Statistical Science

Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors

Jennifer A. Hoeting, David Madigan, Adrian E. Raftery, and Chris T. Volinsky

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Standard statistical practice ignores model uncertainty. Data analysts typically select a model from some class of models and then proceed as if the selected model had generated the data. This approach ignores the uncertainty in model selection, leading to over-confident inferences and decisions that are more risky than one thinks they are. Bayesian model averaging (BMA)provides a coherent mechanism for accounting for this model uncertainty. Several methods for implementing BMA have recently emerged. We discuss these methods and present a number of examples.In these examples, BMA provides improved out-of-sample predictive performance. We also provide a catalogue of currently available BMA software.

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Statist. Sci. Volume 14, Number 4 (1999), 382-417.

First available in Project Euclid: 24 December 2001

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Bayesian model averaging Bayesian graphical models learning model uncertainty Markov chain Monte Carlo


Hoeting, Jennifer A.; Madigan, David; Raftery, Adrian E.; Volinsky, Chris T. Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors. Statist. Sci. 14 (1999), no. 4, 382--417. doi:10.1214/ss/1009212519.

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  • and Kooperberg, 1999). Markov chain Monte Carlo (MCMC) methods provide a stochastic method of obtaining samples from the posterior distributions f Mk Y and f Mk Mk Y and many of the algorithms that the authors mention can be viewed as special cases of reversible jump MCMC algorithms.
  • (Clyde, Parmigiani and Vidakovic, 1998). Sampling models and 2 in conjunction withthe use of Rao-Blackwellized estimators does appear to be more efficient in terms of mean squared error, when there is substantial uncertainty in the error variance (i.e., small sample sizes or low signal-to-noise ratio) or important prior information. Recently, Holmes and Mallick (1998) adapted perfect sampling (Propp and Wilson, 1996) to the context of orthogonal regression. While more computationally intensive per iteration, this may prove to be more efficient for estimation than SSVS or MC3 in problems where the method is applicable and sampling is necessary. While Gibbs and MCMC sampling has worked well in high-dimensional orthogonal problems, Wong, Hansen, Kohn and Smith (1997) found in high-dimensional problems such as nonparametric regression using nonorthogonal basis functions that Gibbs samplers were unsuitable, from both a computational efficiency standpoint as well as for numerical reasons, because the sampler tends to get stuck in local modes. Their proposed sampler "focuses" on variables that are more "active" at each iteration and in simulation studies provided better MSE performance than other classical nonparametric approaches or Bayesian approaches using Gibbs or reversible jump (Holmes and Mallick, 1997) sampling. With the exception of a deterministic search, most methods for implementing BMA rely on algorithms that sample models with replacement and use ergodic averages to compute expectations, as in (7). In problems, suchas linear models, where posterior model probabilities are known up to the normalizing constant, it may be more efficient to devise estimators using renormalized posterior model probabilities (Clyde, DeSimone and Parmigiani, 1996; Clyde, 1999a) and to devise algorithms based on sampling models without replacement. Based on current work withM. Littman, this appears to be a promising direction for implementation of BMA. While many recent developments have greatly advanced the class of problems that can be handled using BMA, implementing BMA in high-dimensional problems withcorrelated variables, suchas nonparametric regression, is still a challenge from both a computational standpoint and the choice of prior distributions.
  • AIC, BIC, and RIC (Clyde and George, 1998, 1999; George and Foster, 1997; Hanson and Yu, 1999) for bothmodel selection and BMA.
  • MODEL AVERAGING, MAYBE This paper offers a good review of one approachto dealing withstatistical model uncertainty, an important topic and one which has only begun to come into focus for us as a profession in this decade (largely because of the availability of Markov chain Monte Carlo computing methods). The authors-who together might be said to have founded the Seattle school of model uncertainty-are to be commended for taking this issue forward so vigorously over the past five years. I have eight comments on the paper, some general and some specific to the body-fat example (Jennifer Hoeting kindly sent me the data, which are well worthlooking at; the data set, and a full description of it, may be obtained by emailing the message send jse/v4n1/datasets.johnson to
