I'm a little confused about how to use Bayes' theorem when I don't have any way to establish an initial prior.
Say I have a sensor that can detect whether there is flouride in a water sample. Let's denote $P(F | M)$ as the probability that a positive measurement is correct, i.e. there is indeed flouride in the water. And let's add some numbers as an example:
- $P(F|M) = 0.7$ (and therefore $P(\bar{F}|M) = 0.3$)
- $P(F|\bar{M}) = 0.1$ (and therefore $P(\bar{F}|\bar{M}) = 0.9$)
where $\bar{M}$ denotes a negative measurement (indicating that there isn't flouride) And $\bar{F}$ denotes that there really is no flouride in the water.
Let's compute $P(M|F)$. Using Bayes:
$$ \begin{align} P(M|F) &= \frac{P(F|M) \cdot P(M)}{P(F)} \\ &= \frac{P(F|M) \cdot P(M)}{P(F|M)P(M) + P(F|\bar{M})P(\bar{M})} \space\space\space\space \text{(total probability on denominator)} \\ &= \frac{0.7 \cdot P(M)}{0.7 \cdot P(M) + 0.1 \cdot P(\bar{M})} \end{align} $$
So now what do I do about the $P(M)$ and $P(\bar{M})$?
This is a classic problem. You can't know what a positive test (for a disease or for fluoride) means if you know only the false positive and true positive rates. You need the incidence of the disease (fluoride) in the population.
See Applied Probability- Bayes theorem