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Suppose I'm interested in the probability that event $A$ will occur. I'm uncertain about $P(A)$, but I believe that it has a uniform distribution on $[0.1,.9]$. Moreover, I know that event $B$ and $C$ make event $A$ more likely, specifically, I know the following:

$$P(B|A)=0.55\hspace{0.2cm} (\text{odds of}\hspace{0.1cm}1.222), P(B|\neg A)=0.5\hspace{0.2cm} (\text{odds of}\hspace{0.1cm}1)$$ $$P(C|A)=0.75\hspace{0.2cm} (\text{odds of}\hspace{0.1cm}3), P(C|\neg A)=0.6 \hspace{0.2cm}(\text{odds of}\hspace{0.1cm}1.5)$$

Moreover, I cannot observe events $B$ and $C$, but believe that $P(B)=0.75$ and $P(C)=0.9$. How should I update my credence regarding $P(A)$? I believe Bayes networks are relevant to this type of question, and I think I have an intuitive sense for how this work, but I'm not too certain.


My attempt, using the the products of Bayes-factors, is as follows. My Bayes factor for the likelihood of $A$ occurring vs not occurring given that $B$ occurs is (in odds) $$B_f(B)=\frac{1.222}{1}=1.222$$

My Bayes factor for the likelihood of $A$ occurring vs not occurring given that $C$ occurs is (in odds) $$B_f(C)=\frac{3}{1.5}=2$$

But I don't know for certain that events $B$ and $C$ occur, so I weight these bayes factors according to their probabilities, giving the following total Bayes Factor:

$$0.75*1.222+(1-0.75)+0.9*2+(1-0.9)= 3.0665$$

I add $1-P(A)$ and $1-P(B)$ because not observing $A$ and $B$ should not update my credence (I think). Since my initial distribution about $P(A)$ was the uniform distribution, which in odds was [0.11111111111, 9], updating this by multiplying by the Bayes factor gives [0.3018,27.59]. In probabilities this is roughly

$$[0.23,0.97].$$

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  • $\begingroup$ Any useful reading material would also be appreciated! $\endgroup$ – pafnuti Oct 20 '18 at 10:02

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