(*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
`$$ \pi(s_{1})=\pi(s_{0}){}^{\frac{s_{1}}{s_{0}}}\sim\text{Beta}(s_{0}/s_{1},1). $$`