Is there a word that means the 'inverse of variance'? That is, if $X$ has high variance, then $X$ has low $\dots$? Not interested in a near antonym (like 'agreement' or 'similarity') but specifically meaning $1/\sigma^2$?
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$1/\sigma^2$ is called precision. You can find it often mentioned in Bayesian software manuals for BUGS and JAGS, where it is used as a parameter for normal distribution instead of variance. It became popular because gamma can be used as a conjugate prior for precision in normal distribution as noticed by Kruschke (2014) and @Scortchi.
Kruschke, J. (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan. Academic Press, p. 454.
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5$\begingroup$ (+1) I don't recall seeing it outside of this context, where it's convenient to be able to say things like "add the precisions of the prior & the data to get the precision of the posterior", & to use the familiar gamma distribution as a conjugate prior for precision. $\endgroup$ Nov 25, 2015 at 11:45
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5$\begingroup$ Also common in multivariate settings, where the precision becomes the inverse of the covariance matrix. (And again, is useful as a parameter for a normal distribution when you need a conjugate prior). $\endgroup$– PeterNov 25, 2015 at 21:02
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4$\begingroup$ Yes, precision matrix is very helpful when dealing with multivariate Gaussians (as @Peter said), e.g. the formulas for conditional distributions are simpler in terms of precision matrices. Bishop spends many pages describing how this works in the Chapter 3 of his Pattern Recognition and Machine Learning, and this then reappears many times throughout the book. $\endgroup$– amoebaNov 25, 2015 at 21:17
[bayesian]tag since, as you can see from my answer and comments, the answer is closely related to Bayesian statistics and it will be easier to find tagged like this. $\endgroup$