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→Log-concave distributions: added independent hypothesis to the sum of log concave rv |
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*If a density is log-concave, so is its [[cumulative distribution function]] (CDF).
*If a multivariate density is log-concave, so is the [[marginal density]] over any subset of variables.
*The sum of two independent log-concave [[random variable]]s is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
*The product of two log-concave functions is log-concave. This means that [[joint distribution|joint]] densities formed by multiplying two probability densities (e.g. the [[normal-gamma distribution]], which always has a shape parameter >= 1) will be log-concave. This property is heavily used in general-purpose [[Gibbs sampling]] programs such as BUGS and JAGS, which are thereby able to use [[adaptive rejection sampling]] over a wide variety of [[conditional distribution]]s derived from the product of other distributions.
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