Bayes-Updating with Uncertain Prior Precision

Let's start with a standard model with Gaussian signals (known
precision s, unknown mean mu) and a Gaussian prior distribution on
mu (known precision t, and mean m). Fine, done. Now let's forget the
sample size that went into creating the current posterior m. Instead,
let's assume that we had a geometric size sample, with sample of size
n with chance a^n(1-a), for some 0<a<1. Is there then an implied
conjugate family to work within, perhaps with an unknown prior
precision on a normal-like variable? DeGroot fails me.

Many thanks, Lones

 

To specify the posterior, one needs to specify the current
mean AND the current variance. It makes no difference how
they got those values.