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 `\(\omega=1\)`

.