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A measure of scientiﬁc advancement in a discipline is the discovery of general laws and principles that govern the phenomenon of interest. As the underlying principles are typically not directly observable, they are formulated in terms of hypotheses. Models are formal expressions of such hypotheses, whose viability must be evaluated against data generated by the phenomenon. In this research paper we review computational approaches to model evaluation, with a special emphasis on the inductive inference perspective. A more in-depth treatment of the topic can be found in Cherkassky and Mulier (1998) and Haykin (1999). For a discussion of some of the issues presented here from the statistical perspective.

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## 1. Modeling As Inductive Inference

The goal of modeling is straightforward: given observed data, identify the underlying system that generated the data. Realistically, modeling is viewed as a learning problem in which we estimate an unknown function from observed data and then use the estimated function to predict the statistics of future data. When modeling, we reconstruct an unknown function based on several sample points in the data space (local subspace) and then apply it globally, throughout the data space. This process of generalizing from a particular instance to all instances is referred to as inductive inference, or predictive inference given the predictive nature of the process and the goals of the modeler. It is at the heart of the scientiﬁc enterprise in modern science as well as in engineering.

### 1.1 Ill-Posedness Of The Modeling Problem

Identifying an unknown function from data samples is an ill-posed problem, meaning that obtaining a unique solution is not in general possible. The reason stems from the reality that there is rarely enough data available (or information in the data) to discriminate between models. Multiple functions may provide equally good descriptions of the data in hand. This problem is known as the curse of dimensionality, which states that for a function deﬁned in high-dimensional space, it becomes increasingly diﬃcult to estimate accurately the function as dimensionality increases, with required sample sizes increasing exponentially with dimensionality (Cherkassky and Mulier 1998). This problem of ill-posedness can be overcome only by introducing external constraints on, and/or prior information about, the data-generating function.

### 1.2 Model Evaluation: Finding Best Approximation To Truth

An implication of the ill-posedness problem is that the ‘true’ model, the one that generated the data, can never be found in practice. Accordingly, the previously deﬁned goal of ﬁnding the true function underlying observations must be modiﬁed. A more realistic goal is to ﬁnd the best possible approximation to the truth, in which the best approximation is deﬁned as the one that generalizes (i.e., infers) best in some deﬁned sense.

## 2. Bias-variance Dilemma

Formally, a model is deﬁned as a parametric family of probability distributions deﬁned on random variables Y, representing data. An observed sample of size n denoted by a lower case vector y =(y_{1}, …, y_{n}) rep- resent realizations (measurements) of Y, which are presumed to be generated from some true distribution g( y) that the scientist seeks to unravel. The scientist, using a family of distributions f( y|θ)’s where θ is a parameter vector on the parameter space Θ, seeks to construct the best approximation to g within the family f.

Let t =E( y) denote the population mean vector of the true distribution g( y), and let z(θ)= (z_{1}, …, z_{n}) denote the vector predicted by the model atθ. Also, let the quadratic function

denote a measure of the discrepancy or error between a vector x and the prediction vector z(θ). Note that θ_{x} denotes the value of the parameter that minimizes the discrepancy measure given a vector x. This function is used to measure the eﬀectiveness or risk of the model family f in approximating the truth g. For example, L( y, z(y)) represents the minimized mean squared error between observed and predicted data, that is, the best possible ﬁt the model provides given data y. Because this quantity changes from sample to sample, a better measure of model eﬀectiveness (or risk) is the expected value of L( y, z(θ_{y})), which can be expressed as the sum of two terms:

In the above equation E denotes the expectation (i.e., average) taken over all data samples of size n which could be realized under the true distribution g, and t is the mean vector of the true distribution g( y) that has generated the data. Note that the value of the prediction vector z(θ_{y}) varies across samples whereas z(θ_{t}) is a ﬁxed constant vector, which can be replaced by E [z(θ_{y})].

The left-hand side of the above equation is the expected risk in predicting all data samples given the model family f, and thus it measures the model’s generalizability. The ﬁrst term of the right-hand side of the equation is the discrepancy between the true mean vector t and the model’s best approximation to t, z(θ_{t}). Accordingly, this term captures the model’s inability to accurately approximate the truth g, and is called the bias or approximation error. If the model family f includes g as a special case, the bias term becomes zero, and takes on a positive value otherwise. The second term is the expected variability of the prediction vector z(θ_{y}) around its mean z(θ_{t}), and is called the variance or estimation error. It captures the lack of information in the sample of ﬁnite size about the truth g. As such, this term decreases to zero as sample size n goes to inﬁnity.

