# Setting variance of an informative prior

I am creating a Bayesian Poisson Regression model and I have access to a dataset and a previous corresponding model. I want to use the previous model to create a prior that I will combine with the data.

To incorporate the previous model as a prior in the Bayesian model I have calculated a point estimate of a parameter $$\hat{\beta}$$. Now, I am planning on using this point estimate as the expectation in a gaussian prior, meaning that I get the following prior: $$N(\hat{\beta}, \sigma)$$.

My question is, does anyone know how to set $$\sigma$$? I know that when a Jeffreys prior is used you could for instance say that the prior should be worth 2 observations. Is there a corresponding method where I could say how many observations my prior is worth in this context?

# Note

Note that I am using point estimations since the previous model is very different from the likelihood and the model cannot be incorporated in a straightforward way.

• How did you estimate $\beta$ in the previous model? Surely it came with a measure of uncertainty attached, be it a posterior variance or a standard error? Commented Jul 3, 2020 at 14:52
• @jacken I estimated it through a linear regression
– MarG
Commented Jul 3, 2020 at 14:55
• so you should have a standard error attached to $\hat{\beta}$ - you could use that as $\sigma$. Howeber, are you using the same data in both models? You shouldn't be basing the prior on the data you're going to use in the analysis ('external' data is absolutely fine though) Commented Jul 3, 2020 at 14:58
• I could try doing that I guess. But, the optimal to me would be to somehow say that my prior is worth 50 % of the observations and the likelihood 50 %. So that the weighting becomes equal between them. @jcken I am not really using any data to estimate $\hat{\beta}$. I am more trying to emulate the previous model and I do not have access to the data that was used to create the previous model.
– MarG
Commented Jul 3, 2020 at 15:03
• Another idea is to set $\sigma$ so that a priori $\beta$ has the same sign as $\hat{\beta}$ with high probability: set $\hat{\beta} - 2*\sigma = 0$ and solve for $\sigma$. Change the $-$ to$+$ if $\hat{\beta}<0$ Commented Jul 3, 2020 at 15:19