Prosecutor's Fallacy in Bayesian Language

"Track 06 framed the prosecutor's fallacy in frequentist Type I terms. Track 08 names it for what it is: the swap of a likelihood for a posterior, performed in front of a jury that has no tools to notice."

The Frequentist View, Recapped

From Track 06: the prosecutor presents a likelihood (e.g., 'the probability of this DNA match if the defendant is innocent is 1 in a billion') and lets the jury translate it into a posterior ('therefore, the probability of innocence is 1 in a billion'). The frequentist criticism is that this is invalid — the likelihood and the posterior are different conditional probabilities — and that the Type I error rate of the resulting system is much higher than the named figure suggests.

The Bayesian View, Explicit

From this track's vocabulary: the prosecutor presents P(E | H₀ = innocent), the likelihood under the null. The jury wants P(H₀ = innocent | E), the posterior. Bayes' rule is the bridge:

P(innocent | E) = P(E | innocent) × P(innocent) / P(E)

A Bayesian calculation requires prior odds and likelihoods for all relevant evidence and hypotheses. That does not justify assigning a defendant a demographic “base rate of guilt,” and legal standards do not require a court to compute a numerical posterior.

The Numerical Demonstration

Imagine a DNA match with a quoted random-match probability of 1 in a million. The prosecution says this is overwhelming evidence of guilt. The Bayesian reads:

P(match | defendant is the source) depends on laboratory sensitivity, sample quality, mixtures, and error.
A quoted random-match probability applies to a specified population and genetic model; it is not the probability of laboratory error or innocence.
P(guilty) prior = depends on the case.

In an idealized search of one million independent profiles with a one-in-a-million random-match probability, the expected number of coincidental matches is about one. The evidential value of a database hit still depends on database size and dependence, search procedure, laboratory error, relatedness, population model, and all other case evidence.

The Citizen Lens

The prosecutor's fallacy reverses P(evidence | innocence) into P(innocence | evidence). Bayes' rule shows why the reversal is invalid, but a defensible posterior also needs a complete model of alternatives, other evidence, and error—not merely one demographic prior. Track 06 dismantled the frame in frequentist language; this lesson does the same in Bayesian. The reversal is invalid; a defensible Bayesian analysis also needs alternative hypotheses, all relevant evidence, and error models.

Code

DNA match posterior depends massively on the prior·python

# DNA match example with explicit Bayes.
def bayes(prior, likelihood_H, likelihood_not_H):
    num = likelihood_H * prior
    den = num + likelihood_not_H * (1 - prior)
    return num / den

P_match_if_guilty = 1.0
P_match_if_innocent = 1 / 1_000_000

# Three different priors for the same defendant case.
for prior_guilty in (0.5, 0.01, 0.0001):
    post = bayes(prior_guilty, P_match_if_guilty, P_match_if_innocent)
    print(f"prior P(guilty) = {prior_guilty:>7.4f}  →  posterior P(guilty | match) = {post:.5f}")

# Prior 0.5 (eyewitness or other strong pre-evidence): posterior near 1.0.
# Prior 0.01 (cold database hit with weak prior evidence): posterior ~99%.
# Prior 0.0001 (random cold hit on a wide database): posterior ~9% — far
#   from 'beyond reasonable doubt' despite the 'one in a million' headline.
# The number depends massively on the prior. That dependence is the
# prosecutor's fallacy when ignored.

Exercise

Find a recent news report involving DNA evidence (or any 'probability' statistic) used in a criminal case. Identify the quoted likelihood (the '1 in N' figure). Then ask: what is the prior P(guilty) being implicitly assumed, and what would the posterior be if the prior were one-tenth or one-thousandth of what the prosecution implies? Most prosecution statistics fall apart under this question.

Hint

The defense's job is partly to make the prior visible. The citizen's job, watching from outside, is to ask which prior is being implicitly assumed and whether it's defensible.

The Frequentist View, Recapped

The Bayesian View, Explicit

The Numerical Demonstration

The Citizen Lens

Code

External links

Exercise

Progress

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