# Bayesian inference, likelihood on positive data

Suppose I have a parameter $$\theta$$, that I know is positive, and some data $$(x_1,x_2,\dots,x_n)$$ on noisy realisations of the $$\theta$$. I then assume a prior with positive support on $$\theta$$ (lognormal for example). How important is it to assume a positive likelihood on $$x|\theta$$ as well, or is it reasonable to assume something like a Gaussian likelihood (which can be negative)?

I ask this because the posterior of $$\theta$$ will have positive support regardless of what the likelihood is because the prior has positive support, so I don't really have to worry about the posterior of $$\theta$$ having probability on negative values. On the other hand, the data should in principle be always positive as well (since it represents measurements of $$\theta$$ with noise).

In real life practical applications what do people usually do in situations like this (any examples would be much appreciated).

• It sounds like you do everything correctly. If noise can make $x_i$ negative, then your model should provide for this possibility. Jul 21, 2022 at 14:54
• And even when all $x_i$'s are positive, assuming a (pseudo-) Gaussian likelihood does not hurt. Jul 21, 2022 at 19:13