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(1)A Note on the Central Limit Theorem for a Posterior Distribution By. t. K6nsang MO and Hiroshi NEGISHI" 1. Introduction Let {X)･:i = 1,2, ...} be independent observations whose individual distribution has a density pip(x) depending on a real parameter ip. Suppose that the parameter ￠ has a density p(ip). It is proved by Bernstein and von Mises that under suitable conditions for pip(x) and p(ip), for almost all (xi, ..., x,,), the posterior distibution of ip for large n is very close to a normal distibution.. Le cam, in (1953)(1958), obtains this result under more general conditions. In (1967), Johnson gives an asymptotic expansion for posterior distribution containing the fact that this distribution converges to the normal when pip(x) is the density of the exponential family suc'h as. pip(x) - C(ip)exp[￠R(x)] (1) and p(ip) is analytic in some neighborhood of the true parameter ipo. In any case, it was assumed that the prior density p(￠) was continuous and positive in a neighborhood of ipo.. In this paper, we shall give an example showing that if p(ip) is not continuous at ipo, the measure of the set of samples (xi, x2, ..., x.) making the. result of Bernstein and von Mises theorem converges to zero as n o oo. 2. A limit theorem for posterior distribution.. Let Xi, X2, ... be a sequence of i.i.d. random variables with density function pip(x) == C(ip)exp[ipR(x)]. Lehmann (1959), has shown that pip is a probability density function for all ip belonging to the some interval I.. Let's denote. K(ip) =- -logC(ip). (2) Then the density eqn.(1) can be written as pip(x) == exp{ipR(x) - K(ip)}.. Now, we consider the asymptotic form of the posterior distribution of ip after standardizing ip by centering at the maximum likelihood estimate di(r) of. - -A. ip and scaling according to a function of ip. Therefore we introduce. e- [ip- di(r)]b(r) (3) * Department of Mathematics, Faculty of Education, Yokohama National University..

(2) i l. K. Mo and H. NEGisHi. 14. i. l. where. t. b(r) =: {K"(di)}ii2, di :di(r). (4) Then E,(g), the posterior distribution of Vfie, can be written as fg-n.-"2 p(r1ip)p(ip)dip. Fh(O =. Soo-- p(rl ￠) p( ￠) d￠. (5). ,. where p(rlip) = pip(xi) ... pip(x.) = exp[n(K(di) - K(di + -ilL) + -il-r]] and ,. r - 12ii=i R(x}). l. n. l. [Proposition 2-1]. !. Let p(ip) is positive in some neighborhood of ipo. If. 1. i. lim p(￠) = p(￠o),. l. (#) T (Po. .1 T ipo as n. oo, then bnl/3. and ip +. oo oo. J-.p(rl￠)p(ip)dip-cJ-.e-'2/2dtasn->oo. (6) 3. Preliminaries Define ro by. ro=Eip,R. (7) The next four lemmas and preliminaries shown below follow Johnson (1967).. [Lemma 3-1] There exists a di > O such that for fixed r with lr - rol s di, exp {￠r - K(￠)} has a unique maximum at di(r) where di(r) satisfise K'(di) = r or. E,fJl-r. (8) [Lemma 3-2] di(r) is continuous function of r satisfying di(ro) = ipo and b(r), defined by eqn.. (4), is a positive continuous function of r.. Let's define a function f(e,r) by. f(e,,)-(sip("(di)-"(di'-il-)'-g-r]''fdi.',,Ii'II,.i,g (g). where b=b(r) and di = ￠"(r).. From eqn. (5), we know that the posterior density of e, given r, is proportional to. lt/.

(3) ANote on the Central Limit Theorem 15. p(di+ 2-) fn (e,r). (io) It is obvious that for fixed r, f(O,r) =- 1, f'(O,r) = O, and f" (O,r) == -1,. where prime denotes differentiation with respect to e. Since fcan be extended to analytic in e for some neighborhood of zero, we could obtain an asymptotic expansion of the posterior distibution of Vfie. Since f(O,r)=1, by the continuity of di and b, there exist a 6o > O and do > O such that the conditions lel s 6o and lr - rol s do imply that 1fte,r) - 11 s g and that p(di + -￡l-) is different from zero. In order to take the principal branch for the log, define a function h(e,r) by. h(e,r) - log f(e,r). (11) From the preceding discussion, there exists an M such that. lh(e,r)l s M for lel s{ 6o and lr -rol f{ do. (12) Define Nb by. Nb-{r:lr-rolgdo}. (13). For fixed r E IVb, h(e,r) is analytic for l61 f{ 6o and its derivatives are given. by the Cauchy formula. ak(r) - hk(kO!'r) - 21,,, J. ht(,ttr,) dt for k- 1,2, (14) liYSceor&eE = (t:ltl = i6o]･ Since ao(r) == ai(r) = o, the Tayior expansion. nh(e,r) == -ge2 + ne3 ,¥.,ooak(r)ek-3 fgr lel s{ 6o and lr - rol ffi do. (15) L. Let. so that. v(e,r) - oo 2 ak(r)ek-3 k==3. (16). p(di + -il-)fn(e,r) = e'S' e2 p(di + -il.),nciv(e,r). for lne31 s 1 and lel E{ 6o.. Since there exists Mo such that lak(r)I s Mo,. oo -oo k=3 k=3. 2 lak(r)llelk-3 s Mo 2 lelk-3 s Mi for lel s 6o. Therefore. (17).

