Bayesian Functional Size (Beta-Binomial Model)

Computes a Bayesian posterior distribution for the functional size of a lineup using a Beta-Binomial conjugate model for the suspect-selection rate.

Usage

func_size_bayes(
  lineup_vec,
  target_pos,
  alpha = 0.5,
  S = 10000,
  credible_mass = 0.95,
  threshold = NULL
)

Arguments

lineup_vec: A numeric vector of lineup choices.
target_pos: A scalar; the position of the suspect in the lineup.
alpha: Dirichlet concentration (prior strength). A positive scalar. Named shortcuts: "jeffreys" (0.5, default), "uniform" (1), "weak" (0.1).
S: Number of posterior draws. Default 10000.
credible_mass: Width of the equal-tailed credible interval. Default 0.95.
threshold: Optional numeric. If supplied, reports $P(F < t)$ and $P(F > t \mid \mathbf{n})$.

Value

An object of class "func_size_bayes" containing:

F_draws: Numeric vector of length S: posterior draws of functional size.
posterior_mean, posterior_median: Posterior point estimates. The mean is exact, and is Inf when it does not exist (see Details).
credible_interval: Named two-element vector (lower, upper).
prior_alpha, n, n_suspect, S, credible_mass: Input metadata.
threshold, threshold_probs: Threshold and probabilities (or NULL).

Details

Functional size $F = N / n_s$ is the reciprocal of the suspect-selection rate $p = n_s / N$. The Beta-Binomial conjugate model gives: $$p \mid \mathbf{n} \sim \mathrm{Beta}(n_s + \alpha,\; N - n_s + \alpha)$$ Posterior draws of $p^{(s)}$ are transformed via $F^{(s)} = 1/p^{(s)}$. The default Jeffreys prior ($\alpha = 0.5$) is recommended. The posterior gives direct probability statements such as $P(F < k \mid \mathbf{n})$, where $k$ is the nominal lineup size.

The posterior mean of a reciprocal is computed exactly, not from the draws: with $a = n_s + \alpha$ and $b = N - n_s + \alpha$, $E[F \mid \mathbf{n}] = (a + b - 1)/(a - 1)$ when $a > 1$. When $a \le 1$ (for example, no suspect choices under the Jeffreys prior) the mean is infinite, and the Monte Carlo average of the draws would not settle on any value as S grew. Report the median and interval instead.

References

Tredoux, C. G. (1998). Statistical inference on measures of lineup fairness. Law and Human Behavior, 22(2), 217-237.

Examples

set.seed(1)
lineup_vec <- round(runif(100, 1, 6))
res <- func_size_bayes(lineup_vec, target_pos = 3)
print(res)
#> Bayesian Functional Size - Beta-Binomial model
#>   Prior: Jeffreys-type Beta(0.50, 0.50)
#>   n = 100, suspect IDs = 28 (rate = 0.280)
#>   Posterior mean F:   3.636
#>   Posterior median F: 3.567
#>   95% credible interval: [2.673, 5.034]
plot(res)


# Threshold: P(F < 6 | data)
res_t <- func_size_bayes(lineup_vec, target_pos = 3, threshold = 6)
print(res_t)
#> Bayesian Functional Size - Beta-Binomial model
#>   Prior: Jeffreys-type Beta(0.50, 0.50)
#>   n = 100, suspect IDs = 28 (rate = 0.280)
#>   Posterior mean F:   3.636
#>   Posterior median F: 3.563
#>   95% credible interval: [2.668, 5.005]
#>   P(F < 6.000 | data): 0.9983
#>   P(F > 6.000 | data): 0.0017