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Nowadays in density estimation, posterior rates of convergence for locationand location-scale mixtures of Gaussians are only known under light-tailassumptions; with better rates achieved by location mixtures. It isconjectured, but not proved, that the situation should be reversed under heavytails assumptions. The conjecture is based on the feeling that there is no needto achieve a good order of approximation in regions with few data (say, in thetails), favoring location-scale mixtures which allow for spatially varyingorder of approximation. Here we test the previous argument on the Gaussianerrors mean regression model with random design, for which the light tailassumption is not required for proofs. Although we cannot invalidate theconjecture due to the lack of lower bound, we find that even with heavy tailsassumptions, location-scale mixtures apparently perform always worst thanlocation mixtures. However, the proofs suggest to introduce hybridlocation-scale mixtures that are find to outperform both location andlocation-scale mixtures, whatever the nature of the tails. Finally, we showthat all tails assumptions can be released at the price of making the priordistribution covariate dependent.

Tails assumptions and posterior concentration rates for mixtures of Gaussians