I’m looking for a reasonable prior for sigma. I’ve read recommendations for half-cauchy or half-t priors. I get the argument for fat tails, but giving the highest prior probability to zero seem unreasonable to me. Why not a lognormal or a gamma? Are there disadvantages to using such distributions as priors?

In hierarchical or random-effects type settings I think the rationale is that more prior mass on zero favors simpler, more parsimonious explanations and penalizes complex explanations. Sigma wants to be big in those settings to fit the data better, so having the prior mode on zero with the density falling off as you move to bigger values counteracts that tendency.

It’s all problem-specific though. If for your problem a sigma of zero doesn’t make sense, by all means, use a zero-avoiding prior like a lognormal. It will usually help things computationally.

Thanks Arya for your quick and clear reply. I understand the parsimony argument. It’s not what I’m after in my current problem. I’m happy to learn when priors with high probability at zero make sense.

You’re going to have to be explicit about what kinds of models you’re talking about. Conventions for parameter names are very weak and sigma could mean a lot of different things.