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24
Reversible jump Markov chain Monte Carlo computation and Bayesian model determination
 Biometrika
, 1995
"... Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some xed standard underlying measure. They have therefore not been available for application to Bayesian model determi ..."
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Cited by 827 (19 self)
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Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some xed standard underlying measure. They have therefore not been available for application to Bayesian model determination, where the dimensionality of the parameter vector is typically not xed. This article proposes a new framework for the construction of reversible Markov chain samplers that jump between parameter subspaces of di ering dimensionality, which is exible and entirely constructive. It should therefore have wide applicability in model determination problems. The methodology is illustrated with applications to multiple changepoint analysis in one and two dimensions, and toaBayesian comparison of binomial experiments.
Bayesian Curve Fitting Using MCMC With Applications to Signal Segmentation
 IEEE Transactions on Signal Processing
, 2002
"... We propose some Bayesian methods to address the problem of fitting a signal modeled by a sequence of piecewise constant linear (in the parameters) regression models, for example, autoregressive or Volterra models. A joint prior distribution is set up over the number of the changepoints/knots, their ..."
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Cited by 54 (0 self)
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We propose some Bayesian methods to address the problem of fitting a signal modeled by a sequence of piecewise constant linear (in the parameters) regression models, for example, autoregressive or Volterra models. A joint prior distribution is set up over the number of the changepoints/knots, their positions, and over the orders of the linear regression models within each segment if these are unknown. Hierarchical priors are developed and, as the resulting posterior probability distributions and Bayesian estimators do not admit closedform analytical expressions, reversible jump Markov chain Monte Carlo (MCMC) methods are derived to estimate these quantities. Results are obtained for standard denoising and segmentation of speech data problems that have already been examined in the literature. These results demonstrate the performance of our methods.
The Equity Premium and Structural Breaks
, 2000
"... A long return history is useftil in estimating the current equity premium even if the historical distribution has experienced structural breaks. The long series helps not only if the timing of breaks is uncertain but also if one believes that large shifts in the premium are unlikely or that the prem ..."
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Cited by 43 (4 self)
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A long return history is useftil in estimating the current equity premium even if the historical distribution has experienced structural breaks. The long series helps not only if the timing of breaks is uncertain but also if one believes that large shifts in the premium are unlikely or that the premium is associated, in part, with volatility. Our framework incorporates these features along with a belief that prices are likely to move opposite to contemporaneous shifts in the premium. The estimated premium since 1834 fluctuates between four and six percent and exhibits its sharpest drop in the last decade.
Bayesian Online Changepoint Detection
"... Changepoints are abrupt variations in the generative parameters of a data sequence. Online detection of changepoints is useful in modelling and prediction of time series in application areas such as finance, biometrics, and robotics. While frequentist methods have yielded online filtering and predic ..."
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Cited by 26 (0 self)
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Changepoints are abrupt variations in the generative parameters of a data sequence. Online detection of changepoints is useful in modelling and prediction of time series in application areas such as finance, biometrics, and robotics. While frequentist methods have yielded online filtering and prediction techniques, most Bayesian papers have focused on the retrospective segmentation problem. Here we examine the case where the model parameters before and after the changepoint are independent and we derive an online algorithm for exact inference of the most recent changepoint. We compute the probability distribution of the length of the current “run, ” or time since the last changepoint, using a simple messagepassing algorithm. Our implementation is highly modular so that the algorithm may be applied to a variety of types of data. We illustrate this modularity by demonstrating the algorithm on three different realworld data sets. 1
Nonlinearity, Structural Breaks Or Outliers In Economic Time Series?
 Nonlinear Econometric Modeling in Time Series Analysis
, 2000
"... This paper has its motivation from discussions at the EC ..."
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Cited by 16 (4 self)
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This paper has its motivation from discussions at the EC
Partition Modelling
"... Introduction This chapter serves as an introduction to the use of partition models to estimate a spatial process z(x) over some pdimensional region of interest X . Partition models can be useful modelling tools as, unlike standard spatial models (e.g. kriging) they allow the correlation structure ..."
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Cited by 15 (3 self)
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Introduction This chapter serves as an introduction to the use of partition models to estimate a spatial process z(x) over some pdimensional region of interest X . Partition models can be useful modelling tools as, unlike standard spatial models (e.g. kriging) they allow the correlation structure between points to vary over the space of interest. Typically, the correlation between points is assumed to be a xed function which is most likely to be parameterised by a few variables that can be estimated from the data (see, for example, Diggle, Tawn and Moyeed (1998)). Partition models avoid the need for preexamination of the data to nd a suitable correlation function to use. This removes the bias necessarily introduced by picking the correlation function and estimating its parameters using the same set of data. Spatial clusters are, by their nature, regions which are not representative of the entire space of intere
Gibbs Sampling
 Journal of the American Statistical Association
, 1995
"... 8> R f(`)d`. To marginalize, say for ` i ; requires h(` i ) = R f(`)d` (i) = R f(`)d` where ` (i) denotes all components of ` save ` i : To obtain Eg(` i ) requires similar integration; to obtain the marginal distribution of say g(`) or its expectation requires similar integration. When p is l ..."
