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Bayes Factors
, 1995
"... In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null ..."
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Cited by 1766 (74 self)
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In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is onehalf. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications in genetics, sports, ecology, sociology and psychology.
Bayesian Model Selection in Social Research (with Discussion by Andrew Gelman & Donald B. Rubin, and Robert M. Hauser, and a Rejoinder)
 SOCIOLOGICAL METHODOLOGY 1995, EDITED BY PETER V. MARSDEN, CAMBRIDGE,; MASS.: BLACKWELLS.
, 1995
"... It is argued that Pvalues and the tests based upon them give unsatisfactory results, especially in large samples. It is shown that, in regression, when there are many candidate independent variables, standard variable selection procedures can give very misleading results. Also, by selecting a singl ..."
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Cited by 545 (21 self)
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It is argued that Pvalues and the tests based upon them give unsatisfactory results, especially in large samples. It is shown that, in regression, when there are many candidate independent variables, standard variable selection procedures can give very misleading results. Also, by selecting a single model, they ignore model uncertainty and so underestimate the uncertainty about quantities of interest. The Bayesian approach to hypothesis testing, model selection and accounting for model uncertainty is presented. Implementing this is straightforward using the simple and accurate BIC approximation, and can be done using the output from standard software. Specific results are presented for most of the types of model commonly used in sociology. It is shown that this approach overcomes the difficulties with P values and standard model selection procedures based on them. It also allows easy comparison of nonnested models, and permits the quantification of the evidence for a null hypothesis...
Bayes factors and model uncertainty
 DEPARTMENT OF STATISTICS, UNIVERSITY OFWASHINGTON
, 1993
"... In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null ..."
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Cited by 121 (6 self)
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In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is onehalf. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of Pvalues, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications. The points we emphasize are: from Jeffreys's Bayesian point of view, the purpose of hypothesis testing is to evaluate the evidence in favor of a scientific theory; Bayes factors offer a way of evaluating evidence in favor ofa null hypothesis; Bayes factors provide a way of incorporating external information into the evaluation of evidence about a hypothesis; Bayes factors are very general, and do not require alternative models to be nested; several techniques are available for computing Bayes factors, including asymptotic approximations which are easy to compute using the output from standard packages that maximize likelihoods; in "nonstandard " statistical models that do not satisfy common regularity conditions, it can be technically simpler to calculate Bayes factors than to derive nonBayesian significance
Inference for Deterministic Simulation Models: The Bayesian Melding Approach
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2000
"... Deterministic simulation models are used in many areas of science, engineering and policymaking. Typically, they are complex models that attempt to capture underlying mechanisms in considerable detail, and they have many userspecified inputs. The inputs are often specified by some form of trialan ..."
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Cited by 54 (4 self)
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Deterministic simulation models are used in many areas of science, engineering and policymaking. Typically, they are complex models that attempt to capture underlying mechanisms in considerable detail, and they have many userspecified inputs. The inputs are often specified by some form of trialanderror approach in which plausible values are postulated, the corresponding outputs inspected, and the inputs modified until plausible outputs are obtained. Here we address the issue of more formal inference for such models. Raftery et al. (1995a) proposed the Bayesian synthesis approach in which the available information about both inputs and outputs was encoded in a probability distribution and inference was made by restricting this distribution to the submanifold specifid by the model. Wolpert (1995) showed that this is subject to the Borel paradox, according to which the results can depend on the parameterization of the model. We show that this dependence is due to the presence of a prior on the outputs. We propose a modified approach, called Bayesian melding, which takes full account of information and uncertainty about both inputs and outputs to the model, while avoiding the Borel paradox. This is done by recognizing the existence of two priors, one implicit and one explicit, on each input and output � these are combined via logarithmic pooling. Bayesian melding is then
Assessing Uncertainty in Urban Simulations Using Bayesian Melding
"... We develop a method for assessing uncertainty about quantities of interest using urban simulation models. The method is called Bayesian melding, and extends a previous method developed for macrolevel deterministic simulation models to agentbased stochastic models. It encodes all the available infor ..."
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Cited by 27 (3 self)
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We develop a method for assessing uncertainty about quantities of interest using urban simulation models. The method is called Bayesian melding, and extends a previous method developed for macrolevel deterministic simulation models to agentbased stochastic models. It encodes all the available information about model inputs and outputs in terms of prior probability distributions and likelihoods, and uses Bayes’s theorem to obtain the resulting posterior distribution of any quantity of interest that is a function of model inputs and/or outputs. It is Monte Carlo based, and quite easy to implement. We applied it to the projection of future household numbers by traffic activity zone in EugeneSpringfield, Oregon, using the UrbanSim model developed at the University of Washington. We compared it with a simpler method that uses repeated runs of the model with fixed estimated inputs. We found that the simple repeated runs method gave distributions of quantities of interest that were too narrow, while Bayesian melding gave well calibrated uncertainty statements.
Benefits of a Bayesian Approach for Synthesizing Multiple Sources of Evidence and Uncertainty Linked by a Deterministic Model
, 1993
"... A Bayesian synthesis approach has been proposed by Raftery, Givens, and Zeh (1992) for making inferences from a deterministic model with many inputs and outputs. The approach was applied to population dynamics models for bowhead whales. The approach consists of establishing a joint prior, or pre ..."
