Information criteria for astrophysical model selection
Abstract
Model selection is the problem of distinguishing competing models, perhaps featuring different numbers of parameters. The statistics literature contains two distinct sets of tools, those based on information theory such as the Akaike Information Criterion (AIC), and those on Bayesian inference such as the Bayesian evidence and Bayesian Information Criterion (BIC). The Deviance Information Criterion combines ideas from both heritages; it is readily computed from Monte Carlo posterior samples and, unlike the AIC and BIC, allows for parameter degeneracy. I describe the properties of the information criteria, and as an example compute them from WMAP3 data for several cosmological models. I find that at present the information theory and Bayesian approaches give significantly different conclusions from that data.
keywords:
cosmology: theory, methods: data analysis, methods: statistical1 Introduction
Although it has been widely recognized only recently, model selection problems are ubiquitous in astrophysics and cosmology. While parameter estimation seeks to determine the values of a parameter set chosen by hand, model selection seeks to distinguish between competing choices of parameter set. A considerable body of statistics literature is devoted to model selection [excellent textbook accounts are given by Jeffreys 1961, Burnham & Anderson 2002, MacKay 2003, and Gregory 2005] and its use is widespread throughout many branches of science. For a nontechnical overview of model selection as applied to cosmology, see Liddle, Mukherjee & Parkinson (2006a), and for an overview of techniques and applications see Lasenby & Hobson (2006).
In general, a model is a choice of parameters to be varied and a prior probability distribution on those parameters. The goal of model selection is to balance the quality of fit to observational data against the complexity, or predictiveness, of the model achieving that fit. This tension is achieved through model selection statistics, which attach a number to each model enabling a rankordered list to be drawn up. Typically, the best model is adopted and used for further inference such as permitted parameter ranges, though the statistics literature has also seen increasing interest in multimodel inference combining a number of adequate models (e.g. Hoeting et al. 1999; Burnham & Anderson 2004).
There are two main schools of thought in model selection. Bayesian inference, particularly as developed by Jeffreys culminating in his classic textbook (Jeffreys 1961) and by many others since, can assign probabilities to models as well as to parameter values, and manipulate these probabilities using rules such as Bayes’ theorem. Informationtheoretic methods, pioneered by Akaike (1974) with his Akaike Information Criterion, instead focus on the Kullback–Leibler information entropy (Kullback & Leibler 1951) as a measure of information lost when a particular model is used in place of the (unknown) true model. Variants on this latter theme include the Takeuchi Information Criterion (TIC, Takeuchi 1976), which extends the AIC by droppinging the assumption that the model set considered includes the true model. Bayesian statistics include the Bayesian evidence and an approximation to it known as the Bayesian Information Criterion (BIC, Schwarz 1978), which, despite the name, does not have an informationtheoretic justification.
Given the plethora of possible statistics, one might despair as to which to use, especially if they give conflicting results. Cosmologists, in particular, tend to ally themselves with a Bayesian methodology, for example the use of Markov Chain Monte Carlo (MCMC) methods to carry out parameter likelihood analyses, and are therefore tempted to adopt methods advertised as such. However, even if one were to side automatically against frequentist approaches, the situation does not appear that clear cut; Burnham & Anderson (2004) have argued that the AIC can be derived in a Bayesian way (and the BIC in a frequentist one), and that one should not casually dismiss a criterion soundly grounded in information theory.
Nevertheless, in my view the Bayesian evidence is the preferred tool; in Bayesian inference it is precisely the quantity which updates the prior model probability to the posterior model probability, and has an unambiguous interpretation in these probabilistic terms. The problem with the evidence is the difficulty in calculating it to the required accuracy, though the situation there has improved with the development of the nested sampling algorithm (Skilling 2006) and its implementation for cosmology in the CosmoNest code (Mukherjee, Parkinson & Liddle 2006; Parkinson, Mukherjee & Liddle 2006). This paper is principally directed at circumstances where the evidence is not readily calculable, and a simpler model selection technique is required.
