Suppose I want to learn the value of \(\omega\in\{0,1\}\).
I observe a sequence of iid signals \((s_n)_{n\ge1}\) with
$$\Pr(s_n=0\,\vert\,\omega=0)=1-\alpha$$
and
$$\Pr(s_n=1\,\vert\,\omega=1)=1-\beta,$$
where \(\alpha\) and \(\beta\) are false positive and false negative rates.
I let \(\pi_n\) denote my belief that \(\omega=1\) after observing \(n\) signals, and update this belief sequentially via Bayes’ formula:
$$\pi_{n}(s)=\frac{\Pr(s_n=s\,\vert\,\omega=1)\pi_{n-1}}{\Pr(s_n=s)}.$$
In particular, if I observe \(s_n=0\) then I update my belief to
$$\pi_n(0)=\frac{\beta\pi_{n-1}}{\beta\pi_{n-1}+(1-\alpha)(1-\pi_{n-1})},$$
whereas if I observe \(s_n=1\) then I update my belief to
$$\pi_n(1)=\frac{(1-\beta)\pi_{n-1}}{(1-\beta)\pi_{n-1}+\alpha(1-\pi_{n-1})}.$$
The chart below shows how my belief \(\pi_n\) changes with \(n\).
Each path in the chart corresponds to the sequence of beliefs \((\pi_0,\pi_1,\ldots,\pi_{100})\) obtained by updating my initial belief \(\pi_0=0.5\) in response to a signal sequence \((s_1,s_2,\ldots,s_{100})\).
I simulate 10 such sequences, fixing \(\omega=1\) and \(\alpha=0.4\) but varying \(\beta\in\{0.2,0.4,0.6,0.8\}\).
If \(\beta\not=0.6\) then my belief converges to \(\pi_n=1\) as \(n\) grows.
However, if \(\beta=0.6\) then \(\pi_n=\pi_0\) for each \(n\); that is, I never update my beliefs regardless of the signals I observe.
This is because if \(\alpha+\beta=1\) then \(\Pr(s_n=s\cap\omega=1)=\Pr(s_n=s)\) for each \(s\in\{0,1\}\), and so signals are uninformative because they are independent of \(\omega\).
The chart below plots the mean of my beliefs \(\pi_n\) across 1,000 realizations of the signals simulated above.
Again, I fix \(\omega=1\) and the false positive rate \(\alpha=0.4\) but vary the false negative rate \(\beta\in\{0.2,0.4,0.6,0.8\}\).
Higher values of \(\beta\) are not always worse: my belief converges to the truth faster when \(\beta=0.8\) than when \(\beta=0.4\).
Intuitively, if I know the false negative rate is close to 100% then observing a signal \(s_n=0\) gives me strong evidence that \(\omega=1\).
