Savage-Dickey Density Ratio Research Articles

Optimizing future dark energy surveys for model selection goals

We demonstrate a methodology for optimizing the ability of future dark energy surveys to answer model selection questions, such as `Is acceleration due to a cosmological constant or a dynamical dark energy model?'. Model selection Figures of Merit are defined, exploiting the Bayes factor, and surveys optimized over their design parameter space via a Monte Carlo method. As a specific example we apply our methods to generic multi-fibre baryon acoustic oscillation spectroscopic surveys, comparable to that proposed for SuMIRe PFS, and present implementations based on the Savage-Dickey Density Ratio that are both accurate and practical for use in optimization. It is shown that whilst the optimal surveys using model selection agree with those found using the Dark Energy Task Force (DETF) Figure of Merit, they provide better informed flexibility of survey configuration and an absolute scale for performance; for example, we find survey configurations with close to optimal model selection performance despite their corresponding DETF Figure of Merit being at only 50% of its maximum. This Bayes factor approach allows us to interpret the survey configurations that will be good enough for the task at hand, vital especially when wanting to add extra science goals and in dealing with time restrictions or multiple probes within the same project.

Post-hoc selection of dynamic causal models

Dynamic causal modelling (DCM) was originally proposed as a hypothesis driven procedure in which a small number of neurobiologically motivated models are compared. Model comparison in this context usually proceeds by individually fitting each model to data and then approximating the corresponding model evidence with a free energy bound. However, a recent trend has emerged for comparing very large numbers of models in a more exploratory manner. This led Friston and Penny (2011) to propose a post-hoc approximation to the model evidence, which is computed by optimising only the largest (full) model of a set of models. The evidence for any (reduced) submodel is then obtained using a generalisation of the Savage-Dickey density ratio (Dickey, 1971). The benefit of this post-hoc approach is a huge reduction in the computational time required for model fitting. This is because only a single model is fitted to data, allowing a potentially huge model space to be searched relatively quickly. In this paper, we explore the relationship between the free energy bound and post-hoc approximations to the model evidence in the context of deterministic (bilinear) dynamic causal models (DCMs) for functional magnetic resonance imaging data.

Using MCMC chain outputs to efficiently estimate Bayes factors

One of the most important methodological problems in psychological research is assessing the reasonableness of null models, which typically constrain a parameter to a specific value such as zero. Bayes factor has been recently advocated in the statistical and psychological literature as a principled means of measuring the evidence in data for various models, including those where parameters are set to specific values. Yet, it is rarely adopted in substantive research, perhaps because of the difficulties in computation. Fortunately, for this problem, the Savage–Dickey density ratio ( Dickey & Lientz, 1970) provides a conceptually simple approach to computing Bayes factor. Here, we review methods for computing the Savage–Dickey density ratio, and highlight an improved method, originally suggested by Gelfand and Smith (1990) and advocated by Chib (1995), that outperforms those currently discussed in the psychological literature. The improved method is based on conditional quantities, which may be integrated by Markov chain Monte Carlo sampling to estimate Bayes factors. These conditional quantities efficiently utilize all the information in the MCMC chains, leading to accurate estimation of Bayes factors. We demonstrate the method by computing Bayes factors in one-sample and one-way designs, and show how it may be implemented in WinBUGS.

Post hoc Bayesian model selection

This note describes a Bayesian model selection or optimization procedure for post hoc inferences about reduced versions of a full model. The scheme provides the evidence (marginal likelihood) for any reduced model as a function of the posterior density over the parameters of the full model. It rests upon specifying models through priors on their parameters, under the assumption that the likelihood remains the same for all models considered. This provides a quick and efficient scheme for scoring arbitrarily large numbers of models, after inverting a single (full) model. In turn, this enables the selection among discrete models that are distinguished by the presence or absence of free parameters, where free parameters are effectively removed from the model using very precise shrinkage priors. An alternative application of this post hoc model selection considers continuous model spaces, defined in terms of hyperparameters (sufficient statistics) of the prior density over model parameters. In this instance, the prior (model) can be optimized with respect to its evidence. The expressions for model evidence become remarkably simple under the Laplace (Gaussian) approximation to the posterior density. Special cases of this scheme include Savage–Dickey density ratio tests for reduced models and automatic relevance determination in model optimization. We illustrate the approach using general linear models and a more complicated nonlinear state-space model.

Dynamic Probabilities of Restrictions in State Space Models: An Application to the Phillips Curve

Using the Savage–Dickey density ratio and an alternative approach that uses more relaxed assumptions, we develop methods to calculate the probability that a restriction holds at a point in time without assuming that the restriction holds at any other points in time. Both approaches use MCMC output only from the unrestricted model to compute the time-varying posterior probabilities for all models of interest. Using U.S. data, we find the probability that the long-run Phillips curve is vertical to be fairly high, but decreases over time. The probability that the NAIRU is not identified fluctuates over time, but increases after 1990.

