Applying the Bayesian theorem to rape statistics

Question

I'm studying the prosecutor's fallacy. Imagine that:

$C = \text{Convicted of rape} \\ I = \text{Innocent of rape} \\ A = \text{Accused of rape}$

If I want to know the probability of being innocent having been accused, I apply Bayes' theorem:

$P(A|I) = \frac{P(I|A)\cdot P(A)}{P(I)}$

So if $P(I|A) = 4\%$ (number of false rape accusations) and $P(A) = \frac{26,3}{100.000}$ (stats for Belgium) and $P(I) = 1-\frac{26,3}{100.000}$ then:

$P(A|I) = 1.052277e^{-05}$

So the probability of being accused while innocent for a random person is far under the probability of being innocent while accused ?

Answer 1

At the moment your problem does not involve the event $C$ at all, and you have not yet specified sufficient information to obtain the probability of interest. Specifically, you have not specified $\mathbb{P}(I|\bar{A})$, which is the probability that a randomly chosen non-accused person is innocent of rape. Using the values you have stipulated, and assuming for simplicity that $\mathbb{P}(I|\bar{A}) \approx 1$ (it would actually be slightly lower than one, since some rapists have never been caught), you should obtain:

$$\begin{equation} \begin{aligned} \mathbb{P}(A|I) &= \frac{\mathbb{P}(I|A) \mathbb{P}(A)}{\mathbb{P}(I|A) \mathbb{P}(A) + \mathbb{P}(I|\bar{A}) \mathbb{P}(\bar{A})} \\[6pt] &\approx \frac{0.04 \cdot 0.00263}{0.04 \cdot 0.00263 + 1 \cdot (1-0.00263)} \\[6pt] &= \frac{0.0001052}{0.0001052 + 0.99737} \\[6pt] &= \frac{0.0001052}{0.9974752} \\[6pt] &= 1.05466 \times 10^{-5}. \\[6pt] \end{aligned} \end{equation}$$

This means that the probability that a randomly selected innocent person is accused of rape is low. Thus, just as you have observed (though you did the calculation wrong) the probability of an accusation against an innocent person may be substantially lower than the probability of innocence conditional on an accusation. This is unsurprising, and is a common result of Bayesian analysis for events with low baseline probability. In this case it is merely a reflection of the fact that the baseline rate of rape, and rape accusations, are low.

Answer 2

Not necessarily. You have incorrectly defined $P(I) = 1 - P(A)$, which does not logically hold to be necessarily true. Is the probability of being innocent equal to the compliment of being accused?

Instead you have defined a $P(I|A)$ and a $P(A)$.

I would think about conditional probability definitions to determine the correct formulation for $P(I)$ with respect to the two terms I have listed above, and then recheck your calculations.