To achieve good generalization performance, both the bias and variance of the model family f would have to be minimized at the same time. It turns out, however, that this is not possible because the bias can be minimized by increasing model complexity, whereas increasing complexity will have the detrimental eﬀect of increasing the variance by capturing random error in the data, thereby reducing generalization performance. Complexity refers to the ﬂexibility inherent in a model that enables it to ﬁt diverse patterns of data. In general, a simple model with few parameters has a large bias but a small variance, whereas a complex model (one with more parameters) has the opposite. We thus have a bias-variance dilemma (Geman et al. 1992), with model complexity as the intervening variable that mediates between the two.

The bias-variance dilemma is a useful conceptual framework for understanding the trade-oﬀ between approximation error and estimation error, both of which contribute to a model’s generalizability. However, the generalizability measure represented by the sum of bias and variance is not directly observable, because the truth g is unknown, and therefore it must be inferred from observed data.

## 3. Computational Approaches To Model Evaluation

In order to make the ill-posed problem of modeling into a well-posed one, additional constraints and prior information must be incorporated into the model evaluation process. Computational approaches to the problem diﬀer from one another in how this is accomplished. Structural risk minimization (Vapnik 1992), regularization theory (Tikhonov 1963) and minimum description length (Rissanen 1989) are three major inductive methods of model evaluation.

Brieﬂy, the structural risk minimization method imposes a constraint requiring that, among a set of candidate models, the one that minimizes an expected risk functional should be chosen. Regularization theory chooses among models based on the prior assumption that the unknown function underlying the data is smooth, in the sense that similar inputs map onto similar outputs. The minimum description length method prescribes the constraint that the description length of data should be minimized. Formalizations of these ideas are described in the following sections.

### 3.1 Structural Risk Minimization

The basic idea of structural risk minimization is to estimate an upper bound of the expected risk, E [L( y, z(θ_{y})], in terms of the observable empirical risk, L_{emp}(y, z(θ_{y})), given data sample y and a conﬁdence interval, the value of which depends on a measure of model complexity and sample size, n. For binary classiﬁcation problems for which y_{i} =0 or 1, the following inequality holds

where I is the conﬁdence interval, α is the alpha level of inferential statistics, and h is a quantity known as the Vapnik–Chervonenkis (VC) dimension (Vapnik and Chervonenkis 1971). The model that provides the lowest value of the upper bound for the expected risk should be chosen.

The VC dimension for a given model family of classiﬁcation functions is deﬁned as the maximum number of distinct binary samples that the model can ﬁt perfectly without error for all possible labelings of the samples. Thus the VC dimension may be viewed as a measure of the complexity or ﬂexibility of a model. Interestingly, the VC dimension depends not only on the number of parameters of a model but also, importantly, on the functional form of the model equation. A model with only one parameter can have inﬁnite VC dimension while a model with a billion parameters can have a low VC dimension. For linear models, the VC dimension is equal to the number of parameters.

Given and n, the empirical risk L_{emp}( y, z(θ_{y})) is a monotonically decreasing function of the VC dimension whereas the conﬁdence interval I(h, n) is a monotonically increasing function of the VC dimension. These relationships along with the above equation reveal how structural risk minimization works. Among a set of competing models, the method chooses the model that describes the data suﬃciently well (i.e., smallest empirical risk) with the least complex structure (i.e., narrowest conﬁdence interval), thereby formalizing the principle of Occam’s razor. In practice, structural risk minimization is implemented by ﬁrst creating a set of k nested models with successively increasing VC dimensions, h_{1} < h_{2} < …,< k, and then calculating the empirical risk for each model given sample data. The model with the smallest expected risk is then selected.

### 3.2 Regularization Theory

Regularization theory’s solution for solving the ill-posed problem is to secure the uniqueness of the solution by requiring it to satisfy some secondary condition that embeds prior information or assumptions about the solution. The form of prior information most often used is the condition that the unknown function being sought is suﬃciently smooth. Formally, the regularization risk for a model is deﬁned as

where the vector z(θ) is the model’s prediction at θ, is known as the regularization parameter, and S(z) is a functional that measures smoothness of the solution. The quantity often used for S(z) is the integral of the squared second derivatives with respect to an in-dependent variable x on which smoothness of the solution is assessed, i.e.,

The regularization theory prescribes that the model with the smallest R(θ) should be chosen.

The ﬁrst term of the above equation is a measure of the discrepancy between the data and the model’s prediction, that is, the empirical risk of the model. The second term, S(z), is a measure of complexity of the model. The larger the value of S(z), the less smooth theprediction,andthusthemorecomplexthemodel.The regularization parameter λ represents the relative importance of the complexity of the model with respect to its empirical risk. With λ_{0} , the solution is primarily determined by observed data and thus tends to be highly complex in structure. On the other hand, with λ →∞, the smoothness condition entirely dictates the form of the solution, which tends to be simple in structure. In practice, a value between these two extremes is chosen based on some principled approach (Haykin 1999). In short, the regularization method, like the structural minimization method, uses complexity to steer the solution toward improved generalization. As such, it is another implementation of the principle of Occam’s razor.

