# Renormalizing a distribution to reduce variance

I have a predictive model $$M$$ that generates an empirical predictive distribution $$P_M$$ via a set of samples. I cannot change the predictive model.

I can evaluate the predictive performance using log score as follows

$$\sum log P_m(x) \text{ for x in validation set}$$

I've noticed that the variance of $$P_m$$ is too large. I can improve the log score by taking

$$\frac{P_m^{\alpha}}{\sum P_m^{\alpha}}$$

this reduces the variance and puts more mass around the center for some $$\alpha > 1$$.

My question is, is there any theory behind this?