Efficient MCMC estimation of inflated beta regression models

This paper introduces a new and computationally efficient Markov chain Monte Carlo (MCMC) estimation algorithm for the Bayesian analysis of zero, one, and zero and one inflated beta regression models. The algorithm is computationally efficient in the sense that it has low MCMC autocorrelations and computational time. A simulation study shows that the proposed algorithm outperforms the slice sampling and random walk Metropolis–Hastings algorithms in both small and large sample settings. An empirical illustration on a loss given default banking model demonstrates the usefulness of the proposed algorithm.     

Adaptive,Delayed-Acceptance MCMC for Targets With Expensive Likelihoods

When conducting Bayesian inference, delayed-acceptance (DA) Metropolis–Hastings (MH) algorithms and DA pseudo-marginal MH algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased estimate thereof, but a computationally cheap approximation is available. A first accept-reject stage is applied, with the cheap approximation substituted for the true posterior in the MH acceptance ratio. Only for those proposals that pass through the first stage is the computationally expensive true posterior (or unbiased estimate thereof) evaluated, with a second accept-reject stage ensuring that detailed balance is satisfied with respect to the intended true posterior. In some scenarios, there is no obvious computationally cheap approximation. A weighted average of previous evaluations of the computationally expensive posterior provides a generic approximation to the posterior. If only the k-nearest neighbors have nonzero weights then evaluation of the approximate posterior can be made computationally cheap provided that the points at which the posterior has been evaluated are stored in a multi-dimensional binary tree, known as a KD-tree. The contents of the KD-tree are potentially updated after every computationally intensive evaluation. The resulting adaptive, delayed-acceptance [pseudo-marginal] Metropolis–Hastings algorithm is justified both theoretically and empirically. Guidance on tuning parameters is provided and the methodology is applied to a discretely observed Markov jump process characterizing predator–prey interactions and an ODE system describing the dynamics of an autoregulatory gene network. Supplementary material for this article is available online.     

Efficient Adaptive MCMC Through Precision Estimation

The performance of Markov chain Monte Carlo (MCMC) algorithms like the Metropolis Hastings Random Walk (MHRW) is highly dependent on the choice of scaling matrix for the proposal distributions. A popular choice of scaling matrix in adaptive MCMC methods is to use the empirical covariance matrix (ECM) of previous samples. However, this choice is problematic if the dimension of the target distribution is large, since the ECM then converges slowly and is computationally expensive to use. We propose two algorithms to improve convergence and decrease computational cost of adaptive MCMC methods in cases when the precision (inverse covariance) matrix of the target density can be well-approximated by a sparse matrix. The first is an algorithm for online estimation of the Cholesky factor of a sparse precision matrix. The second estimates the sparsity structure of the precision matrix. Combining the two algorithms allows us to construct precision-based adaptive MCMC algorithms that can be used as black-box methods for densities with unknown dependency structures. We construct precision-based versions of the adaptive MHRW and the adaptive Metropolis adjusted Langevin algorithm and demonstrate the performance of the methods in two examples. Supplementary materials for this article are available online.     

NONLINEAR WAVELET SMOOTHING OF ERROR DENSITY IN A SEMIPARAMETRIC REGRESSION MODEL

In this paper,a semlparametrie resresaion model in which errors are i. i. d random variables from an unknown density f( ) is considered. Based on Hall et al. (1995),a nonlinear wavelet estimation of f( ) without restrictions of continuity everywhere on f( ) is given,and the convergence rate of the estimators in L2 is obtained.     

A new investigation of the extended Krylov subspace method for matrix function evaluations

For large square matrices A and functions f, the numerical approximation of the action of f(A) to a vector v has received considerable attention in the last two decades. In this paper we investigate theextended Krylov subspace method, a technique that was recently proposed to approximate f(A)v for A symmetric. We provide a new theoretical analysis of the method, which improves the original result for A symmetric, and gives a new estimate for A nonsymmetric. Numerical experiments confirm that the new error estimates correctly capture the linear asymptotic convergence rate of the approximation. By using recent algorithmic improvements, we also show that the method is computationally competitive with respect to other enhancement techniques. Copyright © 2009 John Wiley & Sons, Ltd.     

An algorithm of MCMC method for solving F(X) =0

An algorithm of continuous stage-space MCMC method for solving algebra equation f(x)=0 is given. It is available for the case that the sign of f(x) changes frequently or the derivative f′(x) does not exist in the neighborhood of the root, while the Newton method is hard to work. Let n be the number of random variables created by computer in our algorithm. Then after m=O(n) transactions from the initial value x ₀,x* can be got such that |f(x*)|<e ^−cm |f(x ₀)| by choosing suitable positive constant c. An illustration is also given with the discussion of convergence by adjusting the parameters in the algorithm. Supported by the National Natural Science Foundation of China (70171008).     

