Multi-Witness Bayesian Evidence Combination

Witness Panel

Witness count (N) 6

Individual reliability 0.75

Witness correlation ρ 0.00

Naive confidence (assumes independent)

–

Corrected confidence

–

Effective witnesses N_eff

–

Design effect (deff)

–

How it works

Each witness's testimony is evidence with a log-likelihood ratio (LLR). If a witness is right with probability r (and wrong the rest of the time), their testimony contributes:

LLR = log( r / (1 - r) )

For truly independent witnesses, Bayes' theorem says log-odds simply add up:

log-odds_total = log-odds_prior + Σ LLR_i

That's the naive line below — and it's only correct if no two witnesses influenced each other. In reality, witnesses often share a contamination source: they read the same viral post, talked before filing separate reports, or were primed by the same media coverage. That shared source means their testimonies are not independent draws — they're correlated, and summing raw LLRs double-counts the same underlying "evidence" N times.

The standard fix (Kish's design effect, used for clustered survey samples) shrinks N down to an effective witness count N_eff before combining:

deff = 1 + (N - 1)·ρ
N_eff = N / deff
log-odds_corrected = log-odds_prior + N_eff · mean(LLR)

ρ = 0 — fully independent witnesses; N_eff = N, no correction needed.
ρ = 1 — all witnesses share one common source; N_eff → 1, so 20 "witnesses" carry no more weight than a single report.

Real-world relevance: this is exactly why courts and meta-analyses weight corroborating testimony down when witnesses could have influenced each other, and why "12 people saw it" means far less if all 12 were standing together reading the same rumor thread.

Naive combined confidence (wrong if ρ>0) Correlation-corrected confidence