Editing Conjugate prior (section)

=== When the likelihood function is a discrete distribution ===
{{More citations needed section|date=August 2020}}
{| class="wikitable"
! Likelihood <br> <math>p(x_i|\theta)</math>!! Model parameters <br> <math>\theta</math>!! Conjugate prior (and posterior) distribution <br> <math>p(\theta|\Theta), p(\theta|\mathbf{x},\Theta) = p(\theta|\Theta') </math>!! Prior hyperparameters <br><math>\Theta</math>!! Posterior hyperparameters<ref group=note name="posterior-hyperparameters"/> <br><math>\Theta'</math> !!Interpretation&nbsp;of&nbsp;hyperparameters!! Posterior&nbsp;predictive<ref group=note name=postpred/> <br><math>p(\tilde{x}|\mathbf{x}, \Theta) = p(\tilde{x}|\Theta')</math>
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| [[Bernoulli distribution|Bernoulli]] || ''p'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\in\mathbb{R}\!</math> || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + n - \sum_{i=1}^n x_i\!</math>
| <math>\alpha</math> successes, <math>\beta</math> failures<ref group=note name="beta-interp"/>
| <math>p(\tilde{x}=1) = \frac{\alpha'}{\alpha'+\beta'}</math><br>([[Bernoulli distribution|Bernoulli]])
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| [[binomial distribution|Binomial]]<br />with known number of trials, ''m''  || ''p'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\in\mathbb{R}\!</math>  || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + \sum_{i=1}^nN_i - \sum_{i=1}^n x_i\!</math>
| <math>\alpha</math> successes, <math>\beta</math> failures<ref group=note name="beta-interp"/>
| <math>\operatorname{BetaBin}(\tilde{x}|\alpha',\beta')</math><br />([[beta-binomial distribution|beta-binomial]])
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| [[negative binomial distribution|Negative binomial]]<br />with known failure number, ''r'' || ''p'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\in\mathbb{R}\!</math>  || <math>\alpha + rn ,\, \beta + \sum_{i=1}^n x_i\!</math>
| <math>\alpha</math> total successes, <math>\beta</math> failures<ref group=note name="beta-interp"/> (i.e., <math>\frac{\beta}{r}</math> experiments, assuming <math>r</math> stays fixed)
|<math>\operatorname{BetaNegBin}(\tilde{x}|\alpha',\beta')</math>
[[Beta negative binomial distribution|(beta-negative binomial)]]
|-
| rowspan=2 | [[Poisson distribution|Poisson]]
| rowspan=2 | ''λ'' (rate)
| rowspan=2 | [[Gamma distribution|Gamma]]
| <math>k,\, \theta\in\mathbb{R}\!</math> || <math>k + \sum_{i=1}^n x_i,\ \frac {\theta} {n \theta  + 1}\!</math>
| <math>k</math> total occurrences in <math>\frac{1}{\theta}</math> intervals
| <math>\operatorname{NB}\left(\tilde{x}\mid k', \frac{1}{\theta'+1}\right)</math><br />([[negative binomial distribution|negative binomial]])
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| <math>\alpha,\, \beta\!</math>  <ref group=note name="beta_rate"/> || <math>\alpha + \sum_{i=1}^n x_i ,\ \beta + n\!</math>
| <math>\alpha</math> total occurrences in <math>\beta</math> intervals
| <math>\operatorname{NB}\left(\tilde{x}\mid\alpha', \frac{\beta'}{1 + \beta'}\right)</math><br />([[negative binomial distribution|negative binomial]])
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| [[categorical distribution|Categorical]] || '''''p''''' (probability vector), ''k'' (number of categories; i.e., size of '''''p''''') || [[Dirichlet distribution|Dirichlet]] || <math>\boldsymbol\alpha\in\mathbb{R}^k\!</math> || <math>\boldsymbol\alpha + (c_1, \ldots, c_k),</math> where <math>c_i</math> is the number of observations in category ''i''
| <math>\alpha_i</math> occurrences of category <math>i</math><ref group=note name="beta-interp"/>
| <math>\begin{align}
    p(\tilde{x}=i) &= \frac{{\alpha_i}'}{\sum_i {\alpha_i}'} \\
                   &= \frac{\alpha_i + c_i}{\sum_i \alpha_i + n}
 \end{align}</math><br>([[categorical distribution|categorical]])
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| [[multinomial distribution|Multinomial]] || '''''p''''' (probability vector), ''k'' (number of categories; i.e., size of '''''p''''') || [[Dirichlet distribution|Dirichlet]] || <math>\boldsymbol\alpha\in\mathbb{R}^k\!</math> || <math>\boldsymbol\alpha + \sum_{i=1}^n\mathbf{x}_i\!</math>
| <math>\alpha_i</math> occurrences of category <math>i</math><ref group=note name="beta-interp"/>
| <math>\operatorname{DirMult}(\tilde{\mathbf{x}}\mid\boldsymbol\alpha')</math><br />([[Dirichlet-multinomial distribution|Dirichlet-multinomial]])
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| [[hypergeometric distribution|Hypergeometric]]<br />with known total population size, ''N'' || ''M'' (number of target members) || [[Beta-binomial distribution|Beta-binomial]]<ref name="Fink"/> || <math>n=N, \alpha,\, \beta\!</math> || <math>\alpha + \sum_{i=1}^n x_i,\, \beta + \sum_{i=1}^nN_i - \sum_{i=1}^n x_i\!</math>
| <math>\alpha</math> successes, <math>\beta</math> failures<ref group=note name="beta-interp"/>
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|-
| [[geometric distribution|Geometric]] || ''p<sub>0</sub>'' (probability) || [[Beta distribution|Beta]] || <math>\alpha,\, \beta\in\mathbb{R}\!</math> || <math>\alpha + n,\, \beta + \sum_{i=1}^n x_i\!</math>
| <math>\alpha</math> experiments, <math>\beta</math> total failures<ref group=note name="beta-interp"/>
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