A generalization of bounded variation

Generalized Wiener classes are considered. For these classes the exact order of Fourier coefficients with respect to the trigonometric system is established and the estimation of ‖S _n(·, f)-f(·)‖_{C [0,2π]} where S _n(·, f) are the Fourier partial sums, is given. In particular, a uniform convergence criterion for the Fourier trigonometric series is obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

An Algorithm for Robust Inference for the Cox Model with Frailties

Abstract

This article proposes a robust method of statistical inference for the Cox's proportional hazards model with frailties. We use the Metropolis—Hastings algorithm and the bootstrap method. We present a computationally efficient algorithm with a customized data structure to implement this method and demonstrate this technique with real data.     

Two transformations of series that commute with compositional inversion

It is shown that the compositional inverse of either of two transformations of a given series can be determined from the compositional inverse of the series. Specifically, if t · f(t) and t · g(t) are compositional inverses, then so are t · f_k(t) and $\text{t · g}{}_{\mathrm{k}}{}^1\text{(t)}$, where f_k(t) is the kth Euler transformation of f(t) and $\text{g}{}_{\mathrm{k}}{}^1\text{(t) =}\text{g(t)}\text{(1 ? kt · g(t))}$.     

Structured Markov Chain Monte Carlo

Abstract

This article introduces a general method for Bayesian computing in richly parameterized models, structured Markov chain Monte Carlo (SMCMC), that is based on a blocked hybrid of the Gibbs sampling and Metropolis—Hastings algorithms. SMCMC speeds algorithm convergence by using the structure that is present in the problem to suggest an appropriate Metropolis—Hastings candidate distribution. Although the approach is easiest to describe for hierarchical normal linear models, we show that its extension to both nonnormal and nonlinear cases is straightforward. After describing the method in detail we compare its performance (in terms of run time and autocorrelation in the samples) to other existing methods, including the single-site updating Gibbs sampler available in the popular BUGS software package. Our results suggest significant improvements in convergence for many problems using SMCMC, as well as broad applicability of the method, including previously intractable hierarchical nonlinear model settings.     

Dual gauge programs,with applications to quadratic programming and the minimum-norm problem

A gauge functionf(·) is a nonnegative convex function that is positively homogeneous and satisfiesf(0)=0. Norms and pseudonorms are specific instances of a gauge function. This paper presents a gauge duality theory for a gauge program, which is the problem of minimizing the value of a gauge functionf(·) over a convex set. The gauge dual program is also a gauge program, unlike the standard Lagrange dual. We present sufficient conditions onf(·) that ensure the existence of optimal solutions to the gauge program and its dual, with no duality gap. These sufficient conditions are relatively weak and are easy to verify, and are independent of any qualifications on the constraints. The theory is applied to a class of convex quadratic programs, and to the minimuml _p norm problem. The gauge dual program is shown to provide a smaller duality than the standard dual, in a certain sense discussed in the text.     

A descent method for submodular function minimization

We show a descent method for submodular function minimization based on an oracle for membership in base polyhedra. We assume that for any submodular function f: ?→R on a distributive lattice ?⊆2 ^V with ?,V∈? and f(?)=0 and for any vector x∈R ^V where V is a finite nonempty set, the membership oracle answers whether x belongs to the base polyhedron associated with f and that if the answer is NO, it also gives us a set Z∈? such that x(Z)>f(Z). Given a submodular function f, by invoking the membership oracle O(|V|²) times, the descent method finds a sequence of subsets Z ₁,Z ₂,···,Z _k of V such that f(Z ₁)>f(Z ₂)>···>f(Z _k )=min{f(Y) | Y∈?}, where k is O(|V|²). The method furnishes an alternative framework for submodular function minimization if combined with possible efficient membership algorithms. Received: September 9, 2001 / Accepted: October 15, 2001?Published online December 6, 2001     

Bayesian Calibration and Uncertainty Analysis for Computationally Expensive Models Using Optimization and Radial Basis Function Approximation

We presenta Bayesian approach to model calibration when evaluation of the model is computationally expensive. Here, calibration is a nonlinear regression problem: given a data vector Y corresponding to the regression model f(β), find plausible values of β. As an intermediate step, Y and f are embedded into a statistical model allowing transformation and dependence. Typically, this problem is solved by sampling from the posterior distribution of β given Y using MCMC. To reduce computational cost, we limit evaluation of f to a small number of points chosen on a high posterior density region found by optimization.Then,we approximate the logarithm of the posterior density using radial basis functions and use the resulting cheap-to-evaluate surface in MCMC.We illustrate our approach on simulated data for a pollutant diffusion problem and study the frequentist coverage properties of credible intervals. Our experiments indicate that our method can produce results similar to those when the true “expensive” posterior density is sampled by MCMC while reducing computational costs by well over an order of magnitude.     