(4) 16 K. Mo and H. NEGisHi. i. ene3v(O,r) .. 1 + o(ne3) for lne3i s 1 and lel s 6o･. (18). '. Thus, we obtain. p(di + z2L)fn(e,r) = e-t' e2 p(di + -SL) + p(di + -ilL)o(ne3) (ig). for lne31 s 1 and lel s 6o.. i. [Lemma 3-3] Let f(e,r) be defined by eqn. (9). Then there exist a 63 > O depending on Nb. t ii. e2. such that log f(e,r) g - -2i for all real e with lel sg 63 and rE IVb.. [Lemma 3-4]. Let f(e,r) be defined by eqn. (g) and 6 = min(63, E2tL). Then, there exists. an E (O < E < 1) such that f(e,r) < E for all e with iel }i 6 and r E No. [Proof of Proposition 2-1]. We have Joo p(rlip)p(ip)d￠ - oo. - Loo . fn(e,r)p(di + -8-)de. - JI6. f"(e,r)p(di + -il-)de + fl'g3fn(e,r)p(di + -il-)de +. -1. Jl',lm{ fn(e,r)p(di + -g->de + l,6,-i fn(e,r)p(di + -gL)de + tt. J: f"(e,r)p(di + -il-)de, i. and the lst, 2nd, 4th and 5th terms, by the Lemma 3-3 and Lemma 3-4, converge to zero as n -> oo respectively.. From eqn. (19) -!. L',l-t f"(e･r)p((i) + -S!-)de. 1' 1. - L',l-s e-t' o2 p(di + g)de + fl,i-s p(di + -2-)o(ne3)de.. Therefore.

(5) ANote on the Central Limit Theorem 17 h. -!. J-oo.p(rlip)p(ip)dip - fZ,l-i e-Se2 p(di + -2L)de -> o as n -> oo.. It follows from. '. 1t. -- e-t' e2 p(di + -g-)de - ±= fZls e-S2 p(di + ,J}i= )dt, JZi-i. and. !. fZ6.-2 e-S2 p((i) + beli ) - p((jBo) dt -' O if `i) + b3il> li T ￠o,. that we can get the proof of Prop 2.1.. 4. Example Let. pip(x)=vl;i6ief/2 (￠=ii2)･ Then, b(r) = Rr., and since, by the strong law of large number, r. == ", 2i･i-.i. -> ao2 almost everywhere with respect to the product measure of pip,, di. = - liJt:]I converges to ipo = -2c,3･. Let's consider the following measure pt{din(xi, ･･･, x.) s ipo} - pt(-22/.--n,.7. s - -2-idi] == pt( te., k2' f{ n].. From the properties of chi-square distribution, we can find the fact that b. pa(Ei.., i?i' sn] -> g.. On the other hand u(din + f' E{ ipo] = pa(-±, + JE :f{ -2k ]. -1 )2os} = pa(r. s G- nv-E:}3. == ptC ,2=n, x?･ s (i - NE nk') o&] (2o) = pt[,2.n., x?･ s (n - V2 ni) u8}.

(6) K. Mo and H. NEGism. 18. 2 .. 21z=, :di]EZ, = pt. ut. 1. S -n6. -> O as n -> oo. '. 2 because 2:i== {l71tll.- n g N (o,i) as n -> oo. -1 s dio], we can see the fact Therefore for E. iE ((xi, ..., x.):di. + Z-,. that pt(E,,) -> O as n . oo, that is to say, when p is left-continuous, the measure. of the set of sample points which satisfy Proposition 2-1 converges to O. We also obtain the same result in the case that p is right-continuous.. References [1] JoHNsoN, R.A 1967., "An Asymptotic Expansion for Posterior Distributions", Ann. Math. Statist. 38, 1899-1906. [2] LE cAM, L. 1953., "On some asymptptic properties of maximum likelihood estimates and related Bayes estimates", Univ. of California publ. Statist. 1, 277-330. [3] LE cAM, L 1958., "Les proprietes asymptotiques des solutions de Bayes", Publ. Inst. Statist.. Univ. Paris. 7, 17-35. [4] LEHMANN, E.L. 1959., "Testing Statistical Hypotheses" Wiley, New York. [5] von MisEs, R. 1964., "Mathematical Theory of Probability and Statistics", (Edited by Hilda Geiringer) Academic press, New York.. -. L.

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