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.