proof for posterior predictive of normal-gamma conjugacy

Question

Giving the following equations $$ \mu_n = \frac{\kappa_0 \mu_0 + n \overline{x}}{\kappa_0 + n}, \\ \kappa_n = \kappa_0 + n, \\ \alpha_n = \alpha_0 + n/2, \\ \beta_n = \beta_0 + \frac{1}{2} \sum\limits_{i=1}^n (x_i - \overline{x})^2 + \frac{\kappa_0 n (\overline{x} - \mu_0)^2}{2(\kappa_0 + n)}, $$ where $\overline{x} = \frac{\sum_{i=1}^n x_i }{n}$, i.e., the mean of data. Actually, the above equations are the parameters of the posterior of normal normal-gamma conjugacy (see Equation (85-89) in this paper for details).

Here, I want to prove: $$ \mu_{n+m} = \frac{\kappa_n \mu_n + m \overline{x}}{\kappa_n + m}, \\ \kappa_{n+m} = \kappa_n + m, \\ \alpha_{n+m} = \alpha_n + m/2, \\ \beta_{n+m} = \beta_n + \frac{1}{2} \sum\limits_{i=n+1}^{n+m} (x_i - \overline{x})^2 + \frac{\kappa_n m (\overline{x} - \mu_n)^2}{2(\kappa_n + m)}, $$ where $\overline{x} = \frac{\sum_{i=n+1}^{n+m} \quad x_i}{m}$, i.e., the mean of new observations. The above equations are appeared in Equation (101-104) of this paper, in which the $m$ is set as 1.

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The following effort I have tried $$ \kappa_{n+m} = \kappa_0 + n + m = \kappa_{n} + m \\ \alpha_{n+m} = \alpha_0 + \frac{n+m}{2} = \alpha_n + \frac{m}{2} \\ \mu_{n+m} = \frac{\kappa_0 \mu_0 + \sum_{i=1}^{n+m} x_i }{\kappa_0 + n+ m} = \frac{\kappa_0 \mu_0 + \sum_{i=1}^n x_i + \sum_{i=n+1}^{n+m} x_i }{\kappa_n + m} = \frac{(\kappa_0 + n)\frac{\kappa_0 \mu_0 + \sum_{i=1}^n x_i }{\kappa_0 + n} + \sum_{i=n+1}^{n+m} x_i }{\kappa_n + m} = \frac{\kappa_n \mu_n + m\overline{x}}{\kappa_n + m} $$ where $\overline{x} = \frac{\sum_{i=n+1}^{n+m} \quad x_i}{m}$. But I fail to prove the last equation: $$ \beta_{n+m} = \beta_n + \frac{1}{2} \sum\limits_{i=n+1}^{n+m} (x_i - \overline{x})^2 + \frac{\kappa_n m (\overline{x} - \mu_n)^2}{2(\kappa_n + m)}. $$ So, could anybody give me some suggestions?

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Thanks for Xi'an's help. The complete proof of the posterior predictive of normal-gamma conjugacy is now available in this note (see page 24: Proof of Normal Normal-Gamma Conjugacy)

Answer 1

A direct and elegant way to show these identities is to exploit that the posterior distribution $\pi_{n+m}(\theta|\mathfrak D_{n+m})$ associated with the prior $\pi_0$ and the $n+m$ observations $(x_1,\ldots,x_{n+m})$ is equal to the posterior distribution associated with the "prior" $\pi_n(\theta|\mathfrak D_{n})$ and the $m$ observations $(x_{n+1},\ldots,x_{n+m})$. When the prior is conjugate, e.g., Normal-Gamma, the intermediary posterior $\pi_n(\theta|\mathfrak D_{n})$ is also conjugate and the update of the hyperparameters is immediate.

