# Bayesian estimation via MCMC

Tags:
1. Dec 15, 2015

### witziger_Fuchs

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. Dec 15, 2015

### Staff: Mentor

I use R with the MCMCpack for my Bayesian estimation needs. It has always served me well, however I have never tried the kind of differential equation black box approach like you are describing. I have essentially only used it to do Bayesian linear regression were the residuals were assumed to be normally distributed and the Bayesian approach estimates the posterior of the linear model parameters and the residual variance.

I know that MCMCpack has built in routines for more complicated models, but I just haven't used them. You may need to try something more specialized, such as BUGS, but I have no experience with that.