The Average Information Criterion (AIC) is a model selection criterion used in statistics to evaluate the goodness of fit of a statistical model. It is often used for comparing different models based on their performance in fitting the data while penalizing for model complexity.

In the Bayesian framework, the AIC can be extended to incorporate the uncertainty associated with the model parameters. This leads to the development of Bayesian Information Criterion (BIC) or Schwarz Criterion, which is a Bayesian counterpart to AIC. BIC is derived from a Bayesian perspective and includes a penalty term for model complexity similar to AIC.

The AIC under the Bayesian framework can be expressed as follows:

AIC=-2log(L(\hat{\theta}))+2k

where:

• (L(\hat{\theta})) is the maximized likelihood of the model, given the observed data.
• (\hat{\theta}) represents the maximum likelihood estimates of the model parameters.
• k is the number of parameters in the model.

In the Bayesian context, the AIC can be interpreted as an approximation to the expected log-likelihood under the posterior distribution of the parameters. It incorporates a penalty term (2k) to balance goodness of fit with model complexity, similar to the frequentist AIC.

It's important to note that while the AIC and BIC are widely used for model selection, they have their own assumptions and limitations. Researchers often choose between them based on the specific context of their analysis and the trade-off between model fit and complexity.

The Deviance Information criterion (DIC) is generally used under Bayesian frame work. There as an alternative for DIC is Bayesian model evidence.

In a Bayesian framework, the Akaike Information Criterion (AIC) can be used as a model selection criterion to compare the fit of different Bayesian models, accounting for the trade-off between model complexity and goodness of fit given the posterior distribution.