Toy Problem: 1D Gaussian Width

[drphilmarshall]

Things to do:

• [x] Write down problem, ABC solution, and program of testing in the markdown doc
• [ ] Implement in the simplest possible ipython notebook, whose nbviewer visualistion is linked from the markdown doc
• [x] Or, do the equivalent in some other language. Like R, for example.
• [ ] Explore how to carry out ABC to give accurate inference of the Gaussian width.

[drphilmarshall]

Alright! Looks like @abmantz did this little exercise, but in R instead. In this simplest possible case, the population variance is a sufficient statistic, so ABC can work exactly - and indeed, Adam gets a nice posterior (from the histogram of accepted prior samples). Could be interesting to see if other summary statistics (something KS-related?) give similar results. Otherwise: suggestions for the next example problem?

[abmantz]

I was thinking the next simple problem would be joint constraints on the mean and width of a Gaussian. That way, we would have a slightly meatier case (still with an exact solution) to test methods other than rejection on.

But already it's pretty clear that the hurdle to using ABC in practice will be finding the best summary statistic, and figuring out whether the results can be trusted if it is not sufficient.

[abmantz]

I went ahead and implemented the 2-parameter fit to a 1D Gaussian here. This officially exceeds my procrastination quotient for the month.