How do I perform Bayesian Updating for a function of multiple parameters, each with its own distribution? I have a variable that is a recursive function involving other variables with known distributions (see problem below). 


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*Let $b(t+1) = b(t) + C \sqrt{b(t)}$ where I know $C \sim N(1.82, .0298)$ and the initial value of $b$ [$b_{initial} \sim N(.02,0.0036)$].


My observation for updating is a discrete value of $b$ for a certain '$t$' (say $b(1500) = 0.005$) but I need to find posterior distributions for $C$ and $b_{initial}$.
Any thoughts or resources that can be directed my way would be extremely helpful.
Thanks
 A: Probabilistically, the right way to do this is a posterior joint distribution for C and binitial--if you know b(t) exactly, then you'll get a line of sorts, where if binitial is .022 that implies that C was 1.81, but it binitial was instead .023 then that implies that C was 1.76. Each of those points has a probability associated with it from the prior, and the update just consists of renormalization from the entire 2D space to that line.
If you know b(t) inexactly, then you instead need to calculate the probability of that b(t) measurement for everywhere in the distribution, multiply those, and then renormalize. (That is, you're updating with a continuous likelihood, instead of the 0 or 1 likelihood implied by an exact measurement.)
For your particular function, it doesn't look like it'll be computationally easy to work with, although it probably is smooth (in the sense that the derivatives with respect to C and binitial are well-behaved). Seeing if you can come up with something where you can shift C by $\Delta$C and get the corresponding $\Delta$binitial shift (or the reverse) looks like it'll be very useful.