  • Draper and Fouskakis, 1999). 5. What characteristics of a statistical example predict when BMA will lead to large gains? The only obvious answer I know is the ratio n/p of observations to predictors (withtens of thousands of observations and only dozens of predictors to evaluate, intuitively the price paid for shopping around in the data for a model should be small). Are the authors aware of any other simple answers to this question? As an instance of the n/p effect, in regressionstyle problems like the cirrhosis example where p is in the low dozens and n is in the hundreds, the effect of model averaging on the predictive scale can be modest. HMRV are stretching a bit when they say, in this example, that "the people assigned to the high risk group by BMA had a higher death rate than did those assigned high risk by other methods; similarly those assigned to the low and medium risk groups by BMA had a lower total deathrate"; this can be seen by attaching uncertainty bands to the estimates in Table 5. Over the single random split into build and test data reported in that table, and assuming (at least approximate) independence of the 152 yes/no classifications aggregated in the table, deathrates in the highrisk group, withbinomial standard errors, are 81% ± 5%, 75% ± 6% and 72% ± 6% for the BMA, stepwise, and top PMP methods, and combining the low and medium risk groups yields 18% ± 4%, 19% ± 4% and 17% ± 4% for the three methods, respectively, hardly a rousing victory for BMA. It is probable that by averaging over many random build-test splits a "statistically significant" difference would emerge, but the predictive advantage of BMA in this example is not large in practical terms. 6. Following on from item (4) above, now that the topic of model choice is on the table, why are we doing variable selection in regression at all? People who think that you have to choose a subset of the predictors typically appeal to vague concepts like "parsimony," while neglecting to mention that the "full model" containing all the predictors may well have better out-of-sample predictive performance than many models based on subsets of the xj. Withthe body-fat data, for instance, on the same build-test split used by HMRV, the model that uses all 13 predictors in the authors' Table 7 (fitted by least squares-Gaussian maximum likelihood) has actual coverage of nominal 90% predictive intervals of 95 0 ± 1 8 % and 86 4 ± 3 3 % in the build and test data subsets, respectively; this out-of-sample figure is better than any of the standard variable-selection methods tried by HMRV (though not better than BMA in this example). To make a connection withitem (5) above, I generated a data set 10 times as big but withthe same mean and covariance structure as the body-fat data; with 2,510 total observations the actual coverage of nominal 90% intervals within the 1,420 data values used to fit the model was 90 6 ± 0 8 %, and on the other 1,090 observations it was 89 2 ± 0 9 %. Thus with only 251 data points and 13 predictors, the "full model" overfits the cases used for estimation and underfits the out-of-sample cases, but this effect disappears withlarge n for fixed p (the rate at which this occurs could be studied systematically as a function of n and p). (I put "full model" in quotes because the concept of a full model is unclear when things like quadratics and interactions in the available predictors are considered.) There is another sense in which the "full model" is hard to beat: one can create a rather accurate approximation to the output of the complex, and computationally intensive, HMRV regression machinery in the following closedform Luddite manner. (1) Convert y and all of the xj to standard units, by subtracting off their means and dividing by their SDs, obtaining y and x j (say). This goes some distance toward putting the predictors on a common scale. (2) Use least squares-Gaussian maximum likelihood to regress y on all [or almost all ] th e x j, resolving collinearity problems by simply dropping out of the model altogether any x's that are highly correlated with other x's (when in doubt, drop the x in a pair of suchpredictors that is more weakly correlated with y. This
  • tors in George (1986a, b, c, 1987). However, by going outside the proper prior realm, norming constants
  • George and McCulloch(1998). I am currently developing dilution priors for multiple regression and will report on these elsewhere.
  • 1997). For the purpose of approximating BMA*, I am less sanguine about Occam's window, which is fundamentally a heuristic search algorithm. By restricting attention to the "best" models, the subset of models selected by Occam's Window are unlikely to be representative, and may severely bias the approximation away from BMA*. For example, suppose substantial posterior probability was diluted over a large subset of similar models, as discussed earlier. Although MCMC methods would tend to sample suchsubsets, they would be entirely missed by Occam's Window. A possible correction for this problem might be to base selection on a uniform prior, i.e. Bayes factors, but then use a dilution prior for the averaging. However, in spite of its limitations as an approximation to BMA*, the heuristics which motivate Occam's Window are intuitively very appealing. Perhaps it would simply be appropriate to treat and interpret BMA under Occam's Window as a conditional Bayes procedure.
  • DiCiccio et al., 1997; Oh, 1999). For BMA, it is desirable that the prior on the parameters be spread out enoughthat it is relatively flat over the region of parameter space where the likelihood is substantial (i.e., that we be in the "stable estimation" situation described by Edwards,
  • Lindman and Savage, 1963). It is also desirable that the prior not be much more spread out than is necessary to achieve this. This is because the integrated likelihood for a model declines roughly as -d as
