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Hi I'm working with particle filters at the moment however my maths isnt so strong i was wondering given the $P(X_n|Y_0,....Y_{n-1})$ and $P(Y_n|X_n)$ how do you obtain $P(Y_n|Y_0,...,Y_{n-1})$? i.e. the equations that links the two?

Thanks

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    $\begingroup$ What are your thoughts? What have you tried? $\endgroup$
    – Peter
    Apr 8, 2013 at 15:49
  • $\begingroup$ Hi @Peter honestly my intuition for probability is terrible, I've just spent a good 10 minutes tying to come up with an answer that might not make me look like a complete idiot, but I honestly have no idea, possibly it might be a multiple of the two :S $\endgroup$
    – Jono Brogan
    Apr 8, 2013 at 16:03
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    $\begingroup$ Think of all the set $Y_0 ... Y_{n-1}$ as a single variable $A$. Then, you get the equivalent problem: given $P(C| A)$ and $P(B |C)$ obtain $P(B | A)$ $\endgroup$
    – leonbloy
    Apr 8, 2013 at 16:44

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Hint:

$P(B) = \sum_C P (B C) = \sum_C P(B | C) P(C)$

Hence (conditioning everything over A),

$P(B | A) = \sum_C P(B | C A ) P(C |A)$

You can't simplify this further, but in some cases (I'd bet in yours), depending on extra properties of the variables, you can assume that $P(B|C A) = P(B | C)$

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