Splitting the estimation procedure in two steps

Hi everyone,

So I have the following issue. I want to fit a fairly complex ODE-based model to 5 different datasets A,B,C,D and E. Some parameters \theta_1 are linked to datasets A,…,D while some other parameters \theta_2 are linked to E. I found that trying to estimate everything together is veeery slow, even on a cluster using MPI. So I figured that I could split the procedure in two steps, first inferring \theta_1 from A,…,D and in a second step inferring \theta_2 from E using the posterior estimates of \theta_1. Something like:

\Pr(\theta_{1},\theta_{2}| {A},{B},{C},{D},{E}) = \Pr(\theta_{1}|{A},{B},{C},{D}) \Pr(\theta_{2}|\theta_1,{E})

I feel that this is not ideal, but it’s the only solution I found. Is there any fundamental problem with this approach?

It’s highly unlikely (according to my experience anyway) that the parameters are independent of each other. You could iterate the procedure, EM style, but I imagine then you’re probably back with the ‘veeery slow’ thing.

You can avoid a lot of the problems with this approach if you’re willing to run the second stage separately on each sample from the first posterior of theta.

Thanks for the quick answers!

@Charles_Driver Assuming independence is not particularly problematic in this particular case, as different mechanisms are involved. May I ask what you mean by “EM style”?

@sakrejda Do you mean running the second stage for each posterior sample of the first and then merging everything together?

Yes. The downfall of multistage procedures is that they ignore uncertainty from stage 1, this just happens to be a strategy for including uncertainty that I like and that is easy to parallelize.

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Wouldn’t it be sufficient to approximate the posteriors from stage 1 with a normal (or anything appropriate), and use the obtained mean and variance in stage 2?

Only if there’s no covariance and you already know what the appropriate distribution is

Thank for the advice, I’ll try to find a solution. Have a nice day!

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