# Updating a hypothesis on multiple partitions of uncertain evidence

I want to forecast $$P(A)$$ where $$A$$ is a messy real-world event, for which I have no analytical expression or statistical model.

Assume, however, that for $$b$$ events $$B_i$$ I have forecasts for $$P(B_i)$$ and estimates for $$P(A|B_i)$$ and $$P(A|\neg B_i)$$. For each event I can use Jeffrey's rule to get: $$\begin{equation} P_i(A)=P(A|B_i)*P(B_i)+P(A|\neg B_i)*(1-P(B_i)) \end{equation}$$ (See e.g. Diaconis & Zabell, 1983.)

However, this will give $$b$$ different estimates $$P_i(A)$$.

Is there any way to combine all the information I have to get a single estimate $$P(A)$$?

The $$B_i$$ are not assumed to be either mutually exclusive or jointly exhaustive.

If there is not a straightforward theoretical way like the Jeffrey's rule, I would appreciate reasonably motivated practical heuristics. My use case is getting a good practical estimate, not to prove anything.

• I think that the "Jeffrey's rule" might be otherwise known as the "law of total probability" – NofP Oct 22 '18 at 22:51
• I'm assuming that the $b$ events $B_i$ are a complete disjoint set of events, i.e. that $\sum_i P(B_i) = 1$? – jwimberley Oct 23 '18 at 12:07
• @jwimberley no, they are not. For each single $B_i$ you can take its complement, but nothing beyond that. – Jacob Lagerros Oct 23 '18 at 15:59

You can use the $$B_i$$ to construct a partition, but I'm not sure how useful this is as it might be difficult to estimate the probabilities of the relevant events: $$¬(B_1\cup B_2\dots\cup B_b),\ B_1\cap¬(B_2\cup\dots\cup B_b), \ B_2\cap¬(B_1\cup B_3\cup\dots\cup B_b),\ \dots \ , \ B_b\cap¬(B_1\cup B_2\cup\dots\cup B_{b-1}), \ B_1\cap B_2\cap ¬(B_3\cup\dots\cup B_b),\ \dots \ , \ B_1\cap B_2\cap\dots\cap B_b$$
Then you can apply the law of total probability to get a single estimate of $$P(A)$$.
• That's progress, but still not very practically useful as it requires estimating $>2^b$ quantities if we're not assuming independence. – Jacob Lagerros Oct 23 '18 at 16:52