Model complexity penalties

Information criteria, degrees of freedom etc

April 22, 2015 — June 22, 2020

estimator distribution
model selection

Understanding the “Degrees of freedom” of a model. Estimating that trace penalty matrix. As seen in robust estimation and (?) AIC/BIC.

🏗 Explain AIC, \(C_p\), SURE, and BIC-type degrees of freedom, and whatever variants there are out there.

Figure 1

Complexity penalties crop up in model selection. (i.e. choosing the complexity of model appropriate to your data.) Efron (2004) is an excellent introduction, compressing 30 years of theory into 2 pages. Massart (2000) seems more fashionable in flavour:

The reader who is not familiar with model selection via [complexity] penalization can legitimately ask the question: where does the idea of penalization come from? It is possible to answer this question at two different levels:

  • at some intuitive level by presenting the heuristics of one of the first criterion of this kind which has been introduced by Akaike (1973);

  • at some technical level by explaining why such a strategy of model selection has some chances to succeed.

(Yuan and Lin 2006) are an example of the kind of argumentation I need to use to use linear model approximation for general application of DOF in sparse model selection.

(Zou, Hastie, and Tibshirani 2007):

Degrees of freedom is a familiar phrase for many statisticians. In linear regression the degrees of freedom is the number of estimated predictors. Degrees of freedom is often used to quantify the model complexity of a statistical modeling procedure (Hastie and Tibshirani 1990). However, generally speaking, there is no exact correspondence between the degrees of freedom and the number of parameters in the model (Ye 1998). […] Stein’s unbiased risk estimation (SURE) theory (Stein 1981) gives a rigorous definition of the degrees of freedom for any fitting procedure. […] Efron (Efron 2004) showed that \(C_p\) is an unbiased estimator of the true prediction error, and in some settings it offers substantially better accuracy than cross-validation and related nonparametric methods. Thus degrees of freedom plays an important role in model assessment and selection. Donoho and Johnstone (Donoho and Johnstone 1995) used the SURE theory to derive the degrees of freedom of soft thresholding and showed that it leads to an adaptive wavelet shrinkage procedure called SureShrink. (Ye 1998) and (Shen and Ye 2002) showed that the degrees of freedom can capture the inherent uncertainty in modeling and frequentist model selection. Shen and Ye (Shen and Ye 2002) and (Shen, Huang, and Ye 2004) further proved that the degrees of freedom provides an adaptive model selection criterion that performs better than the fixed-penalty model selection criteria.

1 Information Criteria

Akaike and friends. With M-estimation, (e.g. maximum likelihood estimation and robust estimation) these are marvelous and general shortcuts to do model selection. (i.e. choosing the complexity of model appropriate to your data) without resorting to computionally expensive cross validation.

For all of these, a thing called the number of effective degrees of freedom is important. There are several different definitions for that, and they only sometimes coincide, so I leave that for a different notebook. Claeskens and Hjort (2008) and Konishi and Kitagawa (2008) are probably canonical.

Information criteria can ideally do the same thing cross-validation (i.e. select ideal regularisation given possible models and data) a small fraction of the computational cost. Indeed, they are asymptotically the same - see below.

To learn:

  • How this interacts with robust estimators

  • How to use AIC with nonparametric or high dimensional methods (GIC)

  • How it relates to minimum description length (e.g. Andrew R. Barron et al. (2008))

Influential current English-language texts in this area are Burnham and Anderson (2002), Claeskens and Hjort (2008) and Konishi and Kitagawa (2008). The first of these is highly cited and brought the AIC method into the mainstream in the West from where it had been on the specalised fringes. The latter two focus on extensions such as TIC and GIC.

🏗 general description.

🏗 clarify relationship to Minimum Description Length, Rissanen-style.

Bondell, Krishna, and Ghosh (2010):

In the literature, selection criteria are usually classified into two categories: consistent (e.g., the Bayesian information criterion BIC, Schwarz, 1978) and efficient (e.g., the Akaike information criterion AIC, Akaike, 1974; the generalized cross-validation GCV, Craven and Wahba, 1979). A consistent criterion identifies the true model with a probability that approaches 1 in large samples when a set of candidate models contains the true model. An efficient criterion selects the model so that its average squared error is asymptotically equivalent to the minimum offered by the candidate models when the true model is approximated by a family of candidate models. Detailed discussions on efficiency and consistency can be found in Shibata (1981, 1984), Li (1987), Shao (1997) and McQuarrie and Tsai (1998).

