In Bayesian statistics, with my variable is Gaussian distributed and I have a conjugate prior, I can solve the posterior analytically. I can still use MCMC in the case when things are non-Gaussian so long as I nominate a distribution. But what if I don't know what the appropriate distribution is? I am working with financial data which are known to be non-Gaussian (heavier tails and skewed relative to a Normal distribution). To the best of my knowledge, the exact distribution of financial data is still up for debate in academia. What would be an alternative if I don't want to make a strong assumption on how the data is distributed?