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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).

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    $\begingroup$ 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

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