Inverse Gaussian distribution

1 Conjugate prior

Banerjee and Bhattacharyya (1979) present a reasonably nice conjugate prior, albeit with an alternative parameterization of the distribution.

Write the IG pdf as

$$\begin{array}{rl}& f(x\mid \psi ,\lambda )={\left(\frac{\lambda }{2\pi }\right)}^{1/2}{x}^{-3/2}\\ & \exp \{-\frac{\lambda x}{2}{(\psi -\frac{1}{x})}^2\},x>0,\psi >0,\lambda >0\end{array}$$ the likelihood of a random sample $\mathbf{x}=(x_1,\cdots ,x_n)$ from $\mathrm{IG}(\psi ,\lambda )$, is $$l(\psi ,\lambda \mid \mathbf{x})\propto \exp \{-\frac{nu}{2}[1+\frac{\overline{x}}{u}{(\psi -\frac{1}{\overline{x}})}^2]\lambda \}{\lambda }^{n/2},$$ where $$\overline{x}=\sum x_i/n\text{ and }{\overline{x}}_r=\frac{1}{n}\sum \left(1/x_i\right)$$ are respectively the sample mean of the observations and that of their reciprocals, and $u={\overline{x}}_r-1/\overline{x}$.

Their major result is as follows

[…]a bivariate natural conjugate family for $(\psi ,\lambda )$ can be taken as $$\begin{array}{rl}& p_c(\psi ,\lambda )=K_1\exp \{-\frac{r^{\prime }{\alpha }^{\prime }}{2}[1+\frac{{\beta }^{\prime }}{{\alpha }^{\prime }}\\ & \cdot {(\psi -\frac{1}{{\beta }^{\prime }})}^2]\lambda \}{\lambda }^{r^{\prime \prime 2-1}},\psi >0,\lambda >0\end{array}$$ where $r^{\prime }>1,{\alpha }^{\prime }>0,{\beta }^{\prime }>0$ are parameters and the constant $K_1$ is given by $$\begin{array}{c}{K}_1=\frac{{\left(\frac{{\beta }^{\prime }}{{\alpha }^{\prime }}\right)}^{1/2}{\left(\frac{r^{\prime }{\alpha }^{\prime }}{2}\right)}^{r^{\prime \prime 2}}}{H_{{\nu }^{\prime }\left({\xi }^{\prime }\right)B}^{{\nu }^{\prime }},\frac{1}{2})\mathrm{\Gamma }\left(\frac{r^{\prime }}{2}\right)},\\ {\nu }^{\prime }=r^{\prime }-1,{\xi }^{\prime }={\left(\frac{{\nu }^{\prime }}{{\alpha }^{\prime }{\beta }^{\prime }}\right)}^{1/2}.\end{array}$$ […] $$\begin{array}{rl}nu+r^{\prime }{\alpha }^{\prime }+n\overline{x}{(\psi -\frac{1}{\overline{x}})}^2& +r^{\prime }{\beta }^{\prime }{(\psi -\frac{1}{{\beta }^{\prime }})}^2\\ & =r^{\prime \prime }{\alpha }^{\prime \prime }+r^{\prime \prime }{\beta }^{\prime \prime }{(\psi -\frac{1}{{\beta }^{\prime \prime }})}^2.\end{array}$$

Hence the joint posterior pdf of $\psi$ and $\lambda$ can be reduced to the form $$\begin{array}{rl}& p_c(\psi ,\lambda \mid \mathbf{x})\propto \exp \{-\frac{r^{\prime \prime }{\alpha }^{\prime \prime }}{2}[1+\frac{{\beta }^{\prime \prime }}{{\alpha }^{\prime \prime }}\\ & \cdot {(\psi -\frac{1}{{\beta }^{\prime \prime }})}^2]\lambda \}{\lambda }^{\prime \prime \prime 2-1}\end{array}$$ […] the marginal posterior distribution of $\lambda$ is the modified gamma $G^{\ast }(r^{\prime \prime }{\alpha }^{\prime \prime }/2,{\nu }^{\prime \prime }/2,r^{\prime \prime }/{\beta }^{\prime \prime })$, and the marginal posterior pdf of $\psi$ is the truncated $t$ distribution $t_d(1/{\beta }^{\prime \prime },q^{\prime \prime },{\nu }^{\prime \prime })$ with ${q^{\prime \prime }=\left[{\alpha }^{\prime \prime }/{\nu }^{\prime \prime }{\beta }^{\prime \prime }\right)]}^{1/2}$.

The modified gamma is derived from this guy:

$\begin{array}{rl}& p(\lambda \mid x)=\frac{(nu/2{)}^{\nu /2}}{\mathrm{\Gamma }(\nu /2)}\frac{\mathrm{\Phi }((n\lambda /\overline{x}{)}^{1/2})}{H_{\mathcal{U}}(\xi )}\\ & \cdot \exp (-\frac{nu}{2}\lambda ){\lambda }^{\nu /2-1},\lambda >0\end{array}$

This looks … somewhat tedious, but basically feasible I suppose.

2 References