I have worked out a few successive derivations of Bayes' Theorem...
[imath]\Pr(W) = \Pr(W)[/imath]
[imath]\Pr(W\ |\ C_1)\ =\ \frac{\Pr(C_1\ |\ W)\Pr(W)}{\Pr(C_1)}[/imath]
[imath]\Pr(W\ |\ (C_1\ \cap\ C_2))\ =\ \frac{\Pr(C_2\ |\ (W\ \cap\ C_1))\Pr(W\ |\ C_1)}{\Pr(C_2\ |\ C_1)}[/imath]
[imath]\Pr(W\ |\ (C_1\ \cap\ C_2\ \cap\ C_3)) = \frac{\Pr(C_3\ |\ (W\ \cap\ C_1\ \cap\ C_2))\Pr(W\ |\ (C_1\ \cap\ C_2))}{\Pr(C_3\ |\ (C_1\ \cap\ C_2))}[/imath]
I think the pattern is [imath]\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 1} C_i)\ =\ \frac{\Pr(C_{n - 1}\ |\ (W\ \cap\ \bigcap\limits_{i = 1}^{n - 2} C_i))\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)}{\Pr(C_{n - 1}\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)}[/imath].
I have the intuition that this should be interpretable as a recurrence relation. As far as I can tell, [imath]\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 1} C_i)[/imath] is playing the role of [imath]a_n[/imath], and [imath]\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)[/imath] is playing the role of [imath]a_{n - 1}[/imath]. However, it is less clear what to do with the other factors. Should I leave them as probabilities, yielding [imath]a_n =\ \frac{\Pr(C_{n - 1}\ |\ (W\ \cap\ \bigcap\limits_{i = 1}^{n - 2} C_i))}{\Pr(C_{n - 1}\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)} a_{n - 1}[/imath], 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?
[imath]\Pr(W) = \Pr(W)[/imath]
[imath]\Pr(W\ |\ C_1)\ =\ \frac{\Pr(C_1\ |\ W)\Pr(W)}{\Pr(C_1)}[/imath]
[imath]\Pr(W\ |\ (C_1\ \cap\ C_2))\ =\ \frac{\Pr(C_2\ |\ (W\ \cap\ C_1))\Pr(W\ |\ C_1)}{\Pr(C_2\ |\ C_1)}[/imath]
[imath]\Pr(W\ |\ (C_1\ \cap\ C_2\ \cap\ C_3)) = \frac{\Pr(C_3\ |\ (W\ \cap\ C_1\ \cap\ C_2))\Pr(W\ |\ (C_1\ \cap\ C_2))}{\Pr(C_3\ |\ (C_1\ \cap\ C_2))}[/imath]
I think the pattern is [imath]\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 1} C_i)\ =\ \frac{\Pr(C_{n - 1}\ |\ (W\ \cap\ \bigcap\limits_{i = 1}^{n - 2} C_i))\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)}{\Pr(C_{n - 1}\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)}[/imath].
I have the intuition that this should be interpretable as a recurrence relation. As far as I can tell, [imath]\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 1} C_i)[/imath] is playing the role of [imath]a_n[/imath], and [imath]\Pr(W\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)[/imath] is playing the role of [imath]a_{n - 1}[/imath]. However, it is less clear what to do with the other factors. Should I leave them as probabilities, yielding [imath]a_n =\ \frac{\Pr(C_{n - 1}\ |\ (W\ \cap\ \bigcap\limits_{i = 1}^{n - 2} C_i))}{\Pr(C_{n - 1}\ |\ \bigcap\limits_{i = 1}^{n - 2} C_i)} a_{n - 1}[/imath], 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?