An encompassing prior generalization of the Savage–Dickey density ratio

An encompassing prior (EP) approach to facilitate Bayesian model selection for nested models with inequality constraints has been previously proposed. In this approach, samples are drawn from the prior and posterior distributions of an encompassing model that contains an inequality restricted version as a special case. The Bayes factor in favor of the inequality restriction then simplifies to the ratio of the proportions of posterior and prior samples consistent with the inequality restriction. This formalism has been applied almost exclusively to models with inequality or “about equality” constraints. It is shown that the EP approach naturally extends to exact equality constraints by considering the ratio of the heights for the posterior and prior distributions at the point that is subject to test (i.e., the Savage–Dickey density ratio). The EP approach generalizes the Savage–Dickey ratio method, and can accommodate both inequality and exact equality constraints. The general EP approach is found to be a computationally efficient procedure to calculate Bayes factors for nested models. However, the EP approach to exact equality constraints is vulnerable to the Borel–Kolmogorov paradox, the consequences of which warrant careful consideration.

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

In the field of cognitive psychology, the p-value hypothesis test has established a stranglehold on statistical reporting. This is unfortunate, as the p-value provides at best a rough estimate of the evidence that the data provide for the presence of an experimental effect. An alternative and arguably more appropriate measure of evidence is conveyed by a Bayesian hypothesis test, which prefers the model with the highest average likelihood. One of the main problems with this Bayesian hypothesis test, however, is that it often requires relatively sophisticated numerical methods for its computation. Here we draw attention to the Savage–Dickey density ratio method, a method that can be used to compute the result of a Bayesian hypothesis test for nested models and under certain plausible restrictions on the parameter priors. Practical examples demonstrate the method’s validity, generality, and flexibility.

A BAYESIAN ANALYSIS FOR SHORT-TERM INTEREST RATE MODELS AND ITS APPLICATIONS

This paper presents an analysis of models for Japanese short-term inter- est rate. The models are constructed based on mean reverting model using Bayesian method to capture the dynamics of short-term interest rate. The parameters of our models are estimated by marginal likelihood and posterior expectation and we shall make model selection using the information criterion EIC(extended information cri- terion). An application of the models will be implemented using weekly Japanese average interest rates on certificates of deposit (new issues) less than 30 days in the period from January 2001 to December 2008. 1 Introduction Correct modeling of the short-term interest rate is essential in finace, as it is this rate that is fundamental to the pricing of securities and important for risk man- agement. In the study of the short-term interest rate dynamics, various models have been suggested. There are examples of these models, such as Vasicek model by Vasicek(1977), CIR model by Cox, Ingersoll and Ross(1985) and so on. An empirical comparison of these models was made by Chan, Karolyi, Longstaff and Sanders(1992). In the paper, the param- eters are estimated by generalized method of moments(GMM) and they implemented the hypothesis testing methods developed by Newey and West for evaluation of the models. In the parameter estimation of mean reverting model using GMM, it is known that the results are easy to be influenced by an initial values, and it becomes often unstable. Recently, Ahangarani(2005) employed maximum likelihood method for parameter esti- mation and log likelihood ratio test for model selection, and Kawada(2007) implemented GMM and made model selection using the information criterion EIC(extended information criterion) for various short-term interest rate models. Another articles of the interest rate model, using Bayesian framework, Jones(2003) published study about nonlinear drift of the interest rate model in detail, and Gray(2005) studied continuous time short rate models and the parameters estimated by Markov chain Monte Carlo(MCMC) method. Sanford and Martin(2006) made estimation using MCMC algorithm and model selection is made by Bayes factors for each model calculated using Savage-Dickey density ratio. In addition, a recent study by Hong and Lin(2006) examined a variety of short-term interest rate mod- els including the single-factor diffusion models, GARCH models, Markov regime-switching models and jump-diffusion models for Chinese rates. However, more useful models which capture the dynamics of the short-term interest rate are needed. In this paper, we suggest two hierarchical Bayes models based on mean reverting model. The parameters of the models are treated as random variables so that we have to consider appropriate prior distributions for them. By regarding parameters as random variables, we can consider that model construction and estimation involve necessarily some uncertainty and utilize the prior information of interest rate. In the first hierarchical Bayes model, we assume that volatility is constant, and in the second one, it depends on the

Evidence of the duration-dependence from the stock markets in the Pacific Rim economies

This article investigates the duration-dependent feature of five Pacific Rim economies. The duration-dependent Markov Switching model is employed to achieve this objective. The Savage–Dickey density ratio is also computed in support of the duration-dependent Markov switching model. The possible bull and bear market dates for each stock market are also identified by the posterior probability from the empirical model. It is unambiguous that Japan, South Korea and Hong Kong are all characterized by duration-dependence in a bear market but no duration-dependence in a bull market. In the case of Taiwan and Singapore, the duration-dependence feature holds for both the bear and bull markets.