1.1 Akaike Information Criterion (AIC)

The classic.🏗

1.2 Takeuchi Information Criterion (TIC)

Apparently this one was influential in Japan, but untranslated into English, so only belately common in the west. Good explanations are in Claeskens and Hjort (2008) and Konishi and Kitagawa (2008). Relaxes the assumption that the model is Fisher efficient (i.e. that the true generating process is included in your model, and with enough data you’d discover that.)

1.3 Konishi and Kitegawa’s Generalised Information Criterion (GIC)

Taking information criteria to general (e.g. robust, penalised) M-estimation instead of purely ML estimation; also relaxing the assumption that we even have the “true” model in our class. (Konishi and Kitagawa (1996)); C&C Burman and Nolan (1995), probably others. In paricular, you are no longer trying to fit the midel by minimising least-squares errors, for example. Claeskens and Hjort (2008) mention the “Robustified Information Criterion” in passing, which may relate?

🏗 Explain my laborious reasoning that generalised Akaike information criteria for penalised regression don’t seem work when the penalty term is not differentiable, (cross validation works fine though, abd possibly also BIC) and the issues that therefore arise in model selection for such models in the sparse case.

1.4 Focussed information criterion (FIC)

Claeskens and Hjort define this (Claeskens and Hjort (2008), chapter 6):

The model selection methods presented earlier (such as AIC and the BIC) have one thing in common: they select one single ‘best model’, which should then be used to explain all aspects of the mechanisms underlying the data and predict all future data points. The tolerance discussion in chapter 5 showed that sometimes one model is best for estimating one type of estimand, whereas another model is best for another estimand. The point of view expressed via the [FIC] is that a ‘best model’ should depend on the parameter under focus, such as the mean, or the variance, or the particular covariate values etc Thus the FIC allows and encourages different models to be selected for different parameters of interest.

This sounds very logical; of course, then one must do more work to make it go.

1.5 Network information criterion

Murata, Yoshizawa, and Amari (1994): “an estimator of the expected loss of a loss function \(\ell(\theta)+\lambda H(\theta)\) where \(H(\theta)\) is a regularisation term”.

1.6 Regularization information criterion

Shibata (1989) - is this distinct from GIC?

1.7 Bootstrap information criterion

A compromise between the computational cheapness of information criteria and the practical simplicity of cross-validation.

Konishi and Kitagawa (2008) ch 8. See Claeskens and Hjort (2008) 6.3 for a bootstrap-FIC.

1.8 Consistency of model order selected - AIC

Akaike Information criteria are not asymptotically consistent (see Konishi and Kitagawa (2008).) in the sense that if there is a true model, you do not asymptotically select in the large sample limit with P=1. However, the distribution of model orders does not get worse as n increases. Burnham and Anderson (2002) 6.3 and Konishi and Kitagawa (2008) 3.5.2 discuss this; In a sense it would be surprising if it did do especially well in selecting model order; since our criteria is rather designed to minimise prediction error, not model selection error. Model order is more or less a nuisance parameter in this framework.


1.9 Cross-validation equivalence

Konishi and Kitagawa (2008), 10.1.4 discuss the asymptotic equivalence of AIC/TIC/GIC and cross validation under various circumstances, attributing the equivalence results to Stone (1977) and Shibata (1989). Claeskens and Hjort (2008) proves a similar result.

1.10 Automatic GIC

🏗; I know that Konishi and Kitagawa (1996) give formulae for loss functions for ANY M-estimation and penalisation procedure, but in general the degrees of freedom matrix trace calculation is nasty, and only in-principle estimable from the data, requiring a matrix product of the hessian at every data point. This is not necessarily computationally tractable - I know of formulae only for GLMs and robust regression with \(\ell_2\) penalties. Can we get such penalties for more general ML fits?

1.11 GIC & the LASSO

I thought this didn’t work because we needed the second derivative of the penalty; but see Bondell, Krishna, and Ghosh (2010).