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Cited by 7 (6 self)
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A Bayesian synthesis approach has been proposed by Raftery, Givens, and Zeh (1992) for making inferences from a deterministic model with many inputs and outputs. The approach was applied to population dynamics models for bowhead whales. The approach consists of establishing a joint prior, or premodel distribution, on the model inputs and outputs for which there exists evidence independent of the model. The restriction of this distribution to a subspace defined by the model mapping then constitutes a postmodel distribution, from which inferences are drawn. We briefly review a methodology for implementing the Bayesian synthesis approach, and then consider in detail the potential uses of the results and the strengths and weaknesses of the approach compared to past methodologies. 1 Introduction Complex deterministic models reflect scientists' simplified conceptions of natural mechanisms about which they have incomplete understanding. Such a model depends on a set of input para...
Alternative Bayesian Synthesis Approaches to BeringChukchiBeaufort Seas Bowhead Whale Stock Assessment: Uncertainty in Historic Catch and Hitting with Fixed MSYR
 Whal. Commn
, 1995
"... We report the results of two investigations requested by the Scientific Committee regarding alternative methods for Bayesian synthesis assessment of the BeringBeaufortChukchi Seas stock of bowhead whales (Balaena mysticetus). The first series of studies examines the effect of assuming that the hi ..."
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We report the results of two investigations requested by the Scientific Committee regarding alternative methods for Bayesian synthesis assessment of the BeringBeaufortChukchi Seas stock of bowhead whales (Balaena mysticetus). The first series of studies examines the effect of assuming that the historic pelagic catch record is known without error. We propose a model for variability in the historic catch record and we investigate the potential influence of catch record uncertainty on assessment results. We also examine the influence of systematic bias in the record. Our analyses indicate that variability in the kill record is not a dominant source of uncertainty in the Bayesian synthesis results. Consistent, systematic bias in the record does effect conclusions about initial stock size, depletion, and maximum sustainable yield. There is evidence against the hypothesis that the true historic catch was substantially greater than previously assumed, but the combined analysis can not rule...
Analysis of aerial survey data on Florida manatee using Markov chain Monte Carlo
, 1995
"... This paper analyzes the aerial survey data from the Atlantic coast population only. It is a common assumption that this population does not interact with the Gulf of Mexico population and thus is closed (Reid, Rathbun and Wilcox, 1991). Because a standard survey protocol was not used prior to the 82 ..."
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Cited by 5 (0 self)
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This paper analyzes the aerial survey data from the Atlantic coast population only. It is a common assumption that this population does not interact with the Gulf of Mexico population and thus is closed (Reid, Rathbun and Wilcox, 1991). Because a standard survey protocol was not used prior to the 8283 winter, only the data from that winter on are analyzed. For simplification, the seven sitespecific counts at each survey are collapsed into two; the total of the two most northerly sites and the total of the remaining five. Table 1 summarizes the observed regional counts over the ten year period. Note the considerable variability among counts in each region and year.
. Summary of Results
"... This paper analyzes the aerial survey data from the Atlantic coast population only. Although McClenaghan and O'Shea (1988) provide genetic evidence that manatees from the east and west coasts of Florida intermingle, behavioral studies (e.g., Reid, Rathbun and Wilcox, 1991) suggest that such mix ..."
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Cited by 4 (2 self)
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This paper analyzes the aerial survey data from the Atlantic coast population only. Although McClenaghan and O'Shea (1988) provide genetic evidence that manatees from the east and west coasts of Florida intermingle, behavioral studies (e.g., Reid, Rathbun and Wilcox, 1991) suggest that such mixing is infrequent. closed (Reid, Rathbun and Wilcox, 1991). Because a standard survey protocol was not used prior to the 8283 winter, only the data from that winter on are analyzed. For simplification, the seven sitespecific counts at each survey are collapsed into two; the total of the two most northerly sites and the total of the remaining five. Table 1 summarizes the observed regional counts over the ten year period. Note the considerable variability among counts in each region and year. TABLE 1 HERE In addition to the counts themselves, several covariates were collected at each survey. Two covariates are the northern and southern water temperatures, each averaged over the two days prior to and the day of the survey. Also included is the total number of NOAA degreedays (number of air degrees below 65
Confirmation, Revision, and Sensitivity Analysis of the 1994 Scientific Committee Assessment of the BeringChukchiBeaufort Seas Stock of Bowhead Whales
 Whal. Commn
, 1995
"... Results from the approximate reweighting method used by the Scientific Committee for its 1994 assessment of the BeringChukchiBeaufort Seas stock of bowhead whales are confirmed using the full Bayesian synthesis approach. Sensitivity trials are examined to investigate several areas of interest iden ..."
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Cited by 4 (2 self)
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Results from the approximate reweighting method used by the Scientific Committee for its 1994 assessment of the BeringChukchiBeaufort Seas stock of bowhead whales are confirmed using the full Bayesian synthesis approach. Sensitivity trials are examined to investigate several areas of interest identified during this assessment. The results show that the full analysis gives results which are very close to those obtained by the Scientific Committee in 1994, and which are not very sensitive to changing the distributions of model parameters in ways which are reasonable for bowheads. The Scientific Committee obtained an estimated 5 th percentile for replacement yield of 104 whales; four independent runs of the full analysis give values of 104, 108, 104, and 115. The results show no reason to question the conclusions of the 1994 Scientific Committee assessment. In its 1994 bowhead assessment, the Scientific Committee agreed to use a stock abundance estimate based on data from 1988 because...