A more pedestrian and dreary approach is to expand the expression of $\beta_{n+m}$. For the last update equation, let us introduce the notations $\overline{x}_{1:n+m}$, $\overline{x}_{1:n}$ and $\overline{x}_{(n+1):(n+m)}$ to distinguish between the three averages. Then \begin{align} \beta_{n+m} &= \beta_0 + \frac{1}{2} \sum\limits_{i=1}^{n+m} (x_i - \overline{x}_{1:n+m})^2 + \frac{\kappa_0 (n+m) (\overline{x}_{1:n+m} - \mu_0)^2}{2(\kappa_0 + n+m)}\\ &= \beta_0 + \frac{1}{2} \sum\limits_{i=1}^{n} (x_i - \overline{x}_{1:n}+\overline{x}_{1:n}-\overline{x}_{1:n+m})^2\\ &\quad + \frac{1}{2} \sum\limits_{i=n+1}^{n+m} (x_i - \overline{x}_{(n+1):(n+m)}+ \overline{x}_{(n+1):(n+m)}- \overline{x}_{1:n+m})^2\\ &\quad + \frac{\kappa_0 (n\overline{x}_{1:n}+m\overline{x}_{(n+1):(n+m)} - (n+m)\mu_0)^2}{2(\kappa_0 + n+m)(n+m)}\\ &= \beta_0 + \frac{1}{2} \sum\limits_{i=1}^{n} (x_i - \overline{x}_{1:n})^2+\frac{1}{2}n(\overline{x}_{1:n}-\overline{x}_{1:n+m})^2\\ &\quad + \frac{1}{2} \sum\limits_{i=n+1}^{n+m} (x_i - \overline{x}_{(n+1):(n+m)})^2+ \frac{1}{2}m(\overline{x}_{(n+1):(n+m)}- \overline{x}_{1:n+m})^2\\ &\quad + \frac{\kappa_0 (n\overline{x}_{1:n} - n\mu_0)^2}{2(\kappa_0 + n+m)(n+m)}+ \frac{\kappa_0 (m\overline{x}_{(n+1):(n+m)} - m\mu_0)^2}{2(\kappa_0 + n+m)(n+m)}\\ &\quad + \frac{2\kappa_0 (n\overline{x}_{1:n}-n\mu_0)(m\overline{x}_{(n+1):(n+m)} - m\mu_0)}{2(\kappa_0 + n+m)(n+m)}\\ &= \beta_n + \frac{1}{2} \sum\limits_{i=n+1}^{n+m} (x_i - \overline{x}_{(n+1):(n+m)})^2 + R_{n,m}\end{align} with \begin{align} R_{n,m}&=\frac{1}{2}n(\overline{x}_{1:n}-\overline{x}_{1:n+m})^2+ \frac{1}{2}m(\overline{x}_{(n+1):(n+m)}- \overline{x}_{1:n+m})^2\\ &\quad + \frac{\kappa_0 n^2(\overline{x}_{1:n} - \mu_0)^2}{2(\kappa_0 + n+m)(n+m)}+ \frac{\kappa_0 m^2(\overline{x}_{(n+1):(n+m)} - \mu_0)^2}{2(\kappa_0 + n+m)(n+m)}\\ &\quad + \frac{2\kappa_0 nm(\overline{x}_{1:n}-\mu_0)(\overline{x}_{(n+1):(n+m)} - \mu_0)}{2(\kappa_0 + n+m)(n+m)}- \frac{\kappa_0 n(\overline{x}_{1:n} - \mu_0)^2}{2(\kappa_0 + n)}\\ &=\frac{nm}{2(n+m)}(\overline{x}_{1:n}-\overline{x}_{(n+1):(n+m)})^2\\ &\quad + \frac{\kappa_0 n^2(\overline{x}_{1:n} - \mu_0)^2}{2(\kappa_0 + n+m)(n+m)}+ \frac{\kappa_0 m^2(\overline{x}_{(n+1):(n+m)} - \mu_0)^2}{2(\kappa_0 + n+m)(n+m)}\\ &\quad + \frac{2\kappa_0 nm(\overline{x}_{1:n}-\mu_0)(\overline{x}_{(n+1):(n+m)} - \mu_0)}{2(\kappa_0 + n+m)(n+m)} - \frac{\kappa_0 n(\overline{x}_{1:n} - \mu_0)^2}{2(\kappa_0 + n)}\\ &= \overline{x}_{(n+1):(n+m)}^2 \left\{ \frac{nm}{2(n+m)} + \frac{\kappa_0 m^2}{2(\kappa_0 + n+m)(n+m)}\right\}\\ &\quad + \overline{x}_{(n+1):(n+m)} \left\{ - \frac{2nm}{2(n+m)}\overline{x}_{1:n} - \frac{2\kappa_0 m^2\mu_0}{2(\kappa_0 + n+m)(n+m)}\right.\\ &\qquad\qquad\qquad\left. + \frac{2\kappa_0 nm(\overline{x}_{1:n}-\mu_0)}{2(\kappa_0 + n+m)(n+m)} \right\}\\ &\quad+ \frac{nm}{2(n+m)}\overline{x}_{1:n}^2+ \frac{\kappa_0 n^2(\overline{x}_{1:n} - \mu_0)^2}{2(\kappa_0 + n+m)(n+m)} - \frac{2\kappa_0 nm(\overline{x}_{1:n}-\mu_0)\mu_0}{2(\kappa_0 + n+m)(n+m)}- \frac{\kappa_0 n(\overline{x}_{1:n} - \mu_0)^2}{2(\kappa_0 + n)}+ \frac{\kappa_0 m^2\mu_0^2}{2(\kappa_0 + n+m)(n+m)}\\ &=\overline{x}_{(n+1):(n+m)}^2 \frac{\kappa_nm}{2(\kappa_n +m)}\\ &\quad +\overline{x}_{(n+1):(n+m)}\overbrace{\frac{2m}{2(\kappa_n +m)(n+m)} \left\{ -[\kappa_n+m-\kappa_0]n \overline{x}_{1:n}-[n+m]\kappa_0\mu_0\right\}}^{-2\frac{\kappa_nm}{2(\kappa_n+m)}\mu_n}\\ &\quad +\frac{n\overline{x}_{1:n}^2}{2(\kappa_n+m)\kappa_n(n+m)} \overbrace{\left\{(\kappa_n+m)\kappa_nm+\kappa_0\kappa_nn-\kappa_0(\kappa_n+m)(n+m)\right\}}^{mn^2+nm^2=mn(m+n)}\\ &\quad -2\frac{n\overline{x}_{1:n}\kappa_0\mu_0}{2(\kappa_n+m)\kappa_n(n+m)}\overbrace{\left\{ n\kappa_n+m\kappa_n-(\kappa_n+m)(n+m)\right\}}^{-(n+m)m}\\ &\quad +\frac{\kappa_0\mu_0^2}{2(\kappa_n+m)\kappa_n(n+m)}\overbrace{\left\{ n^2\kappa_