M) = \text{argmin}_{u\in H^r} \big\{\|A u-m\|_{L^2}^2+ \delta^2\|C_U^{-1/2}u\|_{L^2}^2 \big\}. \end{eqnarray*} However, Gaussian white noise does not take values in $L^2$ but in $H^{-s}$ where $s>0$ is big enough. A modification of the above approach to solve the inverse problem is presented, covering the case of white Gaussian measurement noise. Furthermore, the convergence of conditional mean estimate to the correct solution as $\delta\rightarrow 0$ is proven in appropriate function spaces using microlocal analysis. Also the contraction of the credible set is studied.

Abstract:
In recent years, the literature in the area of Bayesian asymptotics has been rapidly growing. It is increasingly important to understand the concept of posterior consistency and validate specific Bayesian methods, in terms of consistency of posterior distributions. In this paper, we build up some conceptual issues in consistency of posterior distributions, and discuss panoramic views of them by comparing various approaches to posterior consistency that have been investigated in the literature. In addition, we provide interesting results on posterior consistency that deal with non-exponential consistency, improper priors and non i.i.d. (independent but not identically distributed) observations. We describe a few examples for illustrative purposes.

Abstract:
Frequentist conditions for asymptotic suitability of Bayesian procedures focus on lower bounds for prior mass in Kullback-Leibler neighbourhoods of the data distribution. The goal of this paper is to investigate the flexibility in criteria for posterior consistency in the context of i.i.d. samples. We formulate a versatile posterior consistency theorem that applies both to well- and mis-specified models and which we use to re-derive Schwartz's theorem, consider Kullback-Leibler consistency and formulate consistency theorems in which priors charge metric balls. We also generalize to sieved models with Barron's negligible prior mass condition and to separable models with variations on Walker's consistency theorem. Results also apply to marginal semi-parametric consistency: support boundary estimation is considered explicitly and posterior consistency is proved in a model where the Kullback-Leibler condition cannot be satisfied by any prior. Other applications include Hellinger consistent density estimation in general mixture models with Dirichlet or Gibbs-type priors of full weak support.

Abstract:
Approximate Bayesian computation (ABC) methods have become increasingly prevalent of late, facilitating as they do the analysis of intractable, or challenging, statistical problems. With the initial focus being primarily on the practical import of ABC, exploration of its formal statistical properties has begun to attract more attention. The aim of this paper is to establish general conditions under which ABC methods are Bayesian consistent, in the sense of producing draws that yield a degenerate posterior distribution at the true parameter (vector) asymptotically (in the sample size). We derive conditions under which arbitrary summary statistics yield consistent inference in the Bayesian sense, with these conditions linked to identification of the true parameters. Using simple illustrative examples that have featured in the literature, we demonstrate that identification, and hence consistency, is unlikely to be achieved in many cases, and propose a simple diagnostic procedure that can indicate the presence of this problem. We also formally explore the link between consistency and the use of auxiliary models within ABC, and illustrate the subsequent results in the Lotka-Volterra predator-prey model.

Abstract:
This paper addresses the estimation of the nonparametric conditional moment restricted model that involves an infinite-dimensional parameter $g_0$. We estimate it in a quasi-Bayesian way, based on the limited information likelihood, and investigate the impact of three types of priors on the posterior consistency: (i) truncated prior (priors supported on a bounded set), (ii) thin-tail prior (a prior that has very thin tail outside a growing bounded set) and (iii) normal prior with nonshrinking variance. In addition, $g_0$ is allowed to be only partially identified in the frequentist sense, and the parameter space does not need to be compact. The posterior is regularized using a slowly growing sieve dimension, and it is shown that the posterior converges to any small neighborhood of the identified region. We then apply our results to the nonparametric instrumental regression model. Finally, the posterior consistency using a random sieve dimension parameter is studied.