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).

  • 1
    $\begingroup$ It sounds like you do everything correctly. If noise can make $x_i$ negative, then your model should provide for this possibility. $\endgroup$
    – frank
    Jul 21, 2022 at 14:54
  • $\begingroup$ And even when all $x_i$'s are positive, assuming a (pseudo-) Gaussian likelihood does not hurt. $\endgroup$
    – Xi'an
    Jul 21, 2022 at 19:13


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.