Do not publish a probability of guilt

What the court held, and whose words are whose

Three kinds of statement appear in the record and they carry different weight. A witness's testimony and an appellant's argument are reported by the court; only the court's own conclusions are holdings.

Read fairly, the record holds both accounts: at trial the figure was presented as showing two natural deaths in such a family would be very rare; on appeal the court accepted it was "no help" ex post, yet dismissed the appeal. These excerpts do not set out the reasons for dismissal, and this page draws no inference about them. Nor does it treat the absence of an identified natural cause as evidence of homicide: the record measures no such thing.

The arithmetic, reproduced exactly

1. Per-birth input. For a family in which the parents do not smoke, at least one has a waged income and the mother is over 26, the cited risk is 1 in 8,543 live births (para 118).
2. Squaring. 8,543 × 8,543 = 72,982,849. The notice reported 1:73,000,000, a rounding of 17,151 in the denominator (para 118, para 122).
3. Denominator. Each 1 in 8,543 is per live birth. The product, 1 in 72,982,849, is per family with two live births and the three characteristics: a per-family figure, not a per-birth one. The testimony described it as "1 in 73 million live births" (para 114).
4. "Once every hundred years." 72,982,849 ÷ 700,000 = 104.26 (using the rounded 73,000,000: 104.29). That divides a per-family-pair frequency by all live births in England, Wales and Scotland in a year, across all families. The yearly number of qualifying two-birth families, which the comparison would need, is not in the record.
5. The conditional probability the squaring hides. Two deaths require P(two SIDS) = P(first) × P(second | first). Squaring sets P(second | first) equal to 1 in 8,543, which is independence. The quoted study gives a subsequent-child range from 1 in 214 to 1 in 8,543 across family profiles, assuming the factors stay the same, and says it "does not take account of possible familial incidence of factors" (para 122). That is not a measured recurrence for this family.

What the statistic counts, and what it leaves unmeasured

The two-hypothesis expression the record supports

P(two SIDS) = (1 / 8,543) × r

Prior odds, two SIDS : two maternal homicides = [ (1 / 8,543) × r ] ÷ h

Here r is the chance a second infant in such a family dies of SIDS given the first did, and h is the incidence of two maternal homicides per family with two live births and the same profile. Both are unknown in this record. No rate for h is invented here.

Bounds that follow from the mathematics alone

• Because 0 < r ≤ 1: P(two SIDS) is above zero and at most 1 in 8,543.
• If one also assumes a first death never lowers a sibling's risk (r at least 1 in 8,543, an assumption the record does not measure), P(two SIDS) lies between 1 in 72,982,849 and 1 in 8,543: a span of 8,543 times.
• For h the record gives nothing narrower than the meaning of a probability. The prior odds can therefore be any positive number.
• These odds compare only two explanations. Even if both were known, they would not give the probability of either, because other explanations exist.

Prior odds, likelihood ratio, posterior: three different things

Posterior odds (two SIDS : two maternal homicides, given E) = Prior odds (two SIDS : two maternal homicides) × P(E | two SIDS) ÷ P(E | two maternal homicides)

E means the actual medical and physical findings, not the absence of an identified natural cause. The likelihood ratio's numerator is under two SIDS, matching the numerator of the odds; using the reciprocal would reverse the direction, and multiplying a ratio by its own reciprocal gives 1, not posterior odds. The record measures neither the prior odds nor this likelihood ratio, so no posterior odds are computed. A posterior probability of guilt would further need every alternative explanation and all the evidence, and the court said this is "not an exercise the courts would perform" (para 160).

Try a hypothetical recurrence

Hypothetical The values below are assumptions for illustration. None is a finding; the record contains no measured recurrence. Whatever you choose, the two maternal homicides comparator stays unknown.

Static equivalent

What would change this call

1. A sourced, measured probability that a second infant in a family with these characteristics dies of SIDS given that a first did, including familial factors.
2. A sourced incidence of two maternal homicides on the same denominator: families with two live births and the same characteristics.
3. Sourced incidences for every other explanation: other natural causes, accident, one death of each kind, and undetermined deaths.
4. Measured probabilities of the actual medical and physical findings under each explanation, giving a likelihood ratio in the correct direction.
5. A basis for combining the items of evidence, including how they depend on one another.
6. Even then, the court's statement that courts would not perform this exercise (para 160) means a published number would be the reporter's construction, not the court's finding.

Chronology to the as-of date

1. 7 October 1999Expert's notice served stating 1:8,543 and 1:73,000,000. (para 118)
2. Trial date not stated in the excerptsTrial begins without a medical statistician for either side. (para 120)
3. Trial date not stated in the excerptsWitness testifies to 1 in 73 million and once every hundred years. (para 114)
4. 2 October 2000Court of Appeal: figure 'no help' ex post; appeal against conviction on each count dismissed. (para 258)

Source trail

Every quotation on this page is taken verbatim from excerpts of SC1, with the paragraph number given in the excerpts. Witnesses not named in the excerpts are not named here.