E L S E V I E R Statistics & Probability Letters 37 (1998) 351-355

STATISTICS& PROBABILITY

LETTERS

A characterisation of Hjort's discrete time beta process

S t e p h e n W a l k e r

epartment o f Mathematics, Imperial College, 180 Queen s Gate, London S W 7 2B , UK

Received 1 July 1996

Abstract

This paper characterises the discrete time version of Hjort's beta process (Hjort, 1990) using Bernoulli random variates. (~) 1998 Elsevier Science B.V. All rights reserved

Kevwords." Bernoulli trips

1. Introduction

We start by defining the discrete time beta process (further details can be found in (Hjort, 1990). Let {~k,fik: k = 1,2 . . . . } be positive numbers and define the independent beta variables {Vk: k = 1,2 . . . . } by

If

V~ ~ beta(ak, flk ).

A k = ~ ' ~, k = l , 2 . . . . .

then {Ak} is a discrete time beta process (Hjort, 1990). Now, define

k-I X]=V~, X k = ~ ( 1 - - h i , k = 2 , 3 . . . .

/=1

and

(1)

-- ~ XJ-- I - (1 - ~ ) . (e) j~<k .i~<k

We prefer to work with {Fk} rather than {Ak} though obtaining Ak from {Fk} is easy enough. The aim of this paper is to obtain a characterisation for the beta process via a Bernoulli sampling scheme

(Bernoulli trips).

0167-7152/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved Pl l S0167-7152(97)00137-5

352 S. Walker / Statistics Probability Let ters 37 (1998) 3 5 1 - 3 5 5

2. ernoulli trips

Let {Yk: k = 1,2 . . . . } be a countable set o f independent Bernoulli random variables with parameters ~k and /~k so that

~k p( rk = 1)-- - - .

~k + flk

A trip along the positive integers involves sampling the Yks in turn, starting at k = 1. A single trip is completed at integer M whenever the event

t~M:(YI = 0 . . . . . YM I = O , YM = 1 }

occurs .

emma 1. The probability that the trip ends is equal to 1 if, and only if, N k=l qk ~ O as N ~ , where

Proof. Let d~k = {YI = 0 . . . . . Yk-1 = 0, Yk = 1} for k = 1,2 . . . . and so the events {d~k} are mutually exclusive. N o w ,

p(tr ip e n d s ) = o~k = P(~k) = 1 - qk, k=l k=l k=l

completing the proof. []

Along the trip whenever Yk = 0 the current flk is replaced by flk + 1 and whenever Yk = 1 the current c~k is replaced by c~k + 1. When a trip is completed a second trip involves returning to k = 1 and repeating the procedure (keeping the updated {~k} and {ilk}).

Suppose that n trips are made and let {~k(n)} be the updated {c~k} so that nk(n )= 7 k ( n ) - ak is the number of times that a trip was terminated at the integer k. Also let m k ( n ) = i l k ( n ) - flk where ilk(n) is the updated flk and qk(n) = flk(n)/(~k(n) + ilk(n)).

emma 2. I f p ( f i r s t trip ends) = 1 then p(nth trip ends) = 1 f o r all n = 2, 3 . . . . .

Proof. Suppose that n trips have been completed and let L(n) be the longest of these trips. Then

qk(n) <~ qk(n) = qk = O. k= 1 k=L(n)+ 1 k=L(n)+ I

This implies that ~-1 qk(n)= 0, that is, the (n + 1 )th trip will end with probability 1, completing the proof. []

3. eta process from ernoulli trips

The required characterisation of the beta process is given in Theorem 1 and a Corollary to it. First a lemma is required.

emma 3. I f Z(n) is a sequence o f random variables defined on some space and X a random variable defined on the same space such that [Z(n)lX] ~ X a.s. then LPZ(n) ~ L/~X.

S. Walker / Statistics Probability Letters 37 (1998) 351-355 3 5 3

Proof. From Fubini's theorem.

Theorem 1. /£ Jor k = 1,2 . . . . and n -- 1,2 . . . . .

Zk (n )=nk (n ) /n

then, .for all ,

S ( Z l ( n ) . . . . . Z (n)) ~ S(X1 . . . . . X ),

where (XI,X2 . . . . ) are obtained via (1).

