Here mu is a set of measurements, the phi are sections of mu (one for each i in {1,2, ..., K}) thought of as parametrizations, and e is a real classification to be estimated. It is pretty easy to show that the average success rate of the estimator is the sum of the volumes of the Vi intersected with the inverse image of i under e( ). So when X has measure 1, the total error is:
I guess it makes sense to call this a best model estimate.
This also could be called "The Philosopher's Equation" because of Berkley's "esse est percipi" which means the essence of being is in being perceived. Here the perception (percipi) is the measurement mu, and the essence (esse) is the choice of ideal representative phi_sub_i. Meanwhile, e and e^ are (respectively) the truth and what we assume to be true about the reality.
ReplyDeleteI pointed out, in a comment to an earlier post, that this activity is actually a cycle that ends where it begins:
ReplyDeleteIn the present philosophical context, this means the "assumption" gets you to the point of attaching a coordinate frame. The "truth" is achieved when that coordinate frame has been attached successfully and new measurements are possible.