most recent 30 from http://mathoverflow.net 2013-05-20T21:19:12Z http://mathoverflow.net/feeds/user/2826 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/9757/measure-of-quality-of-curve-fit/10018#10018 Simon Barthelmé 2009-12-29T10:47:16Z 2009-12-29T10:47:16Z

<p>If I understand correctly, you would like to show that the right-hand dataset better defines the curve? There are many ways you could go about this, here's two:</p> <p>A) You show that one dataset leaves less uncertainty about the parameters of the curve than the other. In a classical setting you'd fit the parameters using some kind of regression, and compare the error bars on the fitted parameters. In a Bayesian setting you could compare the two posterior distributions, maybe showing that one has lower posterior entropy than the other. </p> <p>B) You show that one dataset leads to better predictive ability. Use cross-validation to compute an estimate of the generalisation error in each dataset, and compare that. </p>

http://mathoverflow.net/questions/9854/uniformly-sampling-from-convex-polytopes/9871#9871 Simon Barthelmé 2009-12-27T10:46:01Z 2009-12-27T10:46:01Z

<p>Rejection sampling will definitely work. Take a hypercube that you know contains the polytope, sample from the hypercube, and accept only those samples that belong in the polytope. However if the relative volume of the polytope is small you'll end up rejecting most samples, and the method might get painfully slow. Depending on your needs you might want to find, e.g., the smallest enclosing ball first, so that you can draw your uniform samples from that instead of the hypercube.<br /> See Boyd &amp; Vanderberghe's book on Convex Optimisation (it's online) for finding smallest enclosing sets. </p>

http://mathoverflow.net/questions/9854/uniformly-sampling-from-convex-polytopes/9871#9871 Simon Barthelmé 2009-12-29T10:37:30Z 2009-12-29T10:37:30Z

John: if you want an intrinsic mechanism, you can use MCMC techniques to sample from the density asymptotically. For example, it would be be fairly easy to implement Gibbs sampling, but it might take a long time for the chain to converge. In general the problem you want to solve is very hard, computationally speaking. To find an efficient algorithm, you are going to make some assumptions about the structure of your polytope, or to use some preprocessing to find out more about it. That's what Steve is suggesting.