Revision as of 01:57, 21 June 2014 edit AEdwards92 (talk \| contribs) 1 edit m The partition function has no application (that I know of) in information science, but definitely has application in information theory. I think this was a simple misnomer. I also un-linked the previous "first" occurrence of the hyperlink. Tag: gettingstarted edit ← Previous edit		Revision as of 12:21, 2 November 2014 edit undo Faizan (talk \| contribs) Extended confirmed users 42,227 edits m clean up, typo(s) fixed: Equiping → Equipping using AWB Next edit →
Line 121: :<math>g_{ij}(\beta) = \frac{\partial^2}{\partial \beta^i\partial \beta^j} \left(-\log Z(\beta)\right) = \langle \left(H_i-\langle H_i\rangle\right)\left( H_j-\langle H_j\rangle\right)\rangle</math> This matrix is positive semi-definite, and may be interpreted as a [[metric tensor]], specifically, a [[Riemannian metric]]. ~~Equiping~~Equipping the space of lagrange multipliers with a metric in this way turns it into a [[Riemannian manifold]].<ref>Gavin E. Crooks, "Measuring thermodynamic length" (2007), [http://arxiv.org/abs/0706.0559 ArXiv 0706.0559]</ref> The study of such manifolds is referred to as [[information geometry]]; the metric above is the [[Fisher information metric]]. Here, <math>\beta</math> serves as a coordinate on the manifold. It is interesting to compare the above definition to the simpler [[Fisher information]], from which it is inspired. That the above defines the Fisher information metric can be readily seen by explicitly substituting for the expectation value:

Partition function (mathematics): Difference between revisions