### 3.3 Minimum Description Length

Minimum description length (MDL), which originated from algorithmic coding theory in computer science, regards both model and data as codes. The idea is that any data set can be appropriately encoded with the help of a model, and that the more we compress the data by extracting redundancy from it, the more we uncover underlying regularities in the data. Thus, code length is directly related to the generalization capability of the model, where the model that provides the shortest description of the data should be chosen. Formally, the total description length of a model is given by

where L(data model) is the description length of the data given the model, measured in bits of information, and L(model) is the description length of the model itself.

The above deﬁnition of minimum description length is broad enough to be applied to any well-deﬁned models, even verbally deﬁned, qualitative models. For a statistical model deﬁned as a parametrized family of probability distributions, the minimum description length for such a model takes the following form (Rissanen 1996):

In the equation θ_{y} is the parameter value that maximizes the likelihood function f( y|θ), k is the number of parameters, I(θ) is the Fisher information matrix and det(I ) is the determinant of the matrix. The ﬁrst term of the MDL can be interpreted as a measure of risk or lack of ﬁt, and the second and third terms together are a measure of model complexity. Model evaluation is therefore carried out by trading oﬀ lack of ﬁt for simplicity.

In summary, although the three approaches to model evaluation evolved from diﬀerent conceptual frameworks and philosophies, they all share the common goal of achieving optimum generalizability by striking a balance between risk (or lack of ﬁt) and complexity. Another commonality between these approaches is the aspect of a model that is captured by the complexity measure; they not only include the number of parameters in the model, but also, importantly, its functional form, which refers to the way in which the parameters are combined in the model equation.

## 4. Relations To Statistical Approaches

So far we have considered computational approaches that view model evaluation as an inductive inference problem, which is the predominant view in computer science and engineering. Model evaluation is also a topic of central interest for statisticians (statisticians prefer the term model selection to model evaluation). Approaches which have been developed within the statistical framework include the generalized likelihood ratio test (GLRT), cross-validation, the Akaike information criterion (AIC), the Bayesian information criterion (BIC) and Bayesian model selection. There are many important diﬀerences between these methods, but in essence, they all implement a means of ﬁnding, explicitly or implicitly, the best compromise between lack of ﬁt and complexity by trading oﬀ one for the other. Statistical model selection methods are entirely consistent with the computational modeling approaches discussed above (see Sect. 3).

Here are a few notable diﬀerences and similarities between some of the statistical methods and computational methods. First, GLRT, AIC, and BIC diﬀer from the computational methods such as structural risk minimization and minimum description length in that the two statistical selection criteria consider only the number of parameters as a complexity measure, and thus are insensitive to functional form, which can signiﬁcantly inﬂuence generalizability. Second, application of the statistical methods requires that each model under investigation be a quantitative model deﬁned as a parametric family of probability distributions. On the other hand, the computational methods can be applied to qualitative models as well as quantitative ones. For instance, the minimum description length method deﬁned in Eqn. (4) is applicable to evaluating the eﬀectiveness of decision tree models or even verbal models (Li and Vitanyi 1997). Finally, there exists a close connection between Bayesian model selection and the minimum description length criterion deﬁned in Eqn. (5). The minimum description length criterion can be derived as an asymptotic approximation to the posterior probability in Bayesian model selection for a special form of the parameter prior density. An implication of this connection is that choosing the model that gives the shortest explanation for the observed data is essentially equivalent to choosing the model that is most likely to be true in the sense of probability theory (Li and Vitanyi 1997).

## 5. Conclusion

The question of how one should decide among competing explanations (i.e., models) of data is at the core of the scientiﬁc endeavor. A number of criteria have been proposed for model evaluation (Jacobs and Grainger 1994). They include (a) falsiﬁability: are there potential outcomes inconsistent with the model? (b) explanatory adequacy: is the explanation compatible with established ﬁndings? (c) interpretability: does the model make sense? Is it understandable? (d) descriptive adequacy: does the model provide a good description of the observed data? (e) simplicity: does the model describe the phenomenon in the simplest possible manner? and (f) generalizability: does the model predict well the characteristics of new, as yet unseen, data?

Among these criteria, computational modeling approaches described in this research paper, as well as statistical approaches, consider the last three as they are easier to quantify than the other three. In particular, generalizability has been emphasized as the principal criterion by which the eﬀectiveness of a model be judged. Obviously, this ‘bias’ toward the predictive inference aspect of modeling reﬂects the dominant thinking of the current scientiﬁc community. Clearly one could develop an alternative approach that stresses other criteria, and even incorporates some of the hard-to-quantify criteria. There are plenty of opportunities in this area for the emerging ﬁeld of model evaluation to grow and mature in the decades to come.

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