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Latent Poisson process and Laplace's rule

(Note: This is related to the post on Laplace’s rule).

For each interval length $s$ we have a sequence of binary variables $Y_{1}(s),Y_{2}(s),\ldots Y_{n/s}(s)$. We want to connect these variables, for each $s$, in a meaningful way and apply Laplace’s rule. One way to do so is by assuming a latent Poisson process.

Let $X_{t}$ be a Poisson process with parameter $\lambda$, $t\in[0,n]$. Then the number of observations in the time frame $[t,t+s]$ is distributed as $$ X_{t+s}-X_{t}\sim\text{Poisson}(\lambda s). $$ A natural way to make $X_{t+s}-X_{t}$ into a binary variable is to use $$ Y_{i}(s)=1[X_{t+s}-X_{t}=0]. $$ Then it follows that $$ P(Y_{i}(s)=1)=e^{-\lambda s}. $$ To apply Laplace’s rule on a fixed time scale $s_{0}$, such as days, we’ll have to make the prior over $\pi(s_{0})=e^{-\lambda s_{0}}$ uniform. This is equivalent to $\lambda s_{0}\sim\textrm{Exp}(1)$, or $\lambda\sim\textrm{Exp}(1/s_{0})$.

Now, if we want to look at another time scale $s_{1}$, the induced prior for the success propability $\pi(s_{1})=e^{-\lambda s_{1}}$ is $$ \pi(s_{1})=\pi(s_{0}){}^{\frac{s_{1}}{s_{0}}}\sim\text{Beta}(s_{0}/s_{1},1). $$