How to choose a prior : family for a response with negative values? I’m modeling percentage change in oxygen levels in the blood from a particular experiment. So my prior before seeing the data was an inverse gaussian distribution. But my data (response variable ) has some negative values. The family( ): Inverse.gaussian doesn’t take negative values. How should I go about this? [ https://rdrr.io/cran/brms/]
My min and max range of values is (-23,40).With a mean of 4 and a median of 3.5. (Also this is a repeated measures data).The following is the histogram distribution of the data.
The family function I mentioned here is from the 'brms' package.
 A: Prior is the distribution for the parameter that you assume "before seeing the data", you seem to be talking about likelihood, i.e. the distribution to describe the data itself.
If your outcomes are relative changes, that can be either positive, or negative, choosing inverse Gaussian is a bad idea, because it is a distribution for non-negative outcomes. You didn't give us much details, but given that mean and median are very close and they seem to be lying approximately in the middle between minimum and maximum, it seems that your data is fairly symmetric, isn't it? If that is the case, why not using Gaussian as a likelihood function? If the data has long tails, then you could consider $t$-distribution. To be more precise, we are interested in conditional distribution $E[Y|X]$ of the outcomes, not the marginal distribution, but the above is an educated guess, given the limited information you've given us.
You can also check the How to decide which glm family to use? thread for more details on choosing the family for generalized linear models.
