This paper explores the problem of selecting optimal hyperparameters in the Gibbs algorithm to minimize the population risk, specifically focusing on the inverse temperature. The inverse temperature is a hyperparameter that controls the tradeoff between data fitting and generalization. We present a characterization of the derivative of the population risk with respect to the inverse temperature, expressed in terms of the covariance between empirical risk and test loss. This characterization enables us to identify the optimal inverse temperature that minimizes population risk. Additionally, we provide two illustrative examples—a mean estimate and linear regression—to validate our analytical findings. Notably, our analysis reveals that the optimal inverse temperature exhibits different behaviors in two different regimes based on data quality and prior distribution. These insights contribute to our understanding of linear regression and more general machine learning models.