How to do Bayesian updating for a simple practical problem?

Question

All the previous questions are more sophisticated than I can understand and also slightly different to my puzzle.

Say you observe a crime and you have one possible suspect, so I have a belief about someone who might have done it. I am interested in how to update my beliefs about that suspect, every time a new crime is observed.

Here is how I am working this out.

So given a crime (Event A), what is the probability the Suspect did it (Event B)?

To answer this, we have to specify beliefs, right?

1. We have to say what we believe to be the probability of the crime given the Suspect did not do it. Prob(A). This would just be the baseline probability of such a crime occurring, right? So let's say we have good reason to believe $Prob(A) = .01$.

2. We have to state our prior belief the Suspect did it, prior to what we learned from the crime scene. Let's say the suspect has been on our radar from before the crime, so before the crime we'd estimate he has a 15% chance of committing this crime. $Prob(B) = .15$.

3. Next we have to state the probability of observing the crime if the suspect did it, right? It's not clear to me if I'm supposed to calculate this somehow or is this also just a subjective belief? Like based on whether we think the Suspect would cover it up well or fail to cover it up well? Or should I calculate this as a conditional probability? As in

$Prob(A|B) = Prob(A $ and $ B) / Prob(B) = .0015 / .15 = .01$

Bonus if you can clarify this please but even if I'm wrong perhaps let's just say this probability is .01.

4. Then our posterior probability, the probability the Suspect committed the crime given the crime, would be

$Prob(B|A)$ = $Prob(A|B)$ x $Prob(B)$ / $Prob(A)$

Right? This would be:

.01 * .15 / .01 = .0015

So my first question is, is this correct? If not, could you please explain my mistake and give me the correct answer to this puzzle? It should be pretty easy for you all! :-)

My second question is, if I observe a new crime, how do I update this probability? And so on with each new crime?

Thank you very, very much for your time!

Answer 1

Unfortunately, it it not that simple. Knowing only the marginal probabilities $P(A)$ and $P(B)$ is insufficient to apply Bayes theorem. What you also need to know is the conditional probabilities $P(A\mid B)$ and $P(A\mid \neg B)$, since Bayes theorem is

$$ P(B\mid A) = \frac{P(A\mid B)\,P(B)}{ P(A\mid B) \,P(B) + P(A\mid \neg B) \,P(\neg B)} $$

In your question you mention base rate, in fact Bayes theorem can be applied to cases where you want to correct probabilities for a base rate. This answer describes in detail how results of diagnostic test can be corrected for base rate using Bayes theorem (nice examples are also given on Wikipedia in article Base rate fallacy).

Notice that such usage of Bayes theorem has nothing to do with updating subjective prior probabilities given the data as in Bayesian statistics. Applying Bayes theorem is not the same as using Bayesian statistics. I recently given answer to similar question as yours where I describe in detail that is the difference and how to solve problems similar as yours. In such case you need a prior, i.e. a distribution describing your out-of-data knowledge about some phenomenon (assumed distribution of parameter $\theta$); likelihood, i.e. a distribution describing conditional probability of observing the data you have given parameter $\theta$:

$$ \underbrace{p(\theta \mid X)}_\text{posterior} \propto \underbrace{p(X \mid \theta)}_\text{likelihood} \; \underbrace{p(\theta)}_\text{prior} $$

So it is used in different context, but check the answer I refer to for more details.