Bayesian information criterion

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In statistics, the Bayesian information criterion (BIC) or Schwarz information criterion (also SIC, SBC, SBIC) is a criterion for model selection among a finite set of models; models with lower BIC are generally preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion (AIC).

Contents

When fitting models, it is possible to increase the likelihood by adding parameters, but doing so may result in overfitting. Both BIC and AIC attempt to resolve this problem by introducing a penalty term for the number of parameters in the model; the penalty term is larger in BIC than in AIC.

The BIC was developed by Gideon E. Schwarz and published in a 1978 paper, [1] where he gave a Bayesian argument for adopting it.

Definition

The BIC is formally defined as [2] [lower-alpha 1]

where

Konishi and Kitagawa [4] :217 derive the BIC to approximate the distribution of the data, integrating out the parameters using Laplace's method, starting with the following model evidence:

where is the prior for under model .

The log-likelihood, , is then expanded to a second order Taylor series about the MLE, , assuming it is twice differentiable as follows:

where is the average observed information per observation, and prime () denotes transpose of the vector . To the extent that is negligible and is relatively linear near , we can integrate out to get the following:

As increases, we can ignore and as they are . Thus,

where BIC is defined as above, and either (a) is the Bayesian posterior mode or (b) uses the MLE and the prior has nonzero slope at the MLE. Then the posterior

Usage

When picking from several models, ones with lower BIC values are generally preferred. The BIC is an increasing function of the error variance and an increasing function of k. That is, unexplained variation in the dependent variable and the number of explanatory variables increase the value of BIC. However, a lower BIC does not necessarily indicate one model is better than another. Because it involves approximations, the BIC is merely a heuristic. In particular, differences in BIC should never be treated like transformed Bayes factors.

It is important to keep in mind that the BIC can be used to compare estimated models only when the numerical values of the dependent variable [lower-alpha 2] are identical for all models being compared. The models being compared need not be nested, unlike the case when models are being compared using an F-test or a likelihood ratio test.[ citation needed ]

Properties

Limitations

The BIC suffers from two main limitations [5]

  1. the above approximation is only valid for sample size much larger than the number of parameters in the model.
  2. the BIC cannot handle complex collections of models as in the variable selection (or feature selection) problem in high-dimension. [5]

Gaussian special case

Under the assumption that the model errors or disturbances are independent and identically distributed according to a normal distribution and the boundary condition that the derivative of the log likelihood with respect to the true variance is zero, this becomes (up to an additive constant, which depends only on n and not on the model): [6]

where is the error variance. The error variance in this case is defined as

which is a biased estimator for the true variance.

In terms of the residual sum of squares (RSS) the BIC is

When testing multiple linear models against a saturated model, the BIC can be rewritten in terms of the deviance as: [7]

where is the number of model parameters in the test.

See also

Notes

  1. The AIC, AICc and BIC defined by Claeskens and Hjort [3] are the negatives of those defined in this article and in most other standard references.
  2. A dependent variable is also called a response variable or an outcome variable. See Regression analysis.

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References

  1. Schwarz, Gideon E. (1978), "Estimating the dimension of a model", Annals of Statistics , 6 (2): 461–464, doi: 10.1214/aos/1176344136 , MR   0468014 .
  2. Wit, Ernst; Edwin van den Heuvel; Jan-Willem Romeyn (2012). "'All models are wrong...': an introduction to model uncertainty" (PDF). Statistica Neerlandica. 66 (3): 217–236. doi:10.1111/j.1467-9574.2012.00530.x.
  3. Claeskens, G.; Hjort, N. L. (2008), Model Selection and Model Averaging, Cambridge University Press
  4. Konishi, Sadanori; Kitagawa, Genshiro (2008). Information criteria and statistical modeling. Springer. ISBN   978-0-387-71886-6.
  5. 1 2 Giraud, C. (2015). Introduction to high-dimensional statistics. Chapman & Hall/CRC. ISBN   9781482237948.
  6. Priestley, M.B. (1981). Spectral Analysis and Time Series. Academic Press. ISBN   978-0-12-564922-3. (p. 375).
  7. Kass, Robert E.; Raftery, Adrian E. (1995), "Bayes Factors", Journal of the American Statistical Association , 90 (430): 773–795, doi:10.2307/2291091, ISSN   0162-1459, JSTOR   2291091 .

Further reading