**Stephen E. Fienberg**, Department of Statistics and
Social and Decision Sciences, Carnegie Mellon University,
Pittsburgh PA 15213, U.S.A.; and **D.H. Kaye**, College of
Law, Arizona State University, Tempe AZ 85287-7906, U.S.A.

Note: This paper appears in volume 154 of the Journal of the Royal Statistical Society (series A), at 61-74 (1991). It is available in this journal's format at JSTOR. For permission to copy it, please contact the Royal Statistical Society.

**Key Words and Phrases**: Clusters of events; Expert witnesses;
Law; Latent Variables, Similar events; Statistical testimony.

"It is undoubtedly not competent for the prosecution to adduce evidence tending to show that the accused has been guilty of criminal acts other than those covered by the indictment, for the purpose of leading to the conclusion that the accused is a person likely from his criminal conduct or character to have committed the offence for which he is being tried. On the other hand, the mere fact that the evidence adduced tends to show the commission of other crimes does not render it inadmissible if it be relevant to an issue before the jury, and it may be so relevant if it bears upon the question whether the acts alleged to constitute the crime charged in the indictment were designed or accidental, or to rebut a defence which would otherwise be open to the accused."

That is, where the past crimes are relevant *only* as evidence of defendants' disposition toward
such crimes, they may not be proved in the prosecution's case in chief. Where past crimes are
relevant on some other theory, such as refuting a claim that the allegedly criminal acts at bar were
performed innocently, they may be proved as long as their probative value for this purpose is
sufficient. Thus, "other crimes" evidence is not forbidden because it lacks probative value in
showing the conduct in the particular case. Rather, it is excluded despite its probative
valueeither because the proof of the similar happenings would be unduly time-consuming or
uncertain, because the extrinsic events actually prove less than the jury might think, or because
the jury might be too willing to impose liability on an innocent person who has engaged in other
blameworthy conduct.

(1). P(G=1X_{0}=1,X_{1}=1,...,X_{k}=1) > P(G=1X_{0}=1)

Thus, there is inferential value (see Fienberg and Kadane, 1983; Fienberg and Schervish, 1986) in the knowledge that k other acts "similar" to act 0 occurred, even if we have no direct link between them and G or between them and the accused's commission of acts 1,2,...,k.

"There are about 3,600,000 infants born in the United States every year. Suppose that we took each one of these infants [as] surviving to 6 months and purely by chance assigned them to baby sitters in groups of 20. That means that for each new set of births in the course of a year, we would have about 180,000 baby sitters each with 20 infants in their care. Let's suppose further that those infants were cared for full time, all the time, 24 hours a day from the age of 6 months onward. Then the rate of 9.1 in a trillion means that we would expect to observe as many as three deaths among the charges of one baby sitter about once in 600 years."

His report included several other probability calculations that varied the values of p (.002, .0002 and .00002) and n (20 and 25), and his testimony introduced a Poisson approximation to the binomial probabilities.

Cleary, E. (ed.) (1984) *McCormick on Evidence*, 3rd edn., pp. 549-592. St. Paul, MN: West.

Cross, R. (1979) 5th edn., *Evidence*, London: Butterworths.

Eggleston, R. (1983) *Evidence, Proof and Probability*, 2nd edn., pp. 88-102. London:
Weidenfeld and Nicolson.

Essary, J.D., Proschan, F., and Walkup, D.W. (1967) Association of random variables with
applications, *Annals of Mathematical Statistics*, 38, 1466-1474.

Fienberg, S.E. (1990) Legal likelihoods and a priori assessments: what goes where? In *Bayesian
and Likelihood Methods in Statistics and Econometrics: Essays in Honor of George A. Barnard*.
(S. Geisser, J. S. Hodges, S. J. Press, and A. Zellner, eds.). Amsterdam: North Holland, pp. 141-162.

---------- and Straf, M.L. (1990) Statistical evidence in the U.S. Courts: an appraisal. Presented at this conference.

---------- (ed.) (1989) *The Evolving Role of Statistical Assessments as Evidence in the Courts*.
New York: Springer-Verlag.

Fienberg, S.E. and Kadane, J.B. (1983) The presentation of Bayesian statistical analyses in legal
proceedings, *The Statistician*, 32, 88-98.

---------- and Schervish, M.J. (1986) The relevance of Bayesian inference for the presentation of
statistical evidence and for legal decisionmaking. *Boston University Law Review*, 66, 771-798.

Fleming, I. (1959) *Goldfinger*, p. 123. New York: MacMillan.

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Goodman, L.A. (1974) The analysis of systems of qualitative variables where some variables are
unobservable. Part I: A modified latent structure approach. *American Journal of Sociology*, 75,
1179-1259.

Harrison, K. (1988a), Expert rules out chance in Bolding patient deaths. *Washington Post*, June
7, 1988, B1 & B7.

---------- (1988b), Bolding defense impugns study implicating nurse. *Washington Post*, June 8,
1988, B1 & B7.

--------- (1988c), Judge acquits Nurse Bolding. *Washington Post*, June 21, 1988, A1 & A12.

Imwinkelreid, E. (1984) *Uncharged Misconduct Evidence*. Wilmette, IL: Callaghan.

Istre, G.R., Gustafson, T.L., Baron, R.C., Martin, D.L., and Orlowski, J.P., A mysterious cluster
of deaths and cardiopulmonary arrests in a pediatric intensive care unit. *New England J. Med.*,
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health effects in Woburn, Massachusetts (with discussion). *J. Amer. Statist. Assoc.*, 81, 583-614.

Marjoribanks, E. (1929) *For the Defence: The Life of Sir Edward Marshall Hall*, p. 329. New
York: MacMillan.

Stross, J.K., Shasby, M.D., and Harlan, W.R. (1976) An epidemic of mysterious
cardiopulmonary arrests. *New England J. Med.*, 295, 1107-1110.

UPI (1978), Poisoning charges dropped against two nurses, *New York Times*, Feb. 2, 1978, 16.

Weaver, C. (1988), The chilling case of Nurse 14,
*Regardie's* May 1988, 93-144.

and

We now introduce a latent (unobserved) random variable, Y, that is an indicator for the tendency of the accused to engage in behavior that produces events such as those in question. We let

Clearly, Y is positively correlated with each of the
{X_{i}}, and Y is positively correlated with G if
X_{0}=1.

*Assumption 1*: Given X_{0} and Y, G is
independent of (X_{1},X_{2},...,X_{k}).

*Assumption 2*: X_{0},X_{1},...,X_{k}
are conditionally independent given the latent variable Y.

and

Then we assume that

*Assumption 3*: alpha_{1} > alpha_{0},
ß_{i1} > ß_{i0}, for i = 0,1,...,k.

*Theorem*. Under assumptions 1-3,

*Corollary*: P(G=1X_{0}=1,X_{1}=1,...,X_{k}=1)
> P(G=1X_{0}=1)

where

and

* Last updated 11/1/97 *