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Cited by 10 (0 self)
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8> R f(`)d`. To marginalize, say for ` i ; requires h(` i ) = R f(`)d` (i) = R f(`)d` where ` (i) denotes all components of ` save ` i : To obtain Eg(` i ) requires similar integration; to obtain the marginal distribution of say g(`) or its expectation requires similar integration. When p is large (as it will be in the applications we envision) such integration is analytically infeasible (the socalled curse of dimensionality*). Gibbs sampling provides a Monte Carlo approach for carrying out such integrations. In what sorts of settings would we have need to mar
Prior elicitation in multiple changepoint models
"... This paper discusses Bayesian inference in changepoint models. The main existing approaches either attempt to be noninformative by using a Uniform prior over changepoints or use an informative hierarchical prior. Both these approaches assume a known number of changepoints. We show how they have s ..."
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Cited by 8 (1 self)
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This paper discusses Bayesian inference in changepoint models. The main existing approaches either attempt to be noninformative by using a Uniform prior over changepoints or use an informative hierarchical prior. Both these approaches assume a known number of changepoints. We show how they have some potentially undesirable properties and discuss how these properties relate to the imposition of a …xed number of changepoints. We develop a new Uniform prior which allows some of the changepoints to occur outof sample. This prior has desirable properties, can reasonably be interpreted as “noninformative”and handles the case where the number of changepoints We would like to thank Edward Leamer for useful conversations and also seminar participants at the Federal Reserve Bank of St. Louis and University of Kansas. The views expressed in this paper are those of the authors and do not necessarily re‡ect the views of the Federal Reserve Bank of New York or the Federal Reserve System. 1 is unknown. We show how the general ideas of our approach can be extended to informative hierarchical priors. With arti…cial data and two empirical illustrations, we show how these di¤erent priors can have a substantial impact on estimation and prediction even with moderately large data sets. 1
Bayesian Hierarchical Nonparametric Inference for ChangePoint Problems
, 1995
"... this paper. First, one could avoid the parametric restrictions on the d.f. of the observable random variables. Second, it is often desirable to incorporate information about the direction of the changepoint. A proposal might be to assume X i j F i ¸ F i and F 1 ; F 2 ; : : : ; Fn distributed as a m ..."
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Cited by 6 (2 self)
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this paper. First, one could avoid the parametric restrictions on the d.f. of the observable random variables. Second, it is often desirable to incorporate information about the direction of the changepoint. A proposal might be to assume X i j F i ¸ F i and F 1 ; F 2 ; : : : ; Fn distributed as a mixture of products of Dirichlet processes D(ff i (\Delta; ; ` i )), with mixing distribution H(` 1 ; : : : ; ` n ). The mixing distribution H could incorporate the initial information about the c.p.'s. We plan on investigating this problem in future work.
Estimating the LongTerm Phosphorus Accretion Rate in the Everglades: A Bayesian Approach with Risk Assessment
 Water Resources Research
, 1997
"... Using wetlands as sink of nutrients, phosphorus in particular, is becoming an increasingly attractive alternative to conventional wastewater treatment technology. In this paper, we briefly review the mechanism of phosphorus retention in wetlands, as well as previous modeling efforts. ABayesian metho ..."
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Cited by 6 (3 self)
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Using wetlands as sink of nutrients, phosphorus in particular, is becoming an increasingly attractive alternative to conventional wastewater treatment technology. In this paper, we briefly review the mechanism of phosphorus retention in wetlands, as well as previous modeling efforts. ABayesian method is then proposed for estimating the longterm phosphorus accretion rate in wetlands through a piecewise linear model of outflow phosphorus concentration and phosphorus mass loading rate. The Bayesian approach was used for its simplicity in computation and its ability to accurately represent uncertainty. Applied to an Everglades wetland, the Bayesian method not only produced the probability distribution of the longterm phosphorus accretion rate, but also generated a relationship of acceptable level of "risk" and optimal phosphorus mass loading rate for the proposed constructed wetlands in south Florida. The latter is a useful representation of uncertainty which is of interest to decision makers.