The fine print on tempered posteriors
Konstantinos Pitas Julyan Arbel
We conduct a detailed investigation of tempered posteriors and uncover a
number of crucial and previously undiscussed points. Contrary to previous
results, we first show that for realistic models and datasets and the tightly
controlled case of the Laplace approximation to the posterior, stochasticity
does not in general improve test accuracy. The coldest temperature is often
optimal. One might think that Bayesian models with some stochasticity can at
least obtain improvements in terms of calibration. However, we show empirically
that when gains are obtained this comes at the cost of degradation in test
accuracy. We then discuss how targeting Frequentist metrics using Bayesian
models provides a simple explanation of the need for a temperature parameter
$\lambda$ in the optimization objective. Contrary to prior works, we finally
show through a PAC-Bayesian analysis that the temperature $\lambda$ cannot be
seen as simply fixing a misspecified prior or likelihood.