In this article I describe and apply an additional information criterion, the Deviance Information Criterion (DIC) of Spiegelhalter et al. (2002, henceforth SBCL02), which combines heritage from both Bayesian methods and information theory. It has interesting properties. Firstly, unlike the AIC and BIC it accounts for the situation, common in astrophysics, where one or more parameters or combination of parameters is poorly constrained by the data. Secondly, it is readily calculable from posterior samples, such as those generated by MCMC methods. It has already been used in astrophysics to study quasar clustering (Porciani & Norberg 2006).
2 Model selection statistics
2.1 Bayesian evidence
The Bayesian evidence, also known as the model likelihood and sometimes, less accurately, as the marginal likelihood, comes from a full implementation of Bayesian inference at the model level, and is the probability of the data given the model. Using Bayes theorem, it updates the prior model probability to the posterior model probability. Usually the prior model probabilities are taken as equal, but quoted results can readily be rescaled to allow for unequal ones if required (e.g. Lasenby & Hobson 2006). In many circumstances the evidence can be calculated without simplifying assumptions (though perhaps with numerical errors). It has now been quite widely applied in cosmology; see for example Jaffe (1996), Hobson, Bridle & Lahav (2002), Saini, Weller & Bridle (2004), Trotta (2005), Parkinson et al. (2006), and Lasenby & Hobson (2006).
The evidence is given by
(1) 
where is the vector of parameters being varied in the model and is the properlynormalized prior distribution of those parameters (often chosen to be flat). It is the average value of the likelihood over the entire model parameter space that was allowed before the data came in. It rewards a combination of data fit and model predictiveness. Models which fit the data well and make narrow predictions are likely to fit well over much of their available parameter space, giving a high average. Models which fit well for particular parameter values, but were not very predictive, will fit poorly in most of their parameter space driving the average down. Models which cannot fit the data well will do poorly in any event.
The integral in equation (1) may however be difficult to calculate, as it may have too many dimensions to be amenable to evaluation by gridding, and the simplest MCMC methods such as Metropolis–Hastings produce samples only in the part of parameter space where the posterior probability is high rather than throughout the prior. Nevertheless, many methods exist (e.g. Gregory 2005; Trotta 2005), and the nested sampling algorithm (Skilling 2006) has proven feasible for many cosmology applications (Mukherjee et al. 2006; Parkinson et al. 2006; Liddle et al. 2006b).
A particular property of the evidence worth noting is that it does not penalize parameters (or, more generally, degenerate parameter combinations) which are unconstrained by the data. If the likelihood is flat or nearly flat in a particular direction, it simply factorizes out of the evidence integral leaving it unchanged. This is an appealing property, as it indicates that the model fitting the data is doing so really by varying fewer parameters than at first seemed to be the case, and it is the unnecessary parameters that should be discarded, not the entire model.
2.2 AIC and BIC
Much of the literature, both in astrophysics and elsewhere, seeks a simpler surrogate for the evidence which still encodes the tension between fit and model complexity. In Liddle (2004), I described two such statistics, the AIC and BIC, which have subsequently been quite widely applied to astrophysics problems. They are relatively simple to apply because they require only the maximum likelihood achievable within a given model, rather than the likelihood throughout the parameter space. Of course, such simplification comes at a cost, the cost being that they are derived using various assumptions, particularly gaussianity or neargaussianity of the posterior distribution, that may be poorly respected in realworld situations.
The AIC is defined as
(2) 
where is the maximum likelihood achievable by the model and the number of parameters of the model (Akaike 1974). The best model is the one which minimizes the AIC, and there is no requirement for the models to be nested. The AIC is derived by an approximate minimization of the Kullback–Leibler information entropy, which measures the difference between the true data distribution and the model distribution. An explanation geared to astronomers can be found in Takeuchi (2000), while the full statistical justification is given by Burnham & Anderson (2002).
