Bayesian information criterion


In statistics, the Bayesian information criterion or Schwarz information criterion is a criterion for model selection among a finite set of models; the model with the lowest BIC is preferred. It is based, in part, on the likelihood function and it is closely related to the Akaike information criterion.
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, where he gave a Bayesian argument for adopting it.

Definition

The BIC is formally defined as
where
Konishi and Kitagawa derive the BIC to approximate the distribution of the data, integrating out the parameters using Laplace's method, starting with the following:
where is the prior for under model.
The log,, 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 Big O notation|. Thus,
where BIC is defined as above, and either is the Bayesian posterior mode or uses the MLE and the prior has nonzero slope at the MLE. Then the posterior

Properties

The BIC suffers from two main limitations
  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 problem in high-dimension.

    Gaussian special case

Under the assumption that the model errors or disturbances are independent and identically distributed according to a normal distribution and that the boundary condition that the derivative of the log likelihood with respect to the true variance is zero, this becomes :
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 the BIC is
When testing multiple linear models against a saturated model, the BIC can be rewritten in terms of the
deviance as:
where is the number of model parameters in the test.
When picking from several models, the one with the lowest BIC is 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. Hence, lower BIC implies either fewer explanatory variables, better fit, or both. The strength of the evidence against the model with the higher BIC value can be summarized as follows:
ΔBICEvidence against higher BIC
0 to 2Not worth more than a bare mention
2 to 6Positive
6 to 10Strong
>10Very strong

The BIC generally penalizes free parameters more strongly than the Akaike information criterion, though it depends on the size of n and relative magnitude of n and k.
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 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.

BIC for high-dimensional model

For high dimensional model with the number of potential variables, and the true model size is bounded by a constant, modified BICs has been proposed in Chen and Chen and Gao and Song. For high dimensional model with the number of variables, and the true model size is unbounded, a high dimensional BIC has been proposed in Gao and Carroll. The high dimensional BIC is of the form:
where can be any number greater than zero.
Gao and Carroll proposed a pseudo-likelihood BIC for which the pseudo log-likelihood is used instead of the true log-likelihood. The high dimensional pseudo-likelihood BIC is of the form:
where is an estimated degrees of freedom, and the constant is an unknown constant.
To achieve the theoretical model selection consistency for divergent, the two high dimensional BICs above require the multiplicative factor. However, in practical use, the high dimensional BIC can take a simpler form:
where various choices of the multiplicative factor can be used. In empirical studies, or can be used and it is shown to have good empirical performance.