# Bayesian parameter estimation via MCMC?

#### witziger_Fuchs

##### New member
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?