The BIC was introduced by Schwarz (1978), and is defined as
(3) 
where is the number of datapoints used in the fit. It comes from approximating the evidence ratios of models, known as the Bayes factor (Jeffreys 1961; Kass & Raftery 1995). The BIC assumes that the datapoints are independent and identically distributed, which may or may not be valid depending on the dataset under consideration (e.g. it is unlikely to be good for cosmic microwave anisotropy data, but may well be for supernova luminositydistance data).
Applications of these two criteria have usually shown broad agreement in the conclusions reached, but occasional differences in the detailed ranking of models. One should consider the extent to which the conditions used in the derivation of the criteria are violated in real situations. A particular case in point is the existence of parameter degeneracies; inclusion (inadvertent or otherwise) of unconstrained parameters is penalized by the AIC and BIC, but not by the evidence. Interpretation of the BIC as an estimator of evidence differences is therefore suspect in such cases.
Burnham & Anderson (2002, 2004) have stressed the importance of using a version of the AIC corrected for small sample sizes, AIC. This is given by (Sugiura 1978)
(4) 
Since the correction term anyway disappears for large sample sizes, , there is no reason not to use it even in that case, i.e. it is always preferable to use AIC rather than the original AIC. In typical smallsample cases, e.g. being only a few, the correction term strengthens the penalty, bringing the AIC towards the BIC and potentially mitigating the difference between them.
2.3 Dic
The DIC was introduced by SBCL02. It has already been widely applied outside of astrophysics. Its starting point is a definition of an effective number of parameters of a model. This quantity, known also as the Bayesian complexity, has already been introduced into astrophysics by Kunz, Trotta & Parkinson (2006), with focus on assessing the number of parameters that can be usefully constrained by a particular dataset.
It is defined by
(5) 
Here is a ‘standardizing’ constant depending only on the data which will vanish from any derived quantity, and is the deviance of the likelihood. The bars indicate averages over the posterior distribution. In words, then, is the mean of the deviance, minus the deviance of the mean. If we define an effective chisquared as usual by , we can write
(6) 
Its intent becomes clear from studying a simple onedimensional example, in which the likelihood is a gaussian of zero mean and width , i.e. , and where the prior distribution is flat with width . Care is needed to properly normalize the posterior, which relates the likelihood amplitude to the prior width. In the limit where , so that the posterior is well confined within the prior, one finds (in this case, the averaging is just evaluating the variance of the distribution, but in units of that variance). This corresponds to a wellmeasured parameter. If instead , so that the data are unable to constrain the parameter, then since becomes independent of . Hence indicates the number of parameters actually constrained by the data. Extension of the above argument to an dimensional gaussian, potentially with covariance, indicates if all dimensions are well contained within the prior, and otherwise (SBCL02; Kunz et al. 2006).
One issue of debate in the statistics literature is the choice of the mean parameter value in the definition of . One could alternatively argue for the maximum likelihood in its place. This choice affects the possible reparametrization dependence of the statistic (SBCL02; Celeux et al. 2006). It may be that the best choice depends on the situation under study (e.g. the mean parameter value will be a poor choice if the likelihood has distinct strong peaks).
The DIC is then defined as
(7) 
The first expression is motivated by the form of the AIC, replacing the maximum likelihood with the mean parameter likelihood, and the number of parameters with the effective number. It can therefore be justified on information/decision theory grounds, as discussed by SBCL02. The second form is interesting because the mean deviance can be justified in Bayesian terms, which always deal with modelaveraged quantities rather than maximum values.
The DIC has two attractive properties:

It is determined by quantities readily obtained from Monte Carlo posterior samples. One simply averages the deviances over the samples. If the calculation is being done by whoever generated the chains, they can obtain the deviance at the mean with a single extra likelihood call, but even if using chains generated by others, it should be fine to use the sample closest to that mean value as the estimator, especially bearing in mind the possibility that the mode could have been used in place of the mean. The calculation is also easily done with posterior samples generated by nested sampling, which have noninteger weights (Parkinson et al. 2006).

