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Computing Muhurtham

April 6, 2010

Hmm, information geometry.  As you know, there has been recent work in the distributed learning community on finding coherent probability assignments.  Perhaps for similar problems, there is a way to use your differential equation based methods that operate on manifolds.  As Amari argues, information geometry is also useful for understanding the probabilistic structure of data collected in neuroscience and elsewhere.  Graphical models and variational inference are central concepts.

Rather than talking more about modern methods of inference, though, let me step back to inference problems faced by the ancients.  These mathematical problems were often driven by astronomical calendrics, e.g. computing muhurtham.  As noted on p. 4 of (Kim Plofker, Mathematics in India, Princeton: Princeton University Press, 2008):  “it is not really possible to understand the structure and context of mathematics in India without recognizing its close connections to astronomy.  Most authors of major Sanskrit mathematical works also wrote on astronomy, often in the same work.  Astronomical problems drove the development of many mathematical techniques and practices, from ancient times up through the early modern period.”  Moreover, as argued by Sorenson, “The earliest stimulus for the development of estimation was apparently provided by astronomical studies in which planet and comet motion was studied using telescopic measurement data.  The motion of these bodies can be completely characterized by six parameters, and the estimation problem that was considered was that of inferring the values of these parameters from the measurement data.”

Within the tradition of Newton, Gauss fixed a model of astronomical motion and determined methods to estimate the parameters.  This approach is very different from pure machine learning, where no model with a specified parametrization is provided a priori.  The model-based approach raises the question, however, of how the model of astronomical motion is arrived at in the first place and whether it can be trusted.  Indeed, model-based approaches were often not trusted by Indic thinkers of old, as noted in these two papers on the philosophy of computational positivism.  So let me ask you, how does one learn scientific laws?  Moreover, do you think there is value in connecting concepts from the philosophy of science like falsifiability or positivism with formal notions from learning theory like VC dimension?

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