By D. V. Lindley
A research of these statistical rules that use a likelihood distribution over parameter house. the 1st half describes the axiomatic foundation within the thought of coherence and the results of this for sampling conception facts. the second one half discusses using Bayesian rules in lots of branches of data.
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Extra resources for Bayesian Statistics. A Review
A similar problem with the Poisson process has been examined by Lechner (1962). A group of problems which are related to these are concerned with optional stopping. For example, let xiti = 1,2, ••• , be independent and identically distributed according to a known distribution, p(xi), and JCQ = 0. The cost of each observation x, (i > 0) is c and you get a reward on stopping at the nth stage, according to the problem, either of xn (this is termed, without recall) or of max o^ign x i (with recall).
Notice that the data does not have to be a random sample from a distribution. The result has been known for some time in special cases (for example, with normal and gamma distributions). The idea of shifting an estimate from one value towards another prior value has been discussed by Thompson (1968) and Arnold (1969). 7) is a long way from x, whereas commonsense would suggest that m was "wrong" in some sense and that the estimate should be near, if not at, x. In the case of discrepant means the form of the density p(9) in the tails becomes important because that is where the observations are, surprisingly, to be found.
Its use first seems to have been suggested by Jeffreys (1967). A common special case is that of a sharp hypothesis. This arises when 6 = (£, r\), say, and H specifies the value of ^ = £ 0 , say, without specifying r\. H is simply £ 7^ £ 0 . Then r\ is a nuisance parameter. An obvious example is where we wish to test whether the mean ^ of a normal distribution is £0 without specifying the variance rj. 27) takes on a simple form. ;, rj) and letting p -» 0. Then a natural distribution for // when H is true has density f(^,r\} \f(^0,rj)dt], the usual conditional form.