Editing Sufficient statistic (section)

==Other types of sufficiency==
===Bayesian sufficiency===

An alternative formulation of the condition that a statistic be sufficient, set in a Bayesian context, involves the posterior distributions obtained by using the full data-set and by using only a statistic. Thus the requirement is that, for almost every ''x'',

:<math>\Pr(\theta\mid X=x) = \Pr(\theta\mid T(X)=t(x)). </math>

More generally, without assuming a parametric model, we can say that the statistics ''T'' is ''predictive sufficient'' if

:<math>\Pr(X'=x'\mid X=x) = \Pr(X'=x'\mid T(X)=t(x)).</math>

It turns out that this "Bayesian sufficiency" is a consequence of the formulation above,<ref>{{cite book
 |last1=Bernardo |first1=J.M. |author-link1=José-Miguel Bernardo
 |last2=Smith |first2=A.F.M. |author-link2=Adrian Smith (academic)
 |year=1994
 |title=Bayesian Theory
 |publisher=Wiley
 |isbn=0-471-92416-4
 |chapter=Section 5.1.4
}}</ref> however they are not directly equivalent in the infinite-dimensional case.<ref>{{cite journal
 |last1=Blackwell |first1=D. |author-link1=David Blackwell
 |last2=Ramamoorthi |first2=R. V.
 |title=A Bayes but not classically sufficient statistic.
 |journal=[[Annals of Statistics]]
 |volume=10 |year=1982 |issue=3 |pages=1025–1026
 |doi=10.1214/aos/1176345895 |mr=663456 | zbl = 0485.62004
|doi-access=free }}</ref> A range of theoretical results for sufficiency in a Bayesian context is available.<ref>{{cite journal
 |last1=Nogales |first1=A.G.
 |last2=Oyola |first2=J.A.
 |last3=Perez |first3=P.
 |year=2000
 |title=On conditional independence and the relationship between sufficiency and invariance under the Bayesian point of view
 |journal=Statistics & Probability Letters
 |volume=46 |issue=1 |pages=75–84
 |doi=10.1016/S0167-7152(99)00089-9 |mr=1731351 | zbl = 0964.62003
|url=https://dialnet.unirioja.es/servlet/oaiart?codigo=118597
 }}</ref>

===Linear sufficiency===

A concept called "linear sufficiency" can be formulated in a Bayesian context,<ref>{{cite journal |first1=M. |last1=Goldstein |first2=A. |last2=O'Hagan |year=1996 |title=Bayes Linear Sufficiency and Systems of Expert Posterior Assessments |journal=[[Journal of the Royal Statistical Society]] |series=Series B |volume=58 |issue=2 |pages=301–316 |doi=10.1111/j.2517-6161.1996.tb02083.x |jstor=2345978 }}</ref> and more generally.<ref>{{cite journal |last=Godambe |first=V. P. |year=1966 |title=A New Approach to Sampling from Finite Populations. II Distribution-Free Sufficiency |journal=[[Journal of the Royal Statistical Society]] |series=Series B |volume=28 |issue=2 |pages=320–328 |doi=10.1111/j.2517-6161.1966.tb00645.x |jstor=2984375 }}</ref> First define the best linear predictor of a vector ''Y'' based on ''X'' as <math>\hat E[Y\mid X]</math>. Then a linear statistic ''T''(''x'') is linear sufficient<ref>{{cite journal |last=Witting |first=T. |year=1987 |title=The linear Markov property in credibility theory |journal=ASTIN Bulletin |volume=17 |issue=1 |pages=71–84 |doi= 10.2143/ast.17.1.2014984|doi-access=free |hdl=20.500.11850/422507 |hdl-access=free }}</ref> if

:<math>\hat E[\theta\mid X]= \hat E[\theta\mid T(X)] . </math>