Editing Conjugate prior (section)

== Notes ==
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<ref group=note name="beta-interp">The exact interpretation of the parameters of a [[beta distribution]] in terms of number of successes and failures depends on what function is used to extract a point estimate from the distribution.  The mean of a beta distribution is <math>\frac{\alpha}{\alpha + \beta},</math> which corresponds to <math>\alpha</math> successes and <math>\beta</math> failures, while the mode is <math>\frac{\alpha - 1}{\alpha + \beta - 2},</math> which corresponds to <math>\alpha - 1</math> successes and <math>\beta - 1</math> failures.  Bayesians generally prefer to use the posterior mean rather than the posterior mode as a point estimate, justified by a quadratic loss function, and the use of <math>\alpha</math> and <math>\beta</math> is more convenient mathematically, while the use of <math>\alpha - 1</math> and <math>\beta - 1</math> has the advantage that a uniform <math>{\rm Beta}(1,1)</math> prior corresponds to 0 successes and 0 failures. The same issues apply to the [[Dirichlet distribution]].</ref>

.<ref group=note name="posterior-hyperparameters">Denoted by the same symbols as the prior hyperparameters with primes added ('). For instance <math>\alpha</math> is denoted <math>\alpha'</math></ref>

<ref group=note name=postpred>This is the [[posterior predictive distribution]] of a new data point <math>\tilde{x}</math> given the observed data points, with the parameters [[marginal distribution|marginalized out]].  Variables with primes indicate the posterior values of the parameters.</ref>

<ref group=note name=ppredNt>This is the [[posterior predictive distribution]] of a new data point <math>\tilde{x}</math> given the observed data points, with the parameters [[marginal distribution|marginalized out]].  Variables with primes indicate the posterior values of the parameters. <math>\mathcal{N}</math> and <math>t_n</math> refer to the [[normal distribution]] and [[Student's t-distribution]], respectively, or to the [[multivariate normal distribution]] and [[multivariate t-distribution]] in the multivariate cases.</ref>

<ref group=note name="beta_rate">''β'' is rate or inverse scale. In parameterization of [[gamma distribution]],''θ'' = 1/''β'' and ''k'' = ''α''.</ref>

<ref group=note name="beta_scale">In terms of the [[inverse gamma distribution|inverse gamma]], <math>\beta</math> is a [[scale parameter]]</ref>

<ref group=note name=CG><math>\operatorname{CG}()</math> is a [[compound gamma distribution]]; <math>\operatorname{\beta'}()</math> here is a [[generalized beta prime distribution]].</ref>
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