Howland will forgery trial

From Free net encyclopedia

The Howland will forgery trial was a U.S. court case in 1868 to decide Henrietta Howland Robinson's contest of the will of Sylvia Ann Howland. It is famous for the forensic use of mathematics by Benjamin Peirce as expert witness.

Contents 1 Robinson v. Mandell 2 Statistical analysis 2.1 Peirce's arithmetic error 2.2 Testing hypotheses suggested by the data 2.3 Bayesian considerations 2.4 A modern Bayesian analysis 3 Bibliography

Robinson v. Mandell

Sylvia Ann Howland died in 1865, leaving roughly half her fortune, of some USD 2 million, to various legatees with the residue to be held in trust for the benefit of Robinson, Howland's niece. The principal was to be distributed to various beneficiaries on Robinson's death.

Robinson produced an earlier will, leaving her the whole estate outright. To the will was attached a second and separate page, putatively seeking to invalidate any subsequent wills. Howell's executor, Thomas Mandell, rejected Robinson's claim, insisting that the second page was a forgery, and she sued.

In the ensuing case of Robinson v. Mandell, Charles Sanders Peirce testified that he had made pairwise comparisons of 42 examples of Howland's signature, overlaying them and counting the number of downstrokes that overlapped. Each signature featured 30 downstrokes and he concluded that, on average, 6 of the 30 overlapped, 1 in 5. When the, admitedly genuine, signature on the first page of the contested will was compared with that on the second, all 30 downstrokes coincided, suggesting that the second signature was a tracing of the first.

Benjamin Peirce, Charles' father, then took the stand and asserted that the probability that all 30 downstrokes should coincide in two genuine signatures was 1 divided by 2,666,000,000,000,000,000,000. He went on to observe

So vast improbability is practically an impossibility. Such evanescent shadows of probability cannot belong to actual life. They are unimaginably less than those least things which the law cares not for. ... The coincidence which has occurred here must have had its origin in an intention to produce it. It is utterly repugnant to sound reason to attribute this coincidence to any cause but design.

In the event, the court ruled that Robinson's testimony in support of Howland's signature was inadmissible as she was a party to the will. The statistical evidence was not called upon in judgement.

Statistical analysis

The case is one of a series of attempts to introduce mathematical reasoning into the courts. People v. Collins is a more recent example.

Peirce's arithmetic error

Though Peirce gave 1/5³⁰ as 1/2.666...×10²¹, it is, in fact, 1/9.313...×10²⁰.

Testing hypotheses suggested by the data

Why was the particular metric of overlapping downstrokes chosen, rather than one of many other ways of quantifying the similarity of two signatures? In such situations, there is always a fear that analysts are, perhaps inadvertently, indulging in testing hypotheses suggested by the data. In this case, tiny "probabilities" can be less surprising than where the metric had originated before the data had been seen.

Bayesian considerations

The argument is an example of what is often regarded as the prosecutor's fallacy. The Peirces' analysis attempts to calculate the probability that two signatures would display such a degree of similarity given that they were genuine:

P_Peirce=P(30 coincident downstrokes|genuine signature).

However, it is often contended that what is relevant to the court is the probability that the signatures are genuine given their similarity:

P_{Genuine signature}=P(genuine signature|30 coincident downstrokes).

To relate the two probabilities requires the use of Bayes' theorem:

P_{Genuine signature} α P_Peirce×P(genuine signature),

- where P(genuine signature) is the probability that the signature is genuine given all the other evidence in the case. The ultimate probability of forgery is still a function of just how compelling, or otherwise, is the document's provenance.

Image:Posterior.png

A modern Bayesian analysis

If Peirce simply used the argument above, raising 1/5 to the 30th power (which seems unlikely) then it is an approximate calculation of a Bayes factor, with the approximation being made that the proportion of downstroke matches in a collection of true signatures is exactly 1 in 5. However, this proportion was found from a sample of 42 signatures and is thus subject to some sampling error. A modern Bayesian analysis starts by assigning a vague prior distribution to the downstroke match proportion and learning from the data by conditioning on it with a binomial likelihood function. A suitable prior distribution would be a conjugate prior, which in this case is a beta distribution. Suitable parameters for the vague beta distribution could be either the improper prior alpha = beta = 0, the Jeffreys prior alpha = beta = 0.5, or the uniform prior alpha = beta = 1. (The result does not depend greatly on which is chosen.) The evidence is equivalent to saying that, of the 30 times 42 = 1260 downstroke events, 1 in 5 of them are matches, i.e. there are 252 matches and 1008 non-matches. By conjugacy, the posterior distribution is also a beta distribution, but with parameters (252 plus alpha) and (1008 plus beta). See the figure for a plot of this posterior distribution. Having obtained this posterior distribution, the second stage of the calculation is to compute the probability of observing r = 30 matches, assuming a binomial distribution with N = 30 and a success probability which is unknown, but which follows the previously calculated beta posterior distribution. This is given by

<math>

p(r = 30 | \theta)= \frac{1}{B(252.5,1008.5)}\int_0^1 \theta^{30} \theta^{251.5}(1 - \theta)^{1007.5}d\theta

= 4.153092037700561 \times 10^{-21}

.</math>

where B(a,b) is the beta function.

This gives the probability that 30 matches would be observed, given that the signature on the codicil is genuine. Under the alternate hypothesis that the signature was a traced copy of the signature on the first page, the number of downstroke matches would be 30 i.e. the probability of 30 matches is 1. Thus the numerator and the denominator of the Bayes factor are known. The posterior odds are obtained by multiplying the Bayes factor by the prior odds. Assuming the two hypotheses are equally likely a priori (odds of 1:1), the odds against the hypothesis that the signature is genuine are

<math>

\frac{1}{4.153092037700561 \times 10^{-21}} = 2.40784454311 \times 10^{20} .</math>

In view of the similarity of this result to Peirce's reported result, it is highly likely that he did a Bayesian calculation similar to this one. He may have used a prior other than those listed above. Alternatively, given the computing tools available in 1868, he may have approximated the integral, or he may simply have made an error in its calculation.

Bibliography

• Robinson v. Mandell, 20 F. Cas. 1027 (C.C.D. Mass. 1868) (No. 11,959)
• Menand, L. (2002) The Metaphysical Club: A Story of Ideas in America ISBN 0007147376, pp163-176
• Meier, P. & Zabell, S. (1980) "Benjamin Peirce and the Howland Will", 75 Journal of the American Statistical Association vol. 75 p497
• "The Howland Will Case", American Law Review vol. 4 p625 (1870)
• Eggleston, Richard (1983) Evidence, Truth and Probability ISBN 0297782630