Suppose we know the following about function $f(x)$: (i) There exists $y$ so that $f(x)=\beta$ for all $x\geq y$ and (ii) $f(x)=\alpha+b x$.
We have a prior distribution on $y$ and on $\beta$.
How does Bayesian updating for one period or several periods look like? Or do you have a good reference for sth. like that?
Thank you for any ideas
I'm assuming here that $f$ is a function from the reals to the reals.
There is some number, $y$, with the property that for $x \ge y$, we have $f(x) = \beta$. Letting $x_1 = y + 1$ and $x_2 = y + 2$, we get $$ f(x_1) = \beta \\ f(x_2) = \beta. $$
We are also told there are numbers $a$ and $b$ with $f(x) = a + bx$. Let's look at $f(x_1)$ and $f(x_2)$ given this information:
\begin{align} \beta &= f(x_1) = a + b x_1 = a + b(y+1) \\ \beta &= f(x_2) = a + b x_2 = a + b(y+2) \end{align} subtracting, we get \begin{align} \beta - \beta &= f(x_2) - f(x_1) \\ & = a + b (y+2) - (a + b (y+1))\\ &= b(y+2) - b(y+1) \\ &= b \end{align} whence we see that $b = 0$, from which it's easy to derive that $a = \beta$.
These are facts from algebra, and are completely independent of any assumptions about the distribution on $y$ or $\beta$.
As I said, I still think you're not asking what you meant to ask.
By the way, until you define an "event", the whole notion of Bayesian updating doesn't make a lot of sense. Is the event "we observe $f(c)$ for some randomly chosen $c$?" If so, with respect to what distribution is $c$ drawn? Or is an event something else?
Sigh.