# Can AIC be used on out-of-sample data in cross-validation to select a model over another?

Following Gelman's 2017 publication entitled "Understanding predictive information criteria for Bayesian models" I understand cross-validation and information criteria (Bayesian information criterion and Akaike's information) can be used separately. Usually with enough sample size one would use cross-validation with some measures of predictive accuracy to select a given model over others. With lower sample sizes, AIC and BIC might be preferred on the training data (without cross-validation). My confusion is that whether AIC and BIC can be used along with cross-volition, for example can AIC and BIC be used on the left-out fold in a 10-fold cross-validation? The idea is to use out-of-sample information criteria to penalise for complexity (AIC) as well as model fit (BIC).

1. AIC measures the log-likelihood of the entire sample at once, based on parameters estimated using the entire sample, and subsequently adjusts for overfitting (which occurs when estimating log-likelihood on new data by log-likelihood of the same sample on which the estimation was done) via $$p$$ in $$\text{AIC}=-2(\text{loglik}-p)$$. Here $$\text{loglik}$$ is the log-likelihood of the sample data according to the model and $$p$$ is the number of the model's degrees of freedom (a measure of the model's flexibility);