Learning from noisy signals

Suppose I want to learn the value of $\omega \in \{0,1\}$. I observe a sequence of iid signals $(s_n{)}_{n\ge 1}$ with $$Pr(s_n=0|\omega =0)=1-\alpha$$ and $$Pr(s_n=1|\omega =1)=1-\beta ,$$ 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: $${\pi }_n\left(s\right)=\frac{Pr(s_n=s|\omega =1){\pi }_{n-1}}{Pr(s_n=s)}.$$ In particular, if I observe $s_n=0$ then I update my belief to $${\pi }_n\left(0\right)=\frac{\beta {\pi }_{n-1}}{\beta {\pi }_{n-1}+(1-\alpha )(1-{\pi }_{n-1})},$$ whereas if I observe $s_n=1$ then I update my belief to $${\pi }_n\left(1\right)=\frac{(1-\beta ){\pi }_{n-1}}{(1-\beta ){\pi }_{n-1}+\alpha (1-{\pi }_{n-1})}.$$

The chart below shows how my belief ${\pi }_n$ changes with $n$. Each path in the chart corresponds to the sequence of beliefs $({\pi }_0,{\pi }_1,\dots ,{\pi }_{100})$ obtained by updating my initial belief ${\pi }_0=0.5$ in response to a signal sequence $(s_1,s_2,\dots ,s_{100})$. 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 $\beta \ne 0.6$ 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 $\omega =1$.