I want to fit a model where I am detrending a time series while reweighting my errorbars for the data points multiplicative, i.e. new error bars e_i = sigma_i * c_i, where sigma_i is the old measured errorbar and c_i is the re-weighting factor. I am looking for a correct prior for the c_i’s which makes it more probable that the errorbars are not re-weighted, i.e. c_i’s ~ 1 and down or up re-weightings should be equally probable. Kind of like a normal distribution but here the down-weighting is squished between 0 and 1 while the up-weighting is unbounded from 1 to large values. What kind of prior can I use here?
You could use the lognormal distribution as a prior. Since it’s defined as the exponential of a normal random variable, its support is bounded to be positive. If you set its location parameter to 0, then it will have an expected value close to 1. I am not sure if there is a straightforward way to enforce the condition that up and down-weightings are equally probable; you could adjust the variance so that the probability on either side of 1 integrates to 1/2.
My personal opinion is that if the rescalings in either direction are equally likely in log-space (as is the case here), that’s good enough.
Thank you for your reply. Is there a non-centered parametrization of the log normal?
You could try using a non-centered parameterization of the normal and then exponentiate it. I believe this is identical to multiplying together lognormal components akin to adding normally distributed random variables. There’s more related discussion here but it’s not clear if a satisfying answer was given as to whether or not it’s a good idea.