Efficient Marginal Likelihood Computation for Gaussian Process Regression

In a Bayesian learning setting, the posterior distribution of a predictive model arises from a trade-off between its prior distribution and the conditional likelihood of observed data. Such distribution functions usually rely on additional hyperparameters which need to be tuned in order to achieve optimum predictive performance; this operation can be efficiently performed in an Empirical Bayes fashion by maximizing the posterior marginal likelihood of the observed data. Since the score function of this optimization problem is in general characterized by the presence of local optima, it is necessary to resort to global optimization strategies, which require a large number of function evaluations. Given that the evaluation is usually computationally intensive and badly scaled with respect to the dataset size, the maximum number of observations that can be treated simultaneously is quite limited. In this paper, we consider the case of hyperparameter tuning in Gaussian process regression. A straightforward implementation of the posterior log-likelihood for this model requires O(N^3) operations for every iteration of the optimization procedure, where N is the number of examples in the input dataset. We derive a novel set of identities that allow, after an initial overhead of O(N^3), the evaluation of the score function, as well as the Jacobian and Hessian matrices, in O(N) operations. We prove how the proposed identities, that follow from the eigendecomposition of the kernel matrix, yield a reduction of several orders of magnitude in the computation time for the hyperparameter optimization problem. Notably, the proposed solution provides computational advantages even with respect to state of the art approximations that rely on sparse kernel matrices.

The marginal willingness-to-pay for attributes of a hypothetical HIV vaccine

This paper estimates the marginal willingness-to-pay for attributes of a hypothetical HIV vaccine using discrete choice modeling. We use primary data from 326 respondents from Bangkok and Chiang Mai, Thailand, in 2008–2009, selected using purposive, venue-based sampling across two strata. Participants completed a structured questionnaire and full rank discrete choice modeling task administered using computer-assisted personal interviewing. The choice experiment was used to rank eight hypothetical HIV vaccine scenarios, with each scenario comprising seven attributes (including cost) each of which had two levels. The data were analyzed in two alternative specifications: (1) best-worst; and (2) full-rank, using logit likelihood functions estimated with custom routines in Gauss matrix programming language. In the full-rank specification, all vaccine attributes are significant predictors of probability of vaccine choice. The biomedical attributes of the hypothetical HIV vaccine (efficacy, absence of VISP, absence of side effects, and duration of effect) are the most important attributes for HIV vaccine choice. On average respondents are more than twice as likely to accept a vaccine with 99% efficacy, than a vaccine with 50% efficacy. This translates to a willingness to pay US$383 more for a high efficacy vaccine compared with the low efficacy vaccine. Knowledge of the relative importance of determinants of HIV vaccine acceptability is important to ensure the success of future vaccination programs. Future acceptability studies of hypothetical HIV vaccines should use more finely grained biomedical attributes, and could also improve the external validity of results by including more levels of the cost attribute.

Maximum likelihood estimation of higher-order integer-valued autoregressive processes

In this article, we extend the earlier work of Freeland and McCabe [Journal of time Series Analysis (2004) Vol. 25, pp. 701–722] and develop a general framework for maximum likelihood (ML) analysis of higher-order integer-valued autoregressive processes. Our exposition includes the case where the innovation sequence has a Poisson distribution and the thinning is binomial. A recursive representation of the transition probability of the model is proposed. Based on this transition probability, we derive expressions for the score function and the Fisher information matrix, which form the basis for ML estimation and inference. Similar to the results in Freeland and McCabe (2004), we show that the score function and the Fisher information matrix can be neatly represented as conditional expectations. Using the INAR(2) speci?cation with binomial thinning and Poisson innovations, we examine both the asymptotic e?ciency and ?nite sample properties of the ML estimator in relation to the widely used conditional least
squares (CLS) and Yule–Walker (YW) estimators. We conclude that, if the Poisson assumption can be justi?ed, there are substantial gains to be had from using ML especially when the thinning parameters are large.

Modeling Multivariate Interest Rates Using Time-Varying Copulas and Reducible Nonlinear Stochastic Differential Equations

We propose a new approach for modeling nonlinear multivariate interest rate processes based on time-varying copulas and reducible stochastic differential equations (SDEs). In the modeling of the marginal processes, we consider a class of nonlinear SDEs that are reducible to Ornstein--Uhlenbeck (OU) process or Cox, Ingersoll, and Ross (1985) (CIR) process. The reducibility is achieved via a nonlinear transformation function. The main advantage of this approach is that these SDEs can account for nonlinear features, observed in short-term interest rate series, while at the same time leading to exact discretization and closed-form likelihood functions. Although a rich set of specifications may be entertained, our exposition focuses on a couple of nonlinear constant elasticity volatility (CEV) processes, denoted as OU-CEV and CIR-CEV, respectively. These two processes encompass a number of existing models that have closed-form likelihood functions. The transition density, the conditional distribution function, and the steady-state density function are derived in closed form as well as the conditional and unconditional moments for both processes. In order to obtain a more flexible functional form over time, we allow the transformation function to be time varying. Results from our study of U.S. and UK short-term interest rates suggest that the new models outperform existing parametric models with closed-form likelihood functions. We also find the time-varying effects in the transformation functions statistically significant. To examine the joint behavior of interest rate series, we propose flexible nonlinear multivariate models by joining univariate nonlinear processes via appropriate copulas. We study the conditional dependence structure of the two rates using Patton (2006a) time-varying symmetrized Joe--Clayton copula. We find evidence of asymmetric dependence between the two rates, and that the level of dependence is positively related to the level of the two rates. (JEL: C13, C32, G12) Copyright The Author 2010. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org, Oxford University Press.

Maximum likelihood estimation of higher-order integer-valued autoregressive processes

Corrigendum Vol. 30, Issue 2, 259, Article first published online: 15 MAR 2009 to correct the order of authors names: Bu R., K. Hadri, and B. McCabe.

Predicting the likelihood of carrying a BRCA1 or BRCA2 mutation: validation of BOADICEA, BRCAPRO, IBIS, Myriad and the Manchester scoring system using data from UK genetics clinics

Objectives: Genetic testing for the breast and ovarian cancer susceptibility genes BRCA1 and BRCA2 has important implications for the clinical management of people found to carry a mutation. However, genetic testing is expensive and may be associated with adverse psychosocial effects. To provide a cost-efficient and clinically appropriate genetic counselling service, genetic testing should be targeted at those individuals most likely to carry pathogenic mutations. Several algorithms that predict the likelihood of carrying a BRCA1 or a BRCA2 mutation are currently used in clinical practice to identify such individuals.

Predator diversity enhances secondary production and decreases the likelihood of trophic cascades

We manipulated the diversity of top predators in a three trophic level marine food web. The food web included four top benthic marine fish predators (black goby, rock goby, sea scorpion and shore rockling), an intermediate trophic level of small fish, and a lower trophic level of benthic invertebrates. We kept predator density constant and monitored the response of the lower trophic levels. As top predator diversity increased, secondary production increased. We also observed that in the presence of the manipulated fish predators, the density of small gobiid fish (intermediate consumers) was suppressed, releasing certain groups of benthic invertebrates (caprellid amphipods, copepods, nematodes and spirorbid worms) from heavy intermediate predation pressure. We attribute the mechanism responsible for this trophic cascade to a trait-mediated indirect interaction, with the small gobiid fish changing their use of space in response to altered predator diversity. In the absence of top fish predators, a full-blown trophic cascade occurs. Therefore the diversity of predators reduces the likelihood of trophic cascades occurring and hence provides insurance against the loss of an important ecosystem function (i.e. secondary production).

Social support and the likelihood of maintaining and improving levels of physical activity:the Whitehall II Study

Background: Evidence on the association between social support and leisure time physical activity (LTPA) is scarce and mostly based on cross-sectional data with different types of social support collapsed into a single index. The aim of this study was to investigate whether social support from the closest person was associated with LTPA.

mfx: Marginal Effects, Odds Ratios and Incidence Rate Ratios for GLMs

Estimates probit, logit, Poisson, negative binomial, and beta regression models, returning their marginal effects, odds ratios, or incidence rate ratios as an output.

Nonresident Father Involvement with Children and Divorced Women's Likelihood of Remarriage

Although remarriage is a relatively common transition, little is known about how nonresident fathers affect divorced mothers’ entry into remarriage. Using the 1979–2010 rounds of the National Longitudinal Study of Youth 1979, the authors examined the likelihood of remarriage for divorced mothers (N = 882) by nonresident father contact with children and payment of child support. The findings suggest that maternal remarriage is positively associated with nonresident father contact but not related to receiving child support.