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Lesson 04 of 05 · published

Sally Clark: Conditional Probability, Dependence, and Withheld Evidence

~14 min · sally-clark, prosecutor-fallacy, wrongful-conviction, real-case

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"The Sally Clark case was not one equation gone wrong. Misstated recurrence risk, conditional-probability confusion, and undisclosed microbiological evidence all mattered."

What Happened

Sally Clark, an English solicitor, lost her sons Christopher in 1996 and Harry in 1998. She was convicted in 1999 of murdering both children and sentenced to life imprisonment. Her convictions were quashed in 2003 after a second appeal.

The 1-in-73-Million Figure

At trial, pediatrician Roy Meadow estimated that two sudden infant deaths in a family like Clark's had probability about 1 in 73 million. The calculation squared a single-death estimate of roughly 1 in 8,543, treating the events as independent. The Royal Statistical Society later objected that this independence assumption was not justified and that the number was misleading.

Even a valid P(evidence | innocent) is not P(innocent | evidence). Moving between them requires explicit competing hypotheses, priors, and the probability of the evidence under each. One cannot repair the case by inserting a single guessed prior and announcing a posterior probability of guilt.

Other Evidence Mattered

The successful second appeal also involved microbiological results relating to Harry that had not been disclosed to the defense. The Court of Appeal held that the conviction was unsafe. Statistical error was important, but it should not be presented as the only cause of the conviction or the appeal's only ground.

Clark was released after more than three years in prison. She died in 2007 from acute alcohol intoxication. The case became a central warning about expert statistical testimony, disclosure, dependence, and conditional probabilities in criminal proceedings.

The Mechanism, Carefully Stated

Do not multiply recurrence probabilities without a defensible dependence model, and do not present a likelihood as a posterior. In court, the probability model is only one part of the evidence. Alternative explanations, data quality, disclosure, and the legal burden of proof remain separate questions.

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.

External links

Exercise

Choose a reported court case involving DNA, fingerprints, gunshot residue, or medical statistics. Identify the conditioning direction, the comparison population, dependence assumptions, and any selection or disclosure issues. Do not assume every prosecution made the same fallacy; verify the actual testimony.
Hint
A DNA random-match probability is usually conditional on a specified model and population. It is not automatically P(innocent | match), and it is not the whole evidentiary case.

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