So, I have this problem I am tackling where I am doing a Bayesian scan of a multi-dimensional model. Most of the quantities predicted by the model have likelihood functions which are normal distributions (as functions of the possible data values), however there are some pieces of experimental data I have for which only upper or lower bounds exist, for example experiments have been done which show the observable must be above a certain value with 95% confidence limit or something, with no upper bound.

What is the correct form of the likelihood function to use for such a quantity? In the literature I have seen two possibilities, one being a compound function which is a normal distribution below the limit (if it is a lower bound) and a uniform distribution above this limit, while the the other is an error function centred on the bound. I have seen no theoretical justification for either of these distributions and was wondering if anybody knew of any.

The half-gaussian at first seems sensible because the maximum likelihood is obtained at the value of the limit and for anything above that, with the gaussian falling off as determined by the confidence % value, however on thinking about it some more I now think that the err function is probably more justified, although I can't prove why or even formulate why I think this very well. I guess it seems to me that even if the theory gives a perfect match with whatever value was observed in the experiment then it shouldn't actually be the maximum likelihood value, since it would be quite the fluke that this should happen.

Perhaps I should explain by scenario some more to make this make sense. One particular observable I am concerned with is the relic density of dark matter. If dark matter is only made of one type of particle, then the relic density as calculated by the astrophysics guys can be used to constrain the relic density of a given dark matter candidate particle for ones favourite model with a gaussian likelihood function. However, if dark matter is assumed to be composed of this candidate particle plus other stuff, then the relic density calculated by the astrophysicists can only provide an upper bound on the relic density of the candidate particle, since one can't disfavour the model if it fails to reach the required relic density (because we have allowed for the possibility that other stuff is lurking out there than is taken care of by the model). So in this latter case what likelihood function is appropriate for the relic density of the partial dark matter candidate? We need to kill the model if the relic density gets too large, but we don't want to penalise it if the relic density is lower than the observed astrophysical value.

Sorry if this is a bit incoherent, I could probably have made that all much more clear. If you need me to clarify anything or indeed write an equation down let me know. Mostly I am looking for some nice probability theory justification for which kind of likelihood function makes the most sense here. I have searched around quite a lot and been unable to find anything. Papers I have read where similar things are done seem to skip over the justification part... and I suspect that many authors actually just guess...

Maybe both forms of likelihood function are valid for different cases. I'm a little lost on this one.

In practice it doesn't make a huge different which of the two possibilities I mentioned are used, but I'd like to know which one is really right!