# Is Bayesian nonlinear regression using conjugate priors possible?

I have data for many time series. Each time series can be modeled by a particular nonlinear function of several parameters and, though I want point estimates of the model parameters for each of the many time series, I am more interested in the uncertainty associated with the parameter estimates (i.e., in the strength of the data). I have good prior information that I'd ideally like to include. I have tried MCMC, but am not confident that reasonable quality posterior samples can be obtained for the many time series (I can tweak the parameters of the MCMC algorithm for specific time series, but I'm not convinced my settings generalize well for other time series); additionally, CPU time is an issue.

Rather than use MCMC, I'm wondering about using conjugate priors (a multivariate normal is a good model of my log-transformed parameters). However, I'm having trouble finding examples of Bayesian nonlinear regression with conjugate priors. Is it possible?

• Depends on the sampling distribution (likelihood). A table showing the conjugate prior of various likelihood models is shown on wikipedia: en.wikipedia.org/wiki/Conjugate_prior
– Nick
Apr 2 '12 at 16:12

No, it's not possible. Likelihoods that admit conjugate distributions correspond to data distributions that are members of some exponential family. Having a non-linear function of the parameters in the log-likelihood makes it impossible for the data distribution to belong to an exponential family.

If you specify the model(s?) and data in a bit more detail I might be able to give you advice on the use of a good hands-off adaptive MCMC routine. I have one in mind -- differential evolution Monte Carlo -- but I'm reluctant to claim that it'll address your problem without knowing more.

• Thanks, Cyan. The model is similar to Eqn 4 in this paper, and each time series is similar to that in Figure 1. Apr 2 '12 at 20:00
• What programming language/stats package are you working in?
– Cyan
Apr 3 '12 at 15:46
• I'm using R. (Thanks for your continued interest, BTW.) Apr 4 '12 at 9:40
• @Chris: Let me see if I've got this right. You have time series data for $C_p(t)$ and $C_t(t)$ and the object of the exercise is to estimate the redistribution rate constant $\frac{K_{trans}}{v_e}$ using that funky pharmacokinetic convolution integral (Eq. 4) as the systematic part of the model. If I've got this correct, do you have a model in mind for the error in the time series? Some sensible numbers for simulating data with which to test DE-MC would be helpful too.
– Cyan
Apr 8 '12 at 4:44
• To clarify, my model is not yet published (I can't state it here, I'm afraid), but it is similar to that equation: similar number of parameters; involves an integral; uses an "input function" ($C_t(t)$, here); etc. I'm modeling errors as a weighted sum of normals where one accounts for measurement error and the other for outliers (around 15% of the measurements in each time series have large errors due to things like patient motion, image registration error, and gross but transient changes in physiology). I've managed to port Corey Yanofsky's DE-MC code to R, so will give it a try. Apr 10 '12 at 11:48