# Bayesian inference and curve fitting

I'm a silly scientist, and I've got a process I'm interested in. It gives me lots of time-series data. I've got an explicit, analytical model that is firmly rooted in reality, and basically boils down to fitting exponentials. Unfortunately, the model is under-determined with the experiments I can do; two parameters often end up highly correlated. As ever, it's the parameters I really care about.

At the moment, I'm using matlab's lsqcurvefit to fit my model to data. I estimate appropriate starting values from other means, and fit parts of my data to different models, then use those extracted values on other (later) data to get other parameters. It does a cracking job, and usually converges on sensible values pretty quickly.

I can construct appropriate a priori estimates for the likely distribution of all these parameters -- it's all just chemical kinetics, and I can go and measure some of these parameters in analogous environments in vitro. I'd really love to do something more intelligent than just using lsqcurvefit like a muppet. Am I right in thinking that there are Bayseian methods to tackle this nut? What should I do?

Is this a solved problem in bayseian inference? If you could possibly point me to an appropriate introduction, I'd really love to read it. I'm a physicist working in a biological situation, if that helps you to think of any texts.

• When getting started with Bayesian inference, you can't really go wrong starting from Gelman's Bayesian Data Analysis. It won't help from the chemistry side, but it will help with understanding the underlying approach to this kind of inferential problem. – Sycorax Dec 13 '13 at 22:35
• It sounds like Bayesian methods would be suitable (though it sounds complicated enough that you may need something like MCMC to do it), but Bayesian models are not, of themselves, going to solve the underlying problem of dependence in your parameters (though the prior information might help that issue). Bayesian methods may allow you to do something a bit less ad hoc, however, and to properly incorporate your prior knowledge. – Glen_b Dec 13 '13 at 23:50
• Thanks very much for your suggestions -- I'm checking out Gelman's book. I guess the main thing I need to learn is how to convert everything into a probabilistic framework. – Landak Dec 15 '13 at 22:00
• So, for future reference, I guess what I wanted to know was something like 'MCMC runs, probably with Gibbs sampling', and link to find a convenient implementation of the same. – Landak Dec 21 '13 at 23:14
• Landak: did you get something working? Was it really complicated and written up in a data analysis journal (and if it was it could be :) or simplish and reasonable to describe here? Either way, it'd be great if you answered your question ;) – drevicko Apr 25 '15 at 10:02