Prior probability for transient cooling time constant

I am trying to fit the Transient Cooling equation to data, but I need the prior probability for the time constant $ tau $.

The error function for my data is,

$$
epsilon = tau dot{T} + T – T_{env}
$$

Using max entropy loss function summing over my temperature data,
$$
sum_{T} log Prbig(tau ~big|~ Tbig) propto sum_T log Big[ Prbig(T ~big|~ taubig) Prbig(taubig) Big] = sum_T frac{lambda}{2} big( tau dot{T} + T – T_{env} big)^2 + log Prbig(taubig)
$$

I just don’t know the prior distribution $ Prbig(taubig) $ for the time constant $ tau $ and I’d rather not naively guess the distribution.

Is there a prior probability distribution for the time constant?