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My question results from the subjectivity of priors, and if there are bodies of work that help to create a more objective approach towards prior choices.

My question specifically is to do in the realm of spectroscopic/experimental data, where prior information of the system beforehand is limited. For example, concerning the amplitude of a peak: it is quite difficult to know much prior information about this beforehand.

Other features such as the FWHM and the peak positions have some prior information about them in databanks for known materials, however for newer materials this is also not known. In this case, would you simply assign uniform priors to all parameters? Even with the danger of generating an improper posterior?

Are there alternatives to uniform priors when weakly informative information is known about a parameter, for example the amplitude must be positive, or a certain other parameter must be bounded between 0 and 1. How can you express this in your prior choices?

etcetera
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Are there alternatives to uniform priors when weakly informative information is known about a parameter, for example the amplitude must be positive, or a certain other parameter must be bounded between 0 and 1. How can you express this in your prior choices?

If your prior assigns zero to a particular point in the parameter space, then your posterior will be zero. In this way, you can exclude certain parameter values from ever having a non-zero posterior probability. You should be very careful with this though, limiting it only to cases where such a parameter value is logically or physically impossible (this is sometimes called Cromwell's Rule).

For a prior between 0 and 1, a frequently used choice for an informed prior is a Beta prior; you can play around with the beta distribution here and see how the parameters affect it. This would allow you to have a non-uniform prior which is limited between 0 and 1.

For a non-negative variable, a half-normal or half-cauchy prior might make sense, and will result in a proper posterior. But without more information about your prior knowledge, it's hard to make a concrete recommendation. This will at least make sure your posterior is non-negative, which may be what you're looking for.

Louis Cialdella
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  • Thank you for the explanation. What further information would you require to make a recommendation: What is the methodological process you go through to decide? I am interested in this, as I hope it will help me follow the bayesian process more effectively. – etcetera Feb 16 '21 at 17:37
  • There are a number of approaches to turning expert information into priors. A simple, but often effective method, is to use quantiles of the prior distribution, and construct your prior based off of them. This Q/A (https://stats.stackexchange.com/questions/1/eliciting-priors-from-experts) and this link (https://www.johndcook.com/blog/2010/01/31/parameters-from-percentiles/) might be useful. If you are fitting many parameters at once which are mutually informative, you are probably in a good situation for a hierarchical model (https://en.wikipedia.org/wiki/Bayesian_hierarchical_modeling). – Louis Cialdella Feb 16 '21 at 19:35
  • Thank you for the information. The issue here is that there is little prior expert information when it comes to spectroscopic data such as XPS, X-ray Diffraction, Neutron-Scattering unless you use something like DFT simulation predictions as your informative priors which isn't a simple task. In this way it can be difficult to avoid uniform priors. Are there any resources on priors in chemistry etc based applications? – etcetera Feb 17 '21 at 14:26
  • Unfortunately I'm not sure I can help you there; my knowledge of chemistry is quite limited. A search for "bayesian analysis in spectroscopy" on Google produces examples that might provide case studies for you in how priors are constructed here, but I don't have the domain knowledge of spectroscopic data to point you towards any especially good examples. Alternatively, you could provide more information about your particular problem, in which we might be able to provide so more specific suggestions (what parameters you are estimating, what data you have, etc.). – Louis Cialdella Feb 17 '21 at 15:14