Probability of necessity: bounds for attributing an outcome to a sanction

"Was the sanction necessary for this person's desistance?" With \(a = P(y_x = 1)\) and \(b = P(y_{x'} = 1)\) known (an experiment, or exogenous assignment), the share of people for whom the outcome occurs with the sanction and not without it lies in \([\max(0, a - b), \min(a, 1 - b)]\), both ends attained, and equals \(a - b\) exactly under monotonicity (nobody desists only without the sanction) (Research.P8.necessity_bounds, necessity_of_monotone, necessity_of_disjoint). Dividing by \(a\) gives Pearl's bounds on the probability of necessity and the excess-risk-ratio identity under monotonicity (Pearl, Causality 2e, Thms 9.2.10-9.2.15).

Usage

morie_probability_of_necessity(p_treated, p_control)

Arguments

p_treated

Outcome probability with the sanction, \(a\).

p_control

Outcome probability without it, \(b\).

Value

A list with necessary_share_bounds, pn_bounds (divided by \(a\)), pn_monotone (the excess risk ratio), identified (whether the interval is a point) and theorems.

Examples

morie_probability_of_necessity(p_treated = 0.6, p_control = 0.4)
#> $necessary_share_bounds
#> lower upper 
#>   0.2   0.6 
#> 
#> $pn_bounds
#>     lower     upper 
#> 0.3333333 1.0000000 
#> 
#> $pn_monotone
#> [1] 0.3333333
#> 
#> $identified
#> [1] FALSE
#> 
#> $theorems
#> [1] "Research.P8.necessity_bounds"      "Research.P8.necessity_of_monotone"
#> [3] "Research.P8.necessity_of_disjoint"
#>