Hi,
I recently was able to rewrite my ODE model and simplify it by combining several parameter into a few single parameter.
Since the new parameters don’t bare any direct biological meaning, it is slightly bit tricky to define reasonable priors.
Thus I am wondering: Is there a coherent way of integrating prior knowledge of the “original” parameter into the “new” parameters after re-parametrisation.
Thanks for all your comments already.
Depending on how the parameters are being combined, and how strict you need to be about the actual distribution of the new parameters, you could use the properties of the mean and variance of random variables get some ballpark values for the priors of the new parameters.
If that’s too loose, maybe that can be improved by using a convolution or product of distributions to arrive at a distribution that is formally related to those of the original model.
How exactly would that look like? I had a bit of a read and stumbled upon explenations like this and this.
So following that could I combine the PDFs as described like this:
If
tr = \mathcal{N}(mu1;sigma1)
R = \mathcal{N}(mu2;sigma2)
after re-parametrisation such as q2=tr*R
p1 = \mathcal{N}(mu3;sigma3)
where
mu3 = \frac{sigma1^{-2}*mu1 + sigma2^{-2}*mu2}{sigma1^{-2}+sigma2^{-2}}
sigma3 = \frac{sigma1^{2}*sigma2^{2}}{sigma1^{2}+sigma2^{2}}
If it works like that is it also possible to combine parameters for the following scenario(what would be the rule of combination for it):
g = \mathcal{N}(mu1;sigma1)
k = \mathcal{N}(mu2;sigma2)
p2 = g*2^{k+1}
I guess that’s the intuition behind it, but the product of PDFs you mention seems like it’s equivalent to how the priors for different parameters enter the expression for the posterior (except that when you are sampling you just compute the product of individual values, not the whole distribution). You may multiply the results for that purpose, but it otherwise it may not have a probabilistic interpretation and it’s generally not the same as computing the distribution of a new random variable defined as the sum or product of two (or more) other variables (e.g. the product of two normally-distributed variables is not gaussian).
Considering that you have at least one combination of parameters that is more complicated than a simple product, I’d say it’s probably easier to just work out their mean and variance, and then specify the priors for the new parameters as a gaussian with those values.
Thanks. Do you mean by working out their mean and variance drawing RV from the old parameters combining them according to the new parameter definition and calculate the mean and variance of the new RV and use that for a normal prior?
Yes, something like that. I guess there’s a hierarchy of possibilities here with different levels of accuracy and effort, from highest to lowest:
1. Analytically deriving the actual distribution of the combinations you need (Could be intractable and the resulting distributions are unlikely to be common distributions you could use in Stan as priors)
2. Working out only the mean and variance of the new distributions as the function of old ones from probability rules (this should be more feasible, or not, very likely much easier than 1 but it only gives you the mean and variance)
3. Simulating random draws from all the old distributions you have and combine the samples into the new distributions (this is probably the most straightforward, and will give you the actual shape of the new distributions should you care for that).
I guess I was suggesting option 2, but what you mention on your last comment I see as option 3. I think it’s completely valid, and probably the most straightforward if you are not into computing these things by hand and don’t care too much about being extremely precise about the priors, just want something reasonable to work with.