By using the effective number of parameters, the DIC overcomes the problem of the AIC and BIC that they do not discount parameters which are unconstrained by the data.
Note that in the case of wellconstrained parameters, the DIC approaches the AIC and not the BIC, since and . It is plausible to believe that it too can be corrected for small dataset sizes using the same formula that leads to AIC, though to my knowledge there is currently no proof of this.
2.4 Other criteria
In addition to those already mentioned, the literature contains many other information criteria, but mostly sharing the heritage of those above. The TIC (Takeuchi 1976) generalizes the AIC by dropping the assumption that the true model is in the set considered, but in practice is hard to compute and, where computation has been carried out, tends to give results very similar to the AIC (Burnham & Anderson 2002, 2004). A Bayesian version of the AIC, the Expected AIC (EAIC), where one takes its expected value over the posterior distribution rather than evaluating at the maximum, has been proposed (by Brooks in the comments to SBCL02) but does not appear to have been significantly applied.
Other information criteria, which appear to have been less widely used, include the Network Information Criterion (NIC), the Subspace Information Criterion (SIC, though this abbreviation is sometimes used for Schwarz Information Criterion as another name for the BIC), and the Generalized Information Criterion (GIC). The DIC also comes in many variants, see e.g. Celeux et al. (2006).
An interesting variant was proposed by Sorkin (1983), using a Turing machine construction to define an entropy associated with the theory to be used as a penalty term. This was recently applied to cosmological data by Magueijo & Sorkin (2006). It has not been picked up by the statistics community, but may be related to the widelyused minimum message length paradigm (Wallace & Boulton 1968; Wallace 2005). The idea of interpetting the best model as the one offering maximal algorithmic compression of the data goes all the way back to late 17th century writings by Leibniz.
Model  Parameters  DIC  DIC  AIC  BIC  

Base+  6  5.2  11262.6  11272.9  11262.2  0  0  0 
Base+  6  6.3  11253.3  11265.9  11252.5  7.0  9.7  9.7 
Base++  7  5.6  11253.0  11264.1  11252.6  8.8  7.6  2.3 
Base+++  8  5.4  11254.2  11265.0  11252.6  7.9  5.6  +5.0 
Base+++running  8  6.2  11250.0  11262.3  11249.0  10.6  9.2  +1.4 
2.5 Dimensional consistency and model selection philosophy
Dimensional consistency refers to the behaviour of the model selection statistics in the limit of arbitrarily large datasets. The BIC and evidence are dimensionally consistent, meaning that if one of the considered models is true, they give 100 per cent support to that model as the dataset becomes large. As a necessary consequence, however, they will give 100 per cent support to the best model even if it is not true. By contrast, the AIC is dimensionally inconsistent (Kashyap 1980), sharing its support around the models even with infinite data. As the DIC approaches the AIC in the limit of large datasets, it too is dimensionally inconsistent (SBCL02).
Dimensional consistency does not seem to particularly bother most statisticians, as they are typically seeking models which can explain data and have some predictive power, rather than expecting to represent some underlying truth. Indeed, they commonly quote statistician George Box: “All models are wrong, but some are useful.” The problem of dimensional consistency is therefore mitigated, because they do not expect the set of models to remain static as the dataset evolves. Cosmologists, however, are probably not yet willing to concede that they might be looking for something other than absolute truth specified by a finite number of parameters. Combining this line of argument with statements above, this implies that the Bayesian evidence indeed is the preferred choice for cosmological model selection when it can be calculated.
3 Information criteria for WMAP3
I now apply the information criteria to WMAP3 model fits as compiled by the WMAP team on LAMBDA.^{1}^{1}1Legacy Archive for Microwave Background Data Analysis: http://lambda.gsfc.nasa.gov. Chains were downloaded in December 2006. The subsequent January 2007 update does not allow model selection as the chains were not all generated with the same likelihood code. The DIC calculation is straightforward. The 8 chains for each cosmology are concatenated, the mean deviance found by averaging the likelihoods, and the deviance at the mean estimated by finding the MCMC point located closest to the mean (where the distance in each parameter direction was measured in units of the standard deviation of that parameter).
