BIC balances model fit against model complexity by penalizing the number of parameters. This penalty discourages overly complex models, thereby reducing the risk of overfitting while still rewarding improved likelihood.

Step-by-step Derivation:
Step 1: BIC is defined as BIC = k ln(n) - 2 ln(L), where k is the number of parameters, n is the sample size, and L is the maximized likelihood.
Step 2: The term -2 ln(L) rewards better goodness of fit, while k ln(n) adds a penalty for additional parameters.
Step 3: Because models with too many parameters can fit noise in the training data, the penalty term reduces the likelihood of selecting such models.
Step 4: Therefore, the main purpose of BIC in model selection is not simply to minimize model complexity or bias, but to control the tradeoff between fit and complexity, which minimizes overfitting.