Editing Bayesian inference (section)

===Asymptotic behaviour of posterior===

Consider the behaviour of a belief distribution as it is updated a large number of times with [[independent and identically distributed]] trials. For sufficiently nice prior probabilities, the [[Bernstein–von Mises theorem|Bernstein-von Mises theorem]] gives that in the limit of infinite trials, the posterior converges to a [[Gaussian distribution]] independent of the initial prior under some conditions firstly outlined and rigorously proven by [[Joseph L. Doob]] in 1948, namely if the random variable in consideration has a finite [[probability space]]. The more general results were obtained later by the statistician [[David A. Freedman (statistician)|David A. Freedman]] who published in two seminal research papers in 1963 <ref>{{cite journal| last1=Freedman|first1=DA|title=On the asymptotic behavior of Bayes' estimates in the discrete case|journal=The Annals of Mathematical Statistics|volume=34|issue=4|date=1963|pages=1386–1403|jstor=2238346|doi=10.1214/aoms/1177703871|doi-access=free}}</ref> and 1965 <ref>{{cite journal|last1=Freedman|first1=DA|title=On the asymptotic behavior of Bayes estimates in the discrete case II|journal=The Annals of Mathematical Statistics|date=1965|volume=36|issue=2|pages=454–456|jstor=2238150|doi=10.1214/aoms/1177700155|doi-access=free}}</ref> when and under what circumstances the asymptotic behaviour of posterior is guaranteed. His 1963 paper treats, like Doob (1949), the finite case and comes to a satisfactory conclusion. However, if the random variable has an infinite but countable [[probability space]] (i.e., corresponding to a die with infinite many faces) the 1965 paper demonstrates that for a dense subset of priors the [[Bernstein–von Mises theorem|Bernstein-von Mises theorem]] is not applicable. In this case there is [[almost surely]] no asymptotic convergence. Later in the 1980s and 1990s [[David A. Freedman (statistician)|Freedman]] and [[Persi Diaconis]] continued to work on the case of infinite countable probability spaces.<ref>{{cite journal|first2=Larry|last2= Wasserman |first1 = James|last1 =Robins|journal =   Journal of the American Statistical Association|date = 2000|title = Conditioning, likelihood, and coherence: A review of some foundational concepts|doi=10.1080/01621459.2000.10474344|volume=95|issue=452| pages=1340–1346|s2cid= 120767108 }}</ref> To summarise, there may be insufficient trials to suppress the effects of the initial choice, and especially for large (but finite) systems the convergence might be very slow.