I also quote the values of the differences in AIC and BIC, where the maximum likelihood is taken directly from the most likely posterior sample (in principle this may slightly disadvantage models with more parameters, for which the most likely sample will typically be slightly further from the true maximum, though for the WMAP3 sample sizes this effect will be small). I take to be the number of power spectrum datapoints, (Spergel et al. 2006), this choice to be discussed further below (nothing changes significantly if a slightly larger number is used to allow for the pixelbased treatment of the low likelihood). With this large value, AIC and AIC are indistinguishable.
The available model fits unfortunately do not quite cover all cases that might be of interest. All wellfitting models vary five standard parameters, being the physical baryon density , the physical cold dark matter density , the sound horizon , the perturbation amplitude , and the optical depth (the Hubble constant and dark energy density are derived parameters). However fits varying just these parameters, a Harrison–Zel’dovich model suggested as the best model from firstyear WMAP data in Liddle (2004), are not available. Nevertheless, I will refer to this as the Base model. Instead, there are two different sixparameter models, one adding the spectral index and one adding the phenomenological Sunyaev–Zel’dovich (SZ) marginalization parameter (Spergel et al. 2006). All further available models include ; extra parameters that I then consider are the spectral index (giving the standard CDM model), further addition of tensors to give the standard slowroll inflation model, and inclusion of spectral index running (without tensors).
The main subtlety is the inclusion of . This is poorly constrained by the data and hence is not expected to contribute fully to ; nevertheless the likelihood does have some dependence on it and it must be included in the analysis that determines the deviance at the mean. Of the parameters considered, and are phenomenological parameters which, at least in principle though not yet in practice, can be determined from the others. The remaining four are truly independent according to present understanding.
The uncertainty in the DIC may not be well estimated by analyzing subsamples, as with smaller samples the mean deviance will be less well estimated by the nearest point. Instead I estimated the uncertainty by employing bootstrap resamples of the combined sample list. This showed that the statistical accuracy was limited by the accuracy with which the values were stored, corresponding to in the DIC. As this is a much smaller uncertainty than the level at which differences are significant, the statistical uncertainty in the determination of the DIC is negligible.
The results are shown in Table 1. The values are in good agreement with expectation. Kunz et al. (2006) computed for several models using a compilation of microwave anisotropy data including WMAP3, and always found close to the input number of parameters. However they ran their own chains and did not include the poorlyconstrained parameters and . Models including those parameters return a significantly less than .
While only the Bayesian evidence has the full interpretation as the model likelihood, leading to the posterior model probability, the AIC has also been interpretted as a model likelihood by defining Akaike weights (Akaike 1981; Burnham & Anderson 2004)
(8) 
where there are models and the differences are with respect to any one. The same interpretation can be given to the DIC differences (SBCL02). For the BIC, insofar as it well approximates twice the log of the Bayes factor, it too can be interpreted as a model likelihood. By convention significance is then judged on the Jeffreys’ scale, which rates as ‘strong’ and as ‘decisive’ evidence against the model with higher criterion value. If the interpretation as model likelihoods holds, these points correspond to odds ratios of approximately 13:1 and 150:1 against the weaker model. As with the evidence, these likelihoods can be further weighted by a prior model probability if desired.
Recall that the DIC, like the AIC, is motivated from information theory, while the BIC is not. Indeed, we see that the DIC results quite closely follow the AIC results; both argue quite strongly against the Base+ model, but are then rather inconclusive amongst the remaining models. So information theory methods are neither for nor against inclusion of extra parameters such as and running at this stage. Incidentally, we can also see that if the DIC were defined using rather than , little difference would have arisen in this comparison.
