Part of Advances in Neural Information Processing Systems 13 (NIPS 2000)

*In Myung, Mark Pitt, Shaobo Zhang, Vijay Balasubramanian*

How should we decide among competing explanations of a cognitive process given limited observations? The problem of model selection is at the heart of progress in cognitive science. In this paper, Minimum Description Length (MDL) is introduced as a method for selecting among computational models of cognition. We also show that differential geometry provides an intuitive understanding of what drives model selection in MDL. Finally, adequacy of MDL is demonstrated in two areas of cognitive modeling.

1 Model Selection and Model Complexity

The development and testing of computational models of cognitive processing are a central focus in cognitive science. A model embodies a solution to a problem whose adequacy is evaluated by its ability to mimic behavior by capturing the regularities underlying observed data. This enterprise of model selection is challenging because of the competing goals that must be satisfied. Traditionally, computational models of cognition have been compared using one of many goodness-of-fit measures. However, use of such a measure can result in the choice of a model that over-fits the data, one that captures idiosyncracies in the particular data set (i.e., noise) over and above the underlying regularities of interest. Such models are considered complex, in that the inherent flexibility in the model enables it to fit diverse patterns of data. As a group, they can be characterized as having many parameters that are combined in a highly nonlinear fashion in the model equation. They do not assume a single structure in the data. Rather, the model contains multiple structures; each obtained by finely tuning the parameter values of the model, and thus can fit a wide range of data patterns. In contrast, simple models, frequently with few parameters, assume a specific structure in the data, which will manifest itself as a narrow range of similar data patterns. Only when one of these patterns occurs will the model fit the data well.

The problem of over-fitting data due to model complexity suggests that the goal of model selection should instead be to select the model that generalizes best to all data samples that arise from the same underlying regularity, thus capturing only the regularity, not the noise. To achieve this goal, the selection method must be sensitive to the complexity of a model. There are at least two independent dimensions of model complexity. They are the number of free parameters of a

model and its functional form, which refers to the way the parameters are combined in the model equation. For instance, it seems unlikely that two one-parameter models, y = ex and y = x9, are equally complex in their ability to fit data. The two dimensions of model complexity (number of parameters and functional form) and their interplay can improve a model's fit to the data, without necessarily improving generalizability.

The trademark of a good model selection procedure, then, is its ability to satisfy two opposing goals. A model must be sufficiently complex to describe the data sample accurately, but without over-fitting the data and thus losing generalizability. To achieve this end, we need a theoretically well-justified measure of model complexity that takes into account the number of parameters and the functional form of a model. In this paper, we introduce Minimum Description Length (MDL) as an appropriate method of selecting among mathematical models of cognition. We also show that MDL has an elegant geometric interpretation that provides a clear, intuitive understanding of the meaning of complexity in MDL. Finally, application examples of MDL are presented in two areas of cognitive modeling.

1.1 Minimum Description Length

The central thesis of model selection is the estimation of a model's generalizability. One approach to assessing generalizability is the Minimum Description Length (MDL) principle [1]. It provides a theoretically well-grounded measure of complexity that is sensitive to both dimensions of complexity and also lends itself to intuitive, geometric interpretations. MDL was developed within algorithmic coding theory to choose the model that permits the greatest compression of data. A model family f with parameters e assigns the likelihood f(yle) to a given set of observed data y . The full form of the MDL measure for such a model family is given below.

MDL = -In! (yISA) + ~ln( ; ) + In f dS.jdetl(S)

where SA is the parameter that maximizes the likelihood, k is the number of parameters in the model, N is the sample size and I(e) is the Fisher information matrix. MDL is the length in bits of the shortest possible code that describes the data with the help of a model. In the context of cognitive modeling, the model that minimizes MDL uncovers the greatest amount of regularity (i.e., knowledge) underlying the data and therefore should be selected. The first, maximized log likelihood term is the lack-of-fit measure, and the second and third terms constitute the intrinsic complexity of the model. In particular, the third term captures the effects of complexity due to functional form, reflected through I(e). We will call the latter two terms together the geometric complexity of the model, for reasons that will become clear in the remainder of this paper.

MDL arises as a finite series of terms in an asymptotic expansion of the Bayesian posterior probability of a model given the data for a special form of the parameter prior density [2] . Hence in essence, minimization of MDL is equivalent to maximization of the Bayesian posterior probability. In this paper we present a geometric interpretation of MDL, as well as Bayesian model selection [3], that provides an elegant and intuitive framework for understanding model complexity, a central concept in model selection.

2 Differential Geometric Interpretation of MDL

From a geometric perspective, a parametric model family of probability distributions forms a Riemannian manifold embedded in the space of all probability

distributions [4]. Every distribution is a point in this space, and the collection of points created by varying the parameters of the model gives rise to a hyper-surface in which "similar" distributions are mapped to "nearby" points. The infinitesimal distance between points separated by the infinitesimal parameter differences de; is given by ds 2 = Y' k. g .. (8 )d8 ; d8 j where g ij(e) is the Riemannian metric tensor. The Fisher information, lij(e), is the natural metric on a manifold of distributions in the context of statistical inference [4]. We argue that the MDL measure of model fitness has an attractive interpretation in such a geometric context.

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