# Thread: Bayesian parameter estimation via MCMC?

1. ## Bayesian parameter estimation via MCMC?

Hi folks.

I have the following question.

I have a model M containing 20 adjustable parameters k = {k_j}.
I also have 40-50 measured temporal profiles e = {e_i} at my disposal.

I can use M to predict the experimental values after solving complex systems of differential equations.

Consequently, I get m(k) = {m_i(k)} which I can compare to e = {e_i}.

Now, I want to perform a Bayesian parameter estimation of the system.

I am going to define a (first) prior distribution for the parameters k: p_0(k)
Afterwards, I want to get the posterior probability distribution of k: f_p(k) = p(k|e) = L(e|k)*p_0(k)/p(e).
(Whereby p(e) represents, of course, a very complex multi-dimensional integral of "L(e|k)*p_0(k)".

Naturally, I cannot compute analytically the solution.
It also stands to reason that an approximate calculation of f_p(k) (and integration of "L(e|k)*p_0(k)") would be computationally intractable.

I read that Macrov-Chain-Monte-Carlo (MCMC) methods should be used for computing quantities of interest characterising the posterior (such as the points of highest probability density and high probability density regions, whose bounds can serve as error bars).
To be frank, I am a novice in that field.

Do you know any MCMC software freely available to academic researchers which could carry out all these operations, given a "black box" m(k) relying on solving differential equation systems?
If so, are you also aware of any beginner-friendly introduction into the concrete application of these techniques?

Kind regards.

2. ## Re: Bayesian parameter estimation via MCMC?

Hi,

I am afraid that regression based on differential equations and other statsistical techniques (such as MCMC sampling) are still very poorly connected.

Free software for obtaining posterior samples based on MCMC e.g. are OpenBUGS or Stan. Here, Stan often is a little bit more flexible/faster compared to BUGS.

But I think (if it is possible at all) you have to write most of the code especially for solving the ODEs/PDEs by hand...

Please let me know if you found a solution, I would be very interested!

Best

3. ## Re: Bayesian parameter estimation via MCMC?

Hi, after thinking a little bit about this, I belief that the simplest approach is to use bootstrap methods in this case to get uncertainty estimates for your ODE/PDE model parameters. You can easily combine bootstrap with your differential equation solver by prefixing a resampling of your data before solving the differential equations. You repeat this ~ 2000 times, and thus you obtain a population of resample values for each of your parameters. Based on this, you can calculate confindence intervals for your parameters (e.g. by using quantiles). In case you don't plan to use informative priors, this should lead to the same results compared to he Bayes approach.

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