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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.

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  • $\begingroup$ 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? $\endgroup$
    – jcken
    Commented Jul 3, 2020 at 14:52
  • $\begingroup$ @jacken I estimated it through a linear regression $\endgroup$
    – MarG
    Commented Jul 3, 2020 at 14:55
  • $\begingroup$ 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) $\endgroup$
    – jcken
    Commented Jul 3, 2020 at 14:58
  • $\begingroup$ 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. $\endgroup$
    – MarG
    Commented Jul 3, 2020 at 15:03
  • $\begingroup$ 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 $\hat{\beta}<0$ $\endgroup$
    – jcken
    Commented Jul 3, 2020 at 15:19

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Per a 2005 article, Estimating the mean and variance from the median, range, and the size of a sample, to quote:

We also estimated the variance of an unknown sample using the median, low and high end of the range, and the sample size. Our estimate is performing as the best estimate in our simulations for very small samples (n ≤ 15). For moderately sized samples (15 <n ≤ 70), our simulations show that the formula range/4 is the best estimator for the standard deviation (variance). For large samples (n > 70), the formula range/6 gives the best estimator for the standard deviation (variance).

So, perhaps if you have (say, from the examination of the output from a single old model, and/or use available prior data to derive several contender models) opinions on the possible range of values, and therefrom, the computed midrange as an estimate of the median, combined with knowledge of the historical implied degrees of freedom, you may be able to produce a good estimate of variance (and sigma, all following the guidance above) for your new Bayesian Poisson Regression model.

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