On the greedoid polynomial for rooted graphs and rooted digraphs

We examine some properties of the 2-variable greedoid polynomial f(G·,t,z) when G is the branching greedoid associated to a rooted graph or a rooted directed graph. For rooted digraphs, we show a factoring property of f(G·,t,z) determines whether or not the rooted digraph has a directed cycle. © 1993 John Wiley & Sons, Inc.     

Approximation of integrable functions by general linear operators of their fourier series

The pointwise estimates of the deviation T n,A,B f (·) f (·) in terms of moduli of continuity .f and w.f there are proved. Analogical results on norm approximation with remarks and corollaries are also given. In the results there are used the essentially weaker conditions than these in [Mittal, M. L.: J. Math. Anal. Appl., 220, 434-450 (1998) Theorem 1, p. 437].     

Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·)

We study the Riesz potentials I_αf on the generalized Lebesgue spaces L^p(·)(?^d), where 0 < α < d and I_αf(x) ? ∫equation/tex2gif-inf-3.gif |f(y)| |x – y|^{α – d} dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that I_α : L^p(·)(?^d) → L^p?(·)(?^d), where . If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to on ?^d such that there exists a bounded linear extension operator ? : W^1,p(·)(Ω) ? (?^d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings W^k,p(·)(?^d) ?L^p*(·)(R^d) with and W^1,p(·)(Ω) ? L^p*(·)(Ω) for k = 1. We show compactness of the embeddings W^1,p(·)(Ω) ? L^q(·)(Ω), whenever q(x) ≤ p*(x) – ε for some ε > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution dependence on initial conditions in differential variational inequalities

In the first part of this paper, we establish several sensitivity results of the solution x(t, ξ) to the ordinary differential equation (ODE) initial-value problem (IVP) dx/dt = f(x), x(0) =  ξ as a function of the initial value ξ for a nondifferentiable f(x). Specifically, we show that for $\Xi_T \equiv \{\,x(t,\xi^0): 0 \leq t \leq T\,\}$ , (a) if f is “B-differentiable” on $\Xi_T$ , then so is the solution operator x(t;·) at ξ⁰; (b) if f is “semismooth” on $\Xi_T$ , then so is x(t;·) at ξ⁰; (c) if f has a “linear Newton approximation” on $\Xi_T$ , then so does x(t;·) at ξ⁰; moreover, the linear Newton approximation of the solution operator can be obtained from the solution of a “linear” differential inclusion. In the second part of the paper, we apply these ODE sensitivity results to a differential variational inequality (DVI) and discuss (a) the existence, uniqueness, and Lipschitz dependence of solutions to subclasses of the DVI subject to boundary conditions, via an implicit function theorem for semismooth equations, and (b) the convergence of a “nonsmooth shooting method” for numerically computing such boundary-value solutions.     

A Generalized Markov Sampler

A recent development of the Markov chain Monte Carlo (MCMC) technique is the emergence of MCMC samplers that allow transitions between different models. Such samplers make possible a range of computational tasks involving models, including model selection, model evaluation, model averaging and hypothesis testing. An example of this type of sampler is the reversible jump MCMC sampler, which is a generalization of the Metropolis–Hastings algorithm. Here, we present a new MCMC sampler of this type. The new sampler is a generalization of the Gibbs sampler, but somewhat surprisingly, it also turns out to encompass as particular cases all of the well-known MCMC samplers, including those of Metropolis, Barker, and Hastings. Moreover, the new sampler generalizes the reversible jump MCMC. It therefore appears to be a very general framework for MCMC sampling. This paper describes the new sampler and illustrates its use in three applications in Computational Biology, specifically determination of consensus sequences, phylogenetic inference and delineation of isochores via multiple change-point analysis.     

Estimates for the zeros of differences of meromorphic functions

Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.     

Variance Bounding of Delayed-Acceptance Kernels

Methodology and Computing in Applied Probability - A delayed-acceptance version of a Metropolis–Hastings algorithm can be useful for Bayesian inference when it is computationally expensive to...