Sally Clark: The Prosecutor's Fallacy Convicted an Innocent Mother

"Sally Clark lost two infant children to natural causes. The state convicted her of murdering them, in part because of a statistical mistake the prosecution was permitted to present as evidence. She served three years in prison before the conviction was overturned. She never recovered. She died in 2007."

What Happened

Sally Clark was an English solicitor whose first son, Christopher, died in his sleep at 11 weeks of age in 1996. The death was attributed at the time to natural causes (Sudden Infant Death Syndrome, SIDS). Her second son, Harry, died in similar circumstances at 8 weeks of age in 1998. The repetition of two infant deaths in one family attracted suspicion, and Clark was charged with the murder of both children.

At her 1999 trial, the prosecution called an expert witness, the pediatrician Sir Roy Meadow, who testified that the probability of two SIDS deaths in a family like the Clarks' was about 1 in 73 million. The jury convicted. Clark was sentenced to life imprisonment.

The Statistical Crime in the Statistical Crime

Meadow's '1 in 73 million' figure was wrong in two distinct ways, and the prosecution presented the figure (and the jury accepted it) under the spell of the prosecutor's fallacy.

First, the figure itself was constructed by squaring the single-case SIDS probability (≈ 1 in 8,500 for a non-smoking middle-class family) — implicitly assuming the two deaths were independent. They are not independent: families that suffer one SIDS death have a meaningfully elevated probability of suffering another (genetic, environmental, and behavioral factors are shared). The assumption of independence was statistically illiterate.

Second, even granting the 1-in-73-million figure, presenting it to a jury as 'the probability of innocence' is the prosecutor's fallacy. The figure (had it been correct) would be P(two SIDS deaths | innocent) — the probability of the evidence given the null. The jury translated it as P(innocent | two infant deaths) — the probability of innocence given the evidence. These two numbers can differ by enormous factors. The correct Bayesian translation, accounting for the prior probability of a parent murdering two infants (extremely rare), would have given a posterior probability of guilt far below the threshold for conviction.

The Aftermath

Sally Clark's conviction was overturned in 2003 after the Royal Statistical Society publicly objected to Meadow's testimony and a second appeal revealed that the pathologist who examined Harry had withheld microbiological evidence (a bacterial infection that could plausibly have caused the death). Clark was released after three years in prison. She did not recover. She died in 2007 of acute alcohol poisoning, four years after her release.

The case is now part of the canonical literature on the misuse of statistical evidence in court. Multiple convictions Meadow contributed to were later overturned. The case is studied by statisticians, lawyers, and citizens as a warning about what the prosecutor's fallacy costs when it operates without challenge.

The Mechanism, Compressed

Sally Clark was convicted because the courtroom confused P(evidence | innocent) with P(innocent | evidence). The first number, even when correctly computed, says nothing on its own about the second. Bayes' rule and a prior are required to convert one into the other, and the prior in this case (that a parent murdered two infants) is extraordinarily small. The prosecutor's fallacy is what makes the conversion seem unnecessary. It is the single most consequential statistical mistake the legal system regularly tolerates.

Code

Bayesian decomposition of the Sally Clark case·python

# A Bayesian decomposition of the Sally Clark case.
# We use rough order-of-magnitude numbers — the principles, not the precise figures.

# Hypotheses:
#   H₀ = two natural infant deaths (no murder).
#   H₁ = parent murdered both infants.

# Prior probabilities (rough order of magnitude).
prior_natural = 1 - 1e-6            # natural deaths overwhelmingly more common
prior_murder = 1e-6                  # double infant murder is extremely rare

# Likelihoods (the probability of observing 'two infant deaths' under each hypothesis).
# These are stylized; the actual figures depend on epidemiology.
p_two_deaths_given_natural = 1 / 8500 ** 2 * 10   # adjusted upward to reflect non-independence
p_two_deaths_given_murder = 1.0                    # if H₁ is true, both deaths happened

# Posterior via Bayes' rule.
numerator = p_two_deaths_given_murder * prior_murder
denominator = (
    p_two_deaths_given_murder * prior_murder
    + p_two_deaths_given_natural * prior_natural
)
posterior_murder = numerator / denominator

print(f"P(murder | two infant deaths) ≈ {posterior_murder:.4f}")
print(f"P(natural causes | two infant deaths) ≈ {1 - posterior_murder:.4f}")
print(f"\nThe prosecutor's headline figure (1 in 73 million) was P(evidence | natural)")
print(f"and ignored the prior probability of a parent murdering two infants.")

# Even adjusting the natural-causes probability upward, the posterior for murder
# does not approach 'beyond reasonable doubt.' The prosecution's number
# was technically a likelihood, not a posterior. The jury, untrained in this
# distinction, treated it as the posterior. That conversion error sent an
# innocent woman to prison.

Exercise

Read or recall one more famous case where statistical evidence played a major role (the People v. Collins case, the OJ Simpson case, or any DNA evidence case you've encountered). For each, ask: did the prosecution present P(evidence | hypothesis) and let the jury translate it into P(hypothesis | evidence)? Almost always yes. The fallacy is not rare — it is the default operating mode of statistical evidence in court.

Hint

DNA match probabilities, gunshot residue probabilities, fingerprint match probabilities — all of these are P(evidence | innocent). Translating them into P(innocent | evidence) requires a prior, which juries are almost never given the tools to incorporate.

Sally Clark: The Prosecutor's Fallacy Convicted an Innocent Mother

What Happened

The Statistical Crime in the Statistical Crime

The Aftermath

The Mechanism, Compressed

Code

External links

Exercise

Progress

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