These criteria do not tell you if any of the models is actually

**good**at predicting reality, just which are better or worse.

Definitions in math:

AIC = 2k -2*logL

BIC = k*ln(N) - 2*logL

where k is the number of parameters in the model, logL is the maximized log-likelihood, and N is the sample size.

Again, they are similar and based on the maximum likelihood estimate, but are penalized for number of parameters in different ways. Since you should almost always have a sample size larger than 7 (ln(8) > 2), the penalty for number of parameters is greater using the BIC.

(Adding parameters to a model means making it easier to tweak the model to fit the data-it's a bit like cheating if you don't have a good reason to add the parameters. That's why these two criteria penalize you for having additional parameters.)

In the paper I am currently reading, both AIC and BIC are based on the Yuan-Bentler T2* statistic and a Chi-squared distribution. The T2* statistic is a test statistic, or a function that combines many aspects of the data (such as mean, standard deviation, or number of samples) into one number. A distribution is the set of possible values of the test statistic, and how likely each of them are. The Chi-squared distribution is a particular common distrubition of known form that is used with the assumption of the errors in the sample being independent and normally distributed about zero.

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