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I have a set of differential equations that describes a physical system - actually a ball being heated. I also have some data taken from real experiments on the system. I would like to try to fit these parameters to the data using Bayesian inference. Over the past few days I've been getting to grips with MCMC and I now have something that seems to work well.

The graph below shows both the both the data (in pink) and the fitted model with credible intervals.

model and data

Now I would like to repeat this experiment with a different physical sample in the test machine. The posterior distribution from the first test is my best guess at how the new sample will behave, but after three or four measurements I'll know a lot more about how the new sample behaves. This is where I am uncertain what to do. I think I should use the first posterior as my new prior, but I'm not sure how to do this. I think my options are:

  1. Fit an analytical distribution (or ensemble of distributions) to the posterior data
  2. Carry all of the data from both experiments through to the MCMC algorithm

Is this the correct approach? For the second option I'm worried that carrying all of the data from the first experiment will unduly skew the results for the second experiment - broadly speaking I know the physical sample from the second experiment will be different to the first, so I feel that I should be weighting this data more highly. I could do this by scaling the sum of squares error function to make errors from the second experiment more important than first, but I wondered if there was a more correct way of doing this?

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One thing to consider would be to re-formulate your model to be hierarchical with some dependence structure that allows you to borrow strength across experiments for parameter estimation. There are many ways to this, and how you create these dependence structures is problem specific. But to give you an idea, a simple and ubiquitous example of this kind of thinking is the eight-schools problem presented in the book Bayesian Data Analysis by Gelman et al.

A simpler solution you might consider is using a power prior. In a nutshell, a power prior uses a scalar parameter $a_0$ to weight the historical data relative to the likelihood of the current data. $a_0=0$ corresponds to no weight on the historical data, while $a_0=1$ corresponds to the prior for the new study being the posterior from the previous study. $0 < a_0 < 1$ of course corresponds to something in between. You can also put a prior on the $a_0$ parameter if you wish. The idea would be to down-weight the historical data enough that it didn't skew the results from your second experiment.

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