A posterior distribution has a higher variance than the uniform prior? I am working on some Bayesian inversions with MCMC chains, the posterior of one of the (known to be bad) parameter came out to have (slightly) larger variance than the uniform prior it started with, I am not really sure how to explain this result mathematically, I doubt that it is a numerical error.  If any of you could provide any explanation or related reference would be great!
FYI, the following figure is the posterior (histogram) I am referring to:
histogram of posterior
The variance of these posterior samples is 1.7539
The uniform prior is prior~U[0.002, 4.5], which has a variance of 1.6860
 A: *

*According to The Maximum Variance of Restricted Unimodal Distributions,
Harold I. Jacobson, The Annals of Mathematical Statistics 40 (1969), 1746-1752, the variance of a unimodal distribution constrained to be in a certain interval can be bigger than the variance of the uniform. It rarely happens but it can happen.


*Your histogram seems to imply that the data suggest that the parameter is around 0.4 or so with high probability, however it still is compatible with the parameter being very far away from it; the density at 4.5 isn't really much lower than at 2, say. This would normally have to do with the precise nature of the parameter. It can happen if a parameter is of such a nature that if you generate data from a model with a fixed parameter value, of, say, 4.5, it can occasionally (with low but not very low probability) happen that the data look like as if parameter value 0.4 has produced them. More generally, a parameter may often generate data that corresponds well to what is expected given its value, but with not too low probability it will produce "stray data" that don't look anywhere near to what the parameter typically produces.