The information criteria indicate that WMAP3 has put the Harrison–Zel’dovich model (with SZ marginalization) under considerable, if not yet conclusive, pressure. This is in accord with the conclusions reached by Spergel et al. (2006) using chisquared per degree of freedom arguments, though the information criterion give weaker support to this conclusion by recognizing model dimensionality. The strength of conclusion against Harrison–Zel’dovich could also be weakened by various systematic effects in data analysis choices, e.g. inclusion of gravitational lensing (Lewis 2006), beam modelling (Peiris & Easther 2006), and pointsource subtraction (Eriksen et al. 2006; Huffenberger, Eriksen & Hansen 2006).
By contrast, Bayesian approaches do not put under any kind of pressure. Parkinson et al. (2006) found that the full evidences for the Base model and Base+ were indistinguishable with WMAP3 alone, and still inconclusive with inclusion of other datasets. However that analysis did not include SZ marginalization, and so the equivalent comparison cannot be made here. However the BIC comparison between those models each with added does not show any strong preference, and it seems a safe bet that had the Base model itself been supplied by WMAP3, its BIC difference compared with Base+, the best model in the set as judged by the BIC, would not have been significant.
Further, while the information theory methods are ambivalent about and running, the BIC argues rather strongly against, especially in the case of tensors which offer no improvement at all in datafitting. Full evidence calculations however show that this conclusion is quite prior dependent (Parkinson et al. 2006).
That the two methods give such different answers is due to the way that prior assumptions are treated, in particular the prior widths of the parameter ranges. The AIC does not care about this at all, and the DIC only cares while the data is weak enough that some prior information on the parameter distribution remains. By contrast, in Bayesian model comparison the prior width is a key concept, determining the predictiveness of the model. For the evidence this is reflected in the domain of integration over which the likelihood is averaged, while for the BIC it is in the dependence on the amount of data. Cosmologists are in the fortunate position that for many parameters the likelihood is highly compressed within reasonable priors, forcing a discrepancy between information theory and Bayesian results. This discrepancy will be further enhanced in the future if the data continue to improve without requiring evolution in the model dataset, i.e. the problem of dimensional inconsistency of the AIC/DIC may already be with us.
Concerning the inclusion of in models, it is clear that Bayesian methods don’t like including it as a fit parameter, since it is poorly constrained and does not significantly improve the fit. However the SZ effect is certainly predicted to be in the data at some level, though it ought to be derived from the other parameters rather than fit. It is tempting to try to deal with this by using in the BIC rather than , but there is no existing justification for doing so. The same issue does not arise with the optical depth, also a derived parameter, as it is well constrained by the data in all models.
In computing the BIC above, I adopted the number of datapoints literally. This may not always be the best choice: the derivation of the BIC requires the data to be independent and identically distributed, and it may be that this can be better achieved by binning the data in some suitable way. However to do so would require a whole new likelihood analysis for the binned data, counter to the desire here that the methods should be applicable to preexisting posterior samples. In any case there does not appear to be any welldefined way to judge how much binning, if any, is desirable.
Finally, I note that while here it is the BIC which appears to behave most like the evidence, in their quasar clustering studies Porciani & Norberg (2006) found that the DIC was the only criterion to give precisely the same model ranking order and level of inconclusiveness as the Bayes factors, with the BIC underfitting.
4 Summary
I have described several information criteria that can be used for astrophysical model selection, representing the rival strands of information theory and Bayesian inference. In application to WMAP3 data, the DIC behaves rather similarly to the AIC, despite the presence of parameter degeneracies. The conclusions one would draw from those statistics are rather different from those indicated by Bayesian methods, either the full evidence as computed in Parkinson et al. (2006) or the BIC as calculated in this article.
Acknowledgments
This research was supported by PPARC. Thanks to David Parkinson and Pia Mukherjee for advice on analyzing WMAP chains, and Istvan Szapudi for discussions.
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