Cross Validated Asked by Learning stats by example on October 15, 2020

I’m trying to understand how I might model count data where there’s diminishing marginal utility and a stochastic process.

So, let’s say we’re modeling the number of “useful intelligence tips” given to us by Russian agents dependent on how many shots of vodka we give them. I want to show that past a certain point, the number of useful tips starts to decay.

The way I’d think of this is by making the rate parameter *lambda* dynamic by time *t*.

So at time `t = 1`

we might say we get `rpois(1,10)`

pieces of intel *n*. Poisson distributed *n* pieces of intel, with mean 10.

But let’s say we start to see decay so that by time `t = 100`

only 2 pieces of intel are returned. We can model time `t = 100`

with `rpois(1,100)`

.

My thinking on this is we build a vector so that lambda is dynamic and varies at each point *t*:

`lambda_t[t] = t * rexp(1, rate = z)`

.

We then just need to solve for `z`

.

`for (1:t) Y[t] ~ rpois(1, lambda_t[t]);`

Does this approach make sense? I’ve written in pseudo-code using a bit of stan syntax and R syntax as I imagine I’ll mix the two. But I really am curious how count data is generally modeled when the counts vary over *t*.

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