Bayesian Statistics

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\)

4.3 Conjugate pairs and the exponential family $\offsyl $

For continuous Bayesian models it is known that conjugate priors can only be found when the model family has a particular form. In particular, it must be from the exponential family of distributions. This term does not refer to the exponential distribution, but to a much wider class. We say that a real valued random variable $Y$ is from the exponential family of distributions with parameter $\theta $ if it has a p.d.f. in the form

\begin{equation} \label {eq:exponential_family_pdf} f_{Y}(y) = h(y)g(\theta )\exp \l (\theta T(y)\r ). \end{equation}

Here $h,g$ and $T$ are arbitrary functions, with the restriction $h\geq 0$. Many of the families of distributions that you are familiar with can be fitted into this mould, including the normal, exponential, gamma, chi-squared, and beta distributions.

Take a Bayesian model $(X,\Theta )$ with model family given by (4.3), where both $X$ and $\Theta $ take values in $\R $. That is,

\begin{equation} \label {eq:bayes_exp_family_one_datapoint} f_{M_\theta }(x) = h(x)g(\theta )\exp \l (\theta T(x)\r ) \end{equation}

for $x,\theta \in \R $. We’ll focus on the version of this model that takes $n$ independent items of data, in which case instead

\begin{equation} \label {eq:bayes_exp_family} f_{M_\theta }(x) = \l (\prod _{i=1}^n h(x_i)\r )g(\theta )^n\exp \l (\theta \sum _{i=1}^nT(x_i)\r ). \end{equation}

The prior distribution that provides a conjugate pair to this model is given by

\begin{equation} \label {eq:bayes_exp_family_prior} f_{\Theta }(\theta ) \propto g(\theta )^a\exp \l (b\theta \r ), \end{equation}

where $a>0$ and $b\in \R $ are parameters Note that this distribution is a specialized version of the form in (4.3), and that the function $g$ must be the same as in (4.5). Theorem 3.1.2 allows us to compute the posterior density

\begin{align} f_{\Theta |_{\{X=x\}}}(\theta ) &\propto f_{M_\theta }(x)f_{\Theta }(\theta ) \notag \\ &\propto g(\theta )^{n+a}\exp \l [\theta \l (b+\sum _{i=1}^n T(x_i)\r )\r ]. \label {eq:bayes_exp_family_posterior} \end{align} Therefore the Bayesian update in this case, from the parameters in (4.6) to (4.7) is that

\[(a,b)\mapsto \l (a+n,\,b+\sum _{i=1}^n T(x_i)\r ).\]

Example 4.3.1 Taking $h(x)=1$, $g(\theta )=\theta $ and $T(x)=x$ obtains the $\Exp (\theta )$ in (4.4). Making the same choice for $g$, along with $a=\alpha $ and $b=-\beta $, obtains that $\Gam (\alpha ,\beta )$ distribution in (4.6). Putting these choices for $g$ and $T$ into (4.7) and applying Lemma 1.2.5 gives that the posterior is $\Gam (\alpha +n,\beta +\sum _1^n x_i)$, as we already knew from Lemma 4.2.1.

Remark 4.3.2 There is also a version of this framework for multiple parameters, in which $\theta $ and $b$ are row vectors and $T$ is a column vector, and multiplication of these quantities is done via the dot product. We won’t write down the details of that case here. There is also a version that applies to discrete Bayesian models, but we won’t detail that either.

Bayesian Statistics

4.3 Conjugate pairs and the exponential family \(\offsyl \)