  • els by Raftery, Madigan and Hoeting (1997). A second suchproposal is the unit information prior (UIP), which is a multivariate normal prior centered at the maximum likelihood estimate with variance matrix equal to the inverse of the mean observed Fisher information in one observation. Under regularity conditions, this yields the simple BIC approximation given by equation (13) in our paper
  • (Kass and Wasserman, 1995; Raftery, 1995). The unit information prior, and hence BIC, have been criticized as being too conservative (i.e., too likely to favor simple models). Cox (1995) suggested that the prior standard deviation should decrease withsample size. Weakliem (1999) gave sociological examples where the UIP is clearly too spread out, and Viallefont et al. (1998) have shown how a more informative prior can lead to better performance of BMA in the analysis of epidemiological case-control studies. The UIP is a proper prior but seems to provide a conservative solution. This suggests that if BMA based on BIC favors an "effect," we can feel on solid ground in asserting that the data provide ev
  • idence for its existence (Raftery, 1999). Thus BMA results based on BIC could be routinely reported as a baseline reference analysis, along withresults from other priors if available. A third approach is to allow the data to estimate the prior variance of the parameters. Lindley and Smith (1972) showed that this is essentially what ridge regression does for linear regression, and Volinsky (1997) pointed out that ridge regression has consistently outperformed other estimation methods in simulation studies. Volinsky (1997) proposed combining BMA and ridge regression by using a "ridge regression prior" in BMA. This is closely related to empirical Bayes BMA, which Clyde and George (1999) have shown to work well for wavelets, a special case of orthogonal regression. Clyde, Raftery, Walsh and Volinsky (2000) show that this good performance of empirical Bayes BMA extends to (nonorthogonal) linear regression.
  • income (Featherman and Hauser, 1977). X1 and X2 are highly correlated, but the mechanisms by which they might impact Y are quite different, so all four models are plausible a priori. The posterior model probabilities are saying that at least one of X1 and
  • a LISREL-type model (Bollen, 1989). BMA and Bayesian model selection can still be applied in this
  • context (e.g., Hauser and Kuo, 1998).
  • 1995). Draper says that model choice is a decision problem, and that the use to which the model is to be put should be taken into account explicitly in the model selection process. This is true, of course, but in practice it seems rather difficult to implement. This was first advocated by Kadane and Dickey (1980) but has not been done much in practice, perhaps because specifying utilities and carrying out the full utility maximization is burdensome, and also introduces a whole new set of sensitivity concerns. We do agree with Draper's suggestion that the analysis of the body fat data would be enhanced by a cost- benefit analysis whichtook account of bothpredictive accuracy and data collection costs. In practical decision-making contexts, the choice of statistical model is often not the question of primary interest, and the real decision to be made is something else. Then the issue is decision-making in the presence of model uncertainty, and BMA provides a solution to this. In equation (1) of our article, let be the utility of a course of action, and choose the action for which E D is maximized. Draper does not like our Figure 4. However, we see it as a way of depicting on the same graph the answers to two separate questions: is wrist circumference associated withbody fat after controlling for the other variables? and if so, how strong is the association? The posterior distribution of 13 has two components corresponding to these two questions. The answer to the first question is "no" (i.e., the effect is zero or small) with probability 38%, represented by the solid bar in Figure 4. The answer to the second question is summarized by the continuous curve. Figure 4 shows double shrinkage, withbothdiscrete and continuous components. The posterior distribution of 13, given that 13 = 0, is shrunk continuously towards zero via its prior distribution. Then the posterior is further shrunk (discretely this time) by taking account of the probability that 13 = 0. The displays in Clyde (1999b) convey essentially the same information, and some may find them more appealing than our Figure 4. Draper suggests the use of a practical significance caliper and points out that for one choice, this gives similar results to BMA. Of course the big question here is how the caliper is chosen. BMA can itself be viewed as a significance caliper, where the choice of caliper is based on the data. Draper's Table 1 is encouraging for BMA, because it suggests that BMA does coincide withpractical significance. It has often been observed that P values are at odds with"practical" significance, leading to strong distinctions being made in textbooks between statistical and practical significance. This seems rather unsatisfactory for our discipline: if statistical and practical significance do not at least approximately coincide, what is the use of statistical testing? We have found that BMA often gives results closer to the practical significance judgments of practitioners than do P-values.
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