Appropriate priors for truncated regression model

I have a simple linear regression model with the constraint that my dependent variable y (response time) has to be greater than zero. I want to specify priors for intercept, slope and sigma (the standard deviation of observed values around the predictions) but I don't know how those priors should look like. My model, evaluated with any triple taken from the priors, should result in values strictly greater than zero.

I know how to fit such a model using common software such as JAGS or Stan but I am particularly interested in finding appropriate priors. All parameters taken from those priors should result in plausible values for y.

• There seems to be some confusion there. If your dependent variable is positive, this should be a feature of the sampling model, using for instance a truncated normal,$$y|x\sim\mathcal{N}_+(a+bx,\sigma^2)$$which does not require restrictions on the prior. Commented Aug 9, 2015 at 13:13
• Thank you. What I want to do is compute an "average likelihood" by sampling repeatedly parameters from the priors and compute the likelihood. Finally, I compute the mean of all those likelihoods. Imagine I use your proposed sampling model, what happens if I choose parameters for slope and intercept so that my model predicts a value of say -10 with sigma of say 2? Commented Aug 9, 2015 at 14:03
• Following up: But isn't it the case that priors which would lead to negative model predictions in many cases can be considered as a poor choice? Commented Aug 9, 2015 at 14:19
• Have you considered applying a logarithmic transformation to your dependent variable?
– Cyan
Commented Aug 10, 2015 at 21:44

What it seems you're looking for is a generalized linear model. In brief, a simple linear regression assumes that $y|x \sim N(X\beta, \sigma)$; in words, that $y$ is normally distributed around some linear function of $X$. A generalized linear model treats the data as produced by some other distribution, and transforms this linear function to produce a parameter for that distribution. Logistic regression is a familiar example: The data are Bernoulli generated, with $p = S(X\beta)$, where $S$ is the sigmoid function.
Modeling your data using an appropriate generalized linear model with appropriate priors, rather than a simple linear regression, would solve your problem of negative predictions: All your $y$ values would be positive, because they would all be sampled from a distribution with strictly positive support.
If you need to limit the responses to positive values, you can construct a "tobit" model with the survreg function in Therneau's survival package for R. There is a worked example in the help page for ?survreg. This comes about because censoring and truncation have some commonalities. I realize this isn't the Bayesian answer you were looking for but it might be useful to have a frequentist alternative. It may also be helpful to use "tobit" as an additional search term.