1.12 Information criteria at scale

Big-data information criteria. AIC is already computationally cheaper than cross validation. What about when my data is so large that I would like to select my mode before looking at all of it with such-and-such a guarantee of goodness? Can I do AIC at scale? if I am fitting a model using SGD, can I estimate my model order using partial data? How? I’m interested in doing this in a way that preserves the property of being computationally cheaper than cross-validating.

Here’s an example… Bondell, Krishna, and Ghosh (2010):

In order to avoid complete enumeration of all possible \(2^{p+q}\) models, Wolfinger (1993) and Diggle, Liang and Zeger (1994) recommended the Restricted Information Criterion (denoted by REML.IC), in that, by using the most complex mean structure, selection is first performed on the variance-covariance structure by computing the AIC and/or BIC. Given the best covariance structure, selection is then performed on the fixed effects. Alternatively, Pu and Niu (2006) proposed the EGIC (Extended GIC), where using the BIC, selection is first performed on the fixed effects by including all of the random effects into the model. Once the fixed effect structure is chosen, selection is then performed on the random effects.

In general I’d like to avoid enumerating the models as much as possible and simply select relevant predictors with high probability, compressive-sensing style.

2 Consistent: Bayesian Information Criteria

a.k.a. Schwarz Information Criterion. Also co-invented by the unstoppable Akaike. (Hirotugu Akaike 1978; Schwarz 1978)

This is a different family to the original AIC. This has a justification in terms of MDL and of Bayes risk? Different regularity conditions, something something…

How would this work with regularisation? Apparently Machado (1993) extends the setup to robust inference, much as the GIC extends the AIC. Claeskens and Hjort (2008) give an easy summary and more general settings.

3 Consistent and/or efficient: Nishii’s Generalised Information Criterion

Nishii (1984), commended by Zhang, Li, and Tsai (2010) as a unifying formalism for these efficient/consistent others, includes efficient and consistent-type information penalties as special cases I don’t know much about this.

4 Quasilikielood

See QIC.

