Can a probability distribution have negative values Similar to discussions such as this, I was wondering about the possible values a probability density can have. I get the mathematical reason of why values greater than 1 are possible. However, recently I came across a probability distribution that looked like this:
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Note the dip below zero. The distribution is the result of modeling work that used MCMC sampling to generate parameter estimates. I suspect that there are some problems with the sampling, convergence, or the underlying model, because I don't think negative values were possible, but I am not very certain about it.
Has anyone ever seen something like this and knows whether probability densities with values below zero are possible.
 A: Classical probabilities are always in the range [0, 1]. A probability density cannot have negative values, because integrating over that region would yield a negative probability, which makes no sense - it would seem to imply that something is less likely than "impossible".
One interpretation of probability in the context of repeatable experiments is that it's simply the proportion of times something occurs, calculated as the number of successes divided by the number of trials. Both of the number of successes and number of trials must be non-negative, therefore the probability must as well.
As pointed out in the comments on the question, one could not find a negative probability through Monte Carlo sampling, as that again boils down to a frequency over many trials, which must be non-negative. What we're likely seeing is a failure of interpolation, where all observed values are in fact positive, but the method used to fit the smooth curve "overshoots" the observed low values near the negative dip.
