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)

The bridge requires the prior P(innocent) — the base rate of innocence among all defendants facing this charge. That prior is the explosive component the legal system avoids discussing because it would be politically toxic. (What is the prior probability of guilt for a randomly selected defendant of this race / class / district? Whose number is that?) The legal system refuses to put a number on it; the Bayesian frame refuses to let the question be skipped.

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 | guilty) ≈ 1 (the defendant is the source, of course).
P(match | innocent) = 1 / 1,000,000 (the quoted random-match probability).
P(guilty) prior = depends on the case.

If the police searched a database of one million people for a match, the expected number of matches by chance alone is ~1 — so finding a match tells you almost nothing about guilt without additional context. If the prior P(guilty) was very low (say 0.0001 for an unspecified suspect), the posterior P(guilty | match) might still be far below 'beyond reasonable doubt.' The 'one in a million' figure feels like proof; in Bayesian translation it is a weak signal that needs a strong prior to convict.

The Citizen Lens

The prosecutor's fallacy is the conversion of a likelihood into a posterior without the Bayesian bridge. The bridge requires a prior, and the prior is what the legal system makes invisible. Citizens who can name the missing prior have the tools to read prosecutorial statistics correctly; citizens who cannot are at the mercy of the framing the prosecution chooses. Track 06 dismantled the frame in frequentist language; this lesson does the same in Bayesian. Both languages reach the same conclusion: the missing piece is the prior, and the conversion without it is statistically illiterate.

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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