5 References

Akaike, Hirotogu. 1973. Information Theory and an Extension of the Maximum Likelihood Principle.” In Proceeding of the Second International Symposium on Information Theory.
Akaike, Htrotugu. 1973. Maximum Likelihood Identification of Gaussian Autoregressive Moving Average Models.” Biometrika.
Akaike, Hirotugu. 1978. A New Look at the Bayes Procedure.” Biometrika.
———. 1981. Likelihood of a Model and Information Criteria.” Journal of Econometrics.
Ando, Konishi, and Imoto. 2008. Nonlinear Regression Modeling via Regularized Radial Basis Function Networks.” Journal of Statistical Planning and Inference, Special Issue in Honor of Junjiro Ogawa (1915 - 2000): Design of Experiments, Multivariate Analysis and Statistical Inference,.
Barron, Andrew R. 1986. Entropy and the Central Limit Theorem.” The Annals of Probability.
Barron, Andrew R., Huang, Li, et al. 2008. MDL, Penalized Likelihood, and Statistical Risk.” In Information Theory Workshop, 2008. ITW’08. IEEE.
Barron, A., Rissanen, and Yu. 1998. The Minimum Description Length Principle in Coding and Modeling.” IEEE Transactions on Information Theory.
Bashtannyk, and Hyndman. 2001. Bandwidth Selection for Kernel Conditional Density Estimation.” Computational Statistics & Data Analysis.
Bickel, Li, Tsybakov, et al. 2006. Regularization in Statistics.” Test.
Birgé, and Massart. 2006. Minimal Penalties for Gaussian Model Selection.” Probability Theory and Related Fields.
Bondell, Krishna, and Ghosh. 2010. Joint Variable Selection for Fixed and Random Effects in Linear Mixed-Effects Models.” Biometrics.
Buckland, Burnham, and Augustin. 1997. Model Selection: An Integral Part of Inference.” Biometrics.
Bunea. 2004. Consistent Covariate Selection and Post Model Selection Inference in Semiparametric Regression.” The Annals of Statistics.
Burman, and Nolan. 1995. A General Akaike-Type Criterion for Model Selection in Robust Regression.” Biometrika.
Burnham, and Anderson. 2002. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach.
Burnham, and Anderson. 2004. Multimodel Inference Understanding AIC and BIC in Model Selection.” Sociological Methods & Research.
Cavanaugh. 1997. Unifying the Derivations for the Akaike and Corrected Akaike Information Criteria.” Statistics & Probability Letters.
Cavanaugh, and Shumway. 1998. An Akaike Information Criterion for Model Selection in the Presence of Incomplete Data.” Journal of Statistical Planning and Inference.
Chen, and Chen. 2008. Extended Bayesian Information Criteria for Model Selection with Large Model Spaces.” Biometrika.
Chichignoud, Lederer, and Wainwright. 2014. A Practical Scheme and Fast Algorithm to Tune the Lasso With Optimality Guarantees.” arXiv:1410.0247 [Math, Stat].
Claeskens, and Hjort. 2008. Model Selection and Model Averaging. Cambridge Series in Statistical and Probabilistic Mathematics.
Claeskens, Krivobokova, and Opsomer. 2009. Asymptotic Properties of Penalized Spline Estimators.” Biometrika.
Donoho, and Johnstone. 1995. Adapting to Unknown Smoothness via Wavelet Shrinkage.” Journal of the American Statistical Association.
Dossal, Kachour, Fadili, et al. 2011. The Degrees of Freedom of the Lasso for General Design Matrix.” arXiv:1111.1162 [Cs, Math, Stat].
Efron. 1986. How Biased Is the Apparent Error Rate of a Prediction Rule? Journal of the American Statistical Association.
———. 2004. The Estimation of Prediction Error.” Journal of the American Statistical Association.
———. 2021. Resampling Plans and the Estimation of Prediction Error.” Stats.
Fan, and Li. 2001. Variable Selection via Nonconcave Penalized Likelihood and Its Oracle Properties.” Journal of the American Statistical Association.
Firth. 1993. Bias Reduction of Maximum Likelihood Estimates.” Biometrika.
Hastie, and Tibshirani. 1990. Generalized Additive Models.
Huang, Cong, Cheang, and Barron. 2008. Risk of Penalized Least Squares, Greedy Selection and L1 Penalization for Flexible Function Libraries.”
Huang, Jian, Ma, Xie, et al. 2009. A Group Bridge Approach for Variable Selection.” Biometrika.
Hurvich, Simonoff, and Tsai. 1998. Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion.” Journal of the Royal Statistical Society. Series B (Statistical Methodology).
Hurvich, and Tsai. 1989. Regression and Time Series Model Selection in Small Samples.” Biometrika.
Hu, and Zidek. 2002. The Weighted Likelihood.” The Canadian Journal of Statistics / La Revue Canadienne de Statistique.
Imoto, and Konishi. 1999. Estimation of B-Spline Nonparametric Regression Models Using Information.”
Janson, Fithian, and Hastie. 2015. Effective Degrees of Freedom: A Flawed Metaphor.” Biometrika.
Kato. 2009. On the Degrees of Freedom in Shrinkage Estimation.” Journal of Multivariate Analysis.
Kaufman, and Rosset. 2014. When Does More Regularization Imply Fewer Degrees of Freedom? Sufficient Conditions and Counterexamples.” Biometrika.