Proof. For any integers and M ( ~ > M )

E(XI "~ v"~'+l . X " • . . a M .. ) E ( X ? ' . . . X ; ~ ~ . . . X " )

E(V]'~(1 - VI) m'+' ' ' " v'~t+'(1 - VM)"') _ _ " 3 / /

~ ( v j " , ( l - v ~ ) ~ , . . , v ~ " ( l - ~ , ) m . ) '

where mk = ~ j > ~ nj. Then (3) becomes

×

~k + nk + flk + mk OCM + nM + tim + mM

(3)

o r

p(( i + 1 ) trip ends at M]MI . . . . . Mi) = fik + mk(i) k=J c¢~ + nk(i) + flk + mk(i)

~M + riM(i)

aM + riM(i) + tim + mM(i) '

so that for any (where is greater than the longest trip)

p( ( i + 1 ) trip ends at MIMI . . . . . Mi) = E(X~'{i) '" "XM'li)+l "" "Xn {i)) E(Xj ":~i) . . .X;i ~ l ° . . . y " "~)

' " M ' " ) p(Mi+l = M I M I . . . . , M i ) = E ( X ~ ' " VbM+l " x b E(X,~' . . x ~ " . x b )

f/ where bk = ~ i = l l (Mi = k ) . This implies that for any

p(Ml . . . . . M n ) = E ( X ~ ' - X b ) x ( ) . . I Z b k = n . k = l

(where is greater than the longest trip)

(4)

Now, consider the sampling scheme involving the Bernoulli random variates. After i trips have been completed let nk(i) = ~k(i) - ~k and mk(i) = ilk(i) - ilk. Note that mk(i) = ~ j > k ni(i)" Then

354 S. Walker / Statistics Probability Letters 37 (1998) 351-355

From here it is easy to see that, given X, the ( M I , M 2 . . . . ) are i.i.d, and

p ( M I = k I X ) = X k a.s. k = 1 , 2 . . . . .

If, for k = 1,2 . . . . and n = 1,2 . . . . .

t l

Zk(n)---- -1 Z I ( M / = k ) ' n

i I

then, for all ,

[ ( Z l ( n ) . . . . . Z (n ) ) IX] - - -+(XI . . . . . X ) a . s .

Using Lemma 3 it follows that

S(Z~(n) . . . . ,Z ( n ) ) ~ o~(Xl . . . . . X ),

completing the proof. []

Corollary 1. IJl for k = 1,2 . . . . and n = 1,2 . . . . .

F k ( n ) = ~ - ~ Z j ( n ) j<~k

then, jo r any ,

5 f (Fl (n) , . . . , e (n)) --~ Lf(Fl . . . . . F ),

where F~ = ~-]q <~ k Xi"

Corollary 2. I f Ml . . . . . Mt are i.i.d, f rom F with F obtained f rom (1) and (2) then marginally M1 . . . . . Mt are distributed as the lengths o f the first t Bernoulli trips.

Corollary 3. I f Ml . . . . . Mt are the lengths o f the first t Bernoulli trips then Ml . . . . . Mt are exchangeable.

Additionally, if N1, . . . , Nt are observations on the positive integers with prior F then the predictive law for the next observation coincides with the length of the (t + l )th Bernoulli trip as though N~ . . . . . Nt were the lengths of the first t trips, i.e.,

Lf(Nt+l ] Nl . . . . . Nt ) = c~#(Mt+ 11 MI = Nj . . . . . Mt = Nt).

Starting a sequence of Bernoulli trips from the parameters {~k(t), ilk(t)} where, for example, uk ( t )= cq + ~"~-i l(Mi = k), gives

5f(Fl(n) . . . . . F (n)) ~ of(F1 . . . . . F [Nj . . . . . Nt).

Of course, the beta process is largely concerned with censored observations. This leads naturally to the idea of a censored trip. A Bernoulli trip is said to be censored at the integer k if Yk, Yk+l . . . . are simply not sampled ({Yj = 0: j = 1 . . . . . k - 1} are sampled). This means that the updating mechanism for such a trip can

S. Walker / Statistics Probability Letters 37 (1998) 351 355 355

only be given by

[ 3 i - ~ f l i + l for j = 1 . . . . . k - l .

This also fits in with the updating mechanism of the beta process if a particular observation was only known to be ~>k.

R e f e r e n c e s

Hjort, N.L., 1990. Nonparametric Bayes estimators based on beta processes in models for life history data. Ann. Statist. 18, 1259-1294.