Friday, October 26, 2012

New Paper on General Information Metrics for Experiment Planning

As part of our ongoing project on "turning the scientific method into math", Marc Harper and I have written a paper on expectation potential information as the key measure of information yield from a proposed experiment. Take a look at the paper; we are eager for feedback (e.g. add a comment on this post). The basic idea is:
  • empirical information (\(I_e\)) measures prediction power on observables.
  • potential information (\(I_p\)) measures the maximum additional prediction power possible for a given set of observables, relative to the current model. In other words the theoretical increase in empirical information achievable by the best possible model. The key point is that \(I_p\) can be estimated without in any way searching model space.The value of any experiment is its ability to surprise us, i.e. to demonstrate that our current model is inadequate. Potential information provides a general measure of this, so the value of an experimental dataset is simply its potential information measure. For more details on this previous work, see here.
  • expectation potential information (\(E(I_p)\)) forecasts the expected information value of an experiment, under our current beliefs (uncertainty) about its likely outcomes. That is, adopting the view that our "current model" is always a mix of competing models, the \(E(I_p)\) for a proposed experiment measures its ability to resolve major uncertainties in that mixture.
Two things about this work have been fun:
  • we used an interesting "test problem", RoboMendel: a robot scientist tasked with proposing experiments to discover the laws of genetics. It's been fun working through how the basic \(E(I_p)\) metric addresses not only fine details of experiment planning (e.g. the value of including a specific control) but also the big questions of "what should we look at?"
  • Note that all these metrics are defined strictly in terms of prediction power on observable variables, contrary to the usual focus in statistical inference on our ability to infer hidden variables. Yet the \(E(I_p)\) metric comes full circle; you can prove that as the mixture probabilities converge to the true marginal probabilities of possible "outcomes", the expectation potential information metric converges \(E(I_p) \to I(X;\Omega)\), i.e. the classic information theory metric of how "informative" the observable X is of the true hidden state of the system \(\Omega\).

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