Konishi, and Kitagawa. 1996. Generalised Information Criteria in Model Selection.” Biometrika.
———. 2003. Asymptotic Theory for Information Criteria in Model Selection—Functional Approach.” Journal of Statistical Planning and Inference, C.R. Rao 80th Birthday Felicitation Volume, Part IV,.
Konishi, and Kitagawa. 2008. Information Criteria and Statistical Modeling. Springer Series in Statistics.
Kosmidis. 2014. Bias in Parametric Estimation: Reduction and Useful Side-Effects.” WIREs Computational Statistics.
Kosmidis, and Lunardon. 2020. Empirical Bias-Reducing Adjustments to Estimating Functions.” arXiv:2001.03786 [Math, Stat].
Le, and Clarke. 2017. A Bayes Interpretation of Stacking for M-Complete and M-Open Settings.” Bayesian Analysis.
Leung, and Barron. 2006. Information Theory and Mixing Least-Squares Regressions.” IEEE Transactions on Information Theory.
Li, Ker-Chau. 1987. Asymptotic Optimality for \(C_p, C_L\), Cross-Validation and Generalized Cross-Validation: Discrete Index Set.” The Annals of Statistics.
Li, Jonathan Q., and Barron. 2000. Mixture Density Estimation.” In Advances in Neural Information Processing Systems 12.
Li, Runze, and Liang. 2008. Variable Selection in Semiparametric Regression Modeling.” The Annals of Statistics.
Lim, and Lederer. 2016. Efficient Feature Selection With Large and High-Dimensional Data.” arXiv:1609.07195 [Stat].
Machado. 1993. Robust Model Selection and M-Estimation.” Econometric Theory.
Massart. 2000. Some Applications of Concentration Inequalities to Statistics.” In Annales de La Faculté Des Sciences de Toulouse: Mathématiques.
———. 2007. Concentration Inequalities and Model Selection: Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003. Lecture Notes in Mathematics 1896.
Murata, Yoshizawa, and Amari. 1994. Network Information Criterion-Determining the Number of Hidden Units for an Artificial Neural Network Model.” IEEE Transactions on Neural Networks.
Nishii. 1984. Asymptotic Properties of Criteria for Selection of Variables in Multiple Regression.” The Annals of Statistics.
Qian, and Hans. 1996. Some Notes on Rissanen’s Stochastic Complexity.”
Qian, and Künsch. 1998. On Model Selection via Stochastic Complexity in Robust Linear Regression.” Journal of Statistical Planning and Inference.
Rao, Radhakrishna, and Wu. 1989. A Strongly Consistent Procedure for Model Selection in a Regression Problem.” Biometrika.
Rao, C. R., and Wu. 2001. On Model Selection.” In Institute of Mathematical Statistics Lecture Notes - Monograph Series.
Rissanen. 1978. Modeling by Shortest Data Description.” Automatica.
Saefken, Kneib, Waveren, et al. 2014. A Unifying Approach to the Estimation of the Conditional Akaike Information in Generalized Linear Mixed Models.” Electronic Journal of Statistics.
Schwarz. 1978. Estimating the Dimension of a Model.” The Annals of Statistics.
Shen, and Huang. 2006. Optimal Model Assessment, Selection, and Combination.” Journal of the American Statistical Association.
Shen, Huang, and Ye. 2004. Adaptive Model Selection and Assessment for Exponential Family Distributions.” Technometrics.
Shen, and Ye. 2002. Adaptive Model Selection.” Journal of the American Statistical Association.
Shibata. 1989. Statistical Aspects of Model Selection.” In From Data to Model.
Stein. 1981. Estimation of the Mean of a Multivariate Normal Distribution.” The Annals of Statistics.
Stone. 1977. An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike’s Criterion.” Journal of the Royal Statistical Society. Series B (Methodological).
———. 1979. Comments on Model Selection Criteria of Akaike and Schwarz.” Journal of the Royal Statistical Society. Series B (Methodological).
Sugiura. 1978. Further Analysts of the Data by Akaike’ s Information Criterion and the Finite Corrections.” Communications in Statistics - Theory and Methods.
Taddy. 2013. One-Step Estimator Paths for Concave Regularization.” arXiv:1308.5623 [Stat].
Tharmaratnam, and Claeskens. 2013. A Comparison of Robust Versions of the AIC Based on M-, S- and MM-Estimators.” Statistics.
Tibshirani, Robert. 1996. Regression Shrinkage and Selection via the Lasso.” Journal of the Royal Statistical Society. Series B (Methodological).
Tibshirani, Ryan J. 2015. Degrees of Freedom and Model Search.” Statistica Sinica.
Ye. 1998. On Measuring and Correcting the Effects of Data Mining and Model Selection.” Journal of the American Statistical Association.
Yuan, and Lin. 2006. Model Selection and Estimation in Regression with Grouped Variables.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Zhang, Li, and Tsai. 2010. Regularization Parameter Selections via Generalized Information Criterion.” Journal of the American Statistical Association.
Zou, Hastie, and Tibshirani. 2007. On the ‘Degrees of Freedom’ of the Lasso.” The Annals of Statistics.