Sharp bounds on a true base rate from a noisy recorded one — morie_fairness_base_rate

Rearrest is a proxy for reoffending. With false-positive noise $\alpha = P(\text{recorded } 1 \mid \text{true } 0)$ and false-negative noise $\beta = P(\text{recorded } 0 \mid \text{true } 1)$ the recorded rate is $p_{obs} = p(1-\beta) + (1-p)\alpha$. When only boxes $\alpha \in [0, a]$, $\beta \in [0, b]$ with $a + b < 1$ are credible, the true rate is identified only up to $$\frac{p_{obs} - a}{1 - a} \le p \le \frac{p_{obs}}{1 - b},$$ and both ends are attained (Research.P5.true_base_rate_bounds, lower_bound_attained, upper_bound_attained), so no narrower interval follows from the boxes alone.

Usage

morie_fairness_base_rate_bounds(p_obs, alpha_max, beta_max)

Arguments

p_obs: Recorded (proxy) base rate, one or more values in [0, 1].
alpha_max: Largest credible false-positive noise rate.
beta_max: Largest credible false-negative noise rate; alpha_max + beta_max must be below 1.

Value

A data frame with p_obs, lower, upper (clamped to [0, 1]) and width.

Examples

morie_fairness_base_rate_bounds(c(0.35, 0.55), alpha_max = 0.10, beta_max = 0.20)
#>   p_obs     lower  upper     width
#> 1  0.35 0.2777778 0.4375 0.1597222
#> 2  0.55 0.5000000 0.6875 0.1875000
# can two groups' true base rates still be ordered? only if the boxes do not overlap