Applications of Bayesian model selection to cosmological parameters

Bayesian model selection is a tool to decide whether the introduction of a new parameter is warranted by data. I argue that the usual sampling statistic significance tests for a null hypothesis can be misleading, since they do not take into account the information gained through the data, when updating the prior distribution to the posterior. On the contrary, Bayesian model selection offers a quantitative implementation of Occam’s razor. I introduce the Savage–Dickey density ratio, a computationally quick method to determine the Bayes factor of two nested models and hence perform model selection. As an illustration, I consider three key parameters for our understanding of the cosmological concordance model. By using WMAP 3–year data complemented by other cosmological measurements, I show that a non–scale invariant spectral index of perturbations is favoured for any sensible choice of prior. It is also found that a flat Universe is favoured with odds of 29 : 1 over non–flat models, and that there is strong evidence against a CDM isocurvature component to the initial conditions which is totally (anti)correlated with the adiabatic mode (odds of about 2000 : 1), but that this is strongly dependent on the prior adopted. These results are contrasted with the analysis of WMAP 1–year data, which were not informative enough to allow a conclusion as to the status of the spectral index. In a companion paper, a new technique to forecast the Bayes factor of a future observation is presented.

Bayesian analysis of a self-selection model with multiple outcomes using simulation-based estimation: an application to the demand for healthcare

This paper studies a self-selection model with discrete and continuous outcomes and a treatment variable. The treatment variable is endogenous to the two outcome variables. The approach of the paper is fully parametric and Bayesian. The Bayes factor is calculated with the Savage–Dickey density ratio and used for model selection. The model is applied to two different micro data sets, the 1987–1988 National Medical Expenditure Survey and the 1996 Medical Expenditure Panel Survey. The paper studies the effect of managed care and fee-for-service type of private insurance on the demand for healthcare. It also compares the effects of private insurance and Medicaid in covering health care expenses of elderly Americans.

APPLYING THE SAVAGE-DICKEY DENSITY RATIO TO DEFAULT BAYES FACTORS, WITH AN ILLUSTRATION TO OUTLIER DETECTION IN RANDOM EFFECTS MODELS

When no information is available and hence improper noninformative priors should be used, Bayes factor includes the unspecified constants and can not be calibrated. To solve this problem, we modify the intrinsic Bayes factor (IBF) of Berger and Pericchi 1-2 and the fractional Bayes factor (FBF) of O'Hagan [3] with the generalized Savage-Dickey density ratio of Verdinelli and Wasserman [4]. These modified IBF and FBF are applied to detecting outliers in random effects models with a mean-shift structure. The proposed methodology is exemplified by a simulation experiment with a generated data set and also applied to a real data set, Dyestuff data in Box and Tiao [5]

A Method for Testing Nested Point Null Hypotheses Using Multiple Bayes Factor

As a flexible Bayesian test criterion for nested point null hypotheses, asymmetric and multiple Bayes factors are introduced in the form of a modified Savage-Dickey density ratio. This leads to a simple method for obtaining pairwise comparisons of hypotheses in a statistical experiment with a partition on the parameter space. The method is derived from the fact that in general, the asymmetric Bayes factor can be written as the product of the Savage-Dickey ratio and a correction factor where both terms are easily estimated by means of posterior simulation. Analyses of a censored data problem and a serial correlation problem are illustrated for the method. For these cases, the method is straightforward for specifying distributionally and to implement computationally, with output readily adapted for required tests.

Bayesian inference in a simultaneous equation model with limited dependent variables

This paper develops a Bayesian approach for inference in a simultaneous equation model with limited dependent variables (SLDV). By employing a combination of Gibbs sampling and data augmentation, we can draw from the exact posterior of this SLDV model and avoid direct evaluation of the non-trivial likelihood function. A by-product from our posterior simulation is the Savage—Dickey density ratio which is used for computing the Bayes factor. The practicality and efficiency of the proposed method are illustrated through an example in corporate finance.

Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio

Abstract We present a simple method for computing Bayes factors. The method derives from observing that in general, a Bayes factor can be written as the product of a quantity called the Savage-Dickey density ratio and a correction factor; both terms are easily estimated from posterior simulation. In some cases it is possible to do these computations without ever evaluating the likelihood.

Computing Bayes Factors Using a Generalization of the Savage-Dickey Density Ratio

Abstract We present a simple method for computing Bayes factors. The method derives from observing that in general, a Bayes factor can be written as the product of a quantity called the Savage-Dickey density ratio and a correction factor; both terms are easily estimated from posterior simulation. In some cases it is possible to do these computations without ever evaluating the likelihood.