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Not an answer. But perhaps you benefit from having a peek at my Bayesian tutorial. You can read most of it online under "Read Sample." amazon.com/dp/B0BTNVFR65
The tails in your data are wider than that of your normal distribution. So you could try to use a distribution with a "fatter" tail: logistic distribution, Student's T distribution, ..., up to a Cauchy distribution.
Not an answer. But perhaps you benefit from having a peek at my Bayesian tutorial. You can read most of it online under "Read Sample." amazon.com/dp/B0BTNVFR65
Not an answer. But perhaps you benefit from having a peek at my Bayesian tutorial. You can read most of it online under "Read Sample." amazon.com/dp/B0BTNVFR65
John P. Nolan's book "Univariate Stable Distributions" (Springer) gives only four families of stable distributions: Gaussian, Cauchy, Levy, and Stable Distributions (in 2 types).
You may benefit from reading the Exercise in David MacKay's book (p. 59) which starts with "When spun on edge 250 times, a Belgian one-euro coin came up heads 140 times and tails 110. ‘It looks very suspicious to me’, said ..."
In my opinion, comparing marginal distributions can be very(!) misleading. My approach would be: first to find the maximum of the Posterior, for both cases. Next, integrate for the zeroth (!), first, and second moment for all parameters, possibly using the Laplace (Gaussian) approximation (far from trivial!). Then compare the "error bars": Sqrt[secondMoment - firstMoment^2]. And finally see whether you can identify some discrepancy. Good luck.
Not an answer. But perhaps you benefit from having a peek at my Bayesian tutorial. You can read most of it online under "Read Sample." amazon.com/dp/B0BTNVFR65
This question is not well posed. You may benefit from having a peek on our tutorial paper for the meaning of the KL divergence: mdpi.com/2673-9984/5/1/22
I like your example for illustrating the OPs question. But I would like to remark that your P-s are not probabilities but are ratio's (or frequencies). In my opinion, this is not a Bayesian problem.
You may benefit from Googling "Nested Sampling" to read about practical difficulties in computing the normalisation factor P(X), sometimes called the Evidence.
Not an answer. But perhaps you benefit from having a peek at my Bayesian tutorial (Ch. 11.4). You can read most of it online under "Read Sample." amazon.com/dp/B0BTNVFR65
