I have worked out a few successive derivations of Bayes' Theorem...
Pr(W)=Pr(W)
Pr(W ∣ C1) = Pr(C1)Pr(C1 ∣ W)Pr(W)
Pr(W ∣ (C1 ∩ C2)) = Pr(C2 ∣ C1)Pr(C2 ∣ (W ∩ C1))Pr(W ∣ C1)
Pr(W ∣ (C1 ∩ C2 ∩ C3))=Pr(C3 ∣ (C1 ∩ C2))Pr(C3 ∣ (W ∩ C1 ∩ C2))Pr(W ∣ (C1 ∩ C2))
I think the pattern is Pr(W ∣ i=1⋂n−1Ci) = Pr(Cn−1 ∣ i=1⋂n−2Ci)Pr(Cn−1 ∣ (W ∩ i=1⋂n−2Ci))Pr(W ∣ i=1⋂n−2Ci).
I have the intuition that this should be interpretable as a recurrence relation. As far as I can tell, Pr(W ∣ i=1⋂n−1Ci) is playing the role of an, and Pr(W ∣ i=1⋂n−2Ci) is playing the role of an−1. However, it is less clear what to do with the other factors. Should I leave them as probabilities, yielding an= Pr(Cn−1 ∣ i=1⋂n−2Ci)Pr(Cn−1 ∣ (W ∩ i=1⋂n−2Ci))an−1, or is there some transformation I need to apply to the coefficient as well? Or should I leave everything in probability notation? The coefficient is certainly not constant in this form, yet I believe the solution should be simple (probably related to Bayesean updating formulas). How can I massage this into an analytically solvable recurrence relation?
Pr(W)=Pr(W)
Pr(W ∣ C1) = Pr(C1)Pr(C1 ∣ W)Pr(W)
Pr(W ∣ (C1 ∩ C2)) = Pr(C2 ∣ C1)Pr(C2 ∣ (W ∩ C1))Pr(W ∣ C1)
Pr(W ∣ (C1 ∩ C2 ∩ C3))=Pr(C3 ∣ (C1 ∩ C2))Pr(C3 ∣ (W ∩ C1 ∩ C2))Pr(W ∣ (C1 ∩ C2))
I think the pattern is Pr(W ∣ i=1⋂n−1Ci) = Pr(Cn−1 ∣ i=1⋂n−2Ci)Pr(Cn−1 ∣ (W ∩ i=1⋂n−2Ci))Pr(W ∣ i=1⋂n−2Ci).
I have the intuition that this should be interpretable as a recurrence relation. As far as I can tell, Pr(W ∣ i=1⋂n−1Ci) is playing the role of an, and Pr(W ∣ i=1⋂n−2Ci) is playing the role of an−1. However, it is less clear what to do with the other factors. Should I leave them as probabilities, yielding an= Pr(Cn−1 ∣ i=1⋂n−2Ci)Pr(Cn−1 ∣ (W ∩ i=1⋂n−2Ci))an−1, or is there some transformation I need to apply to the coefficient as well? Or should I leave everything in probability notation? The coefficient is certainly not constant in this form, yet I believe the solution should be simple (probably related to Bayesean updating formulas). How can I massage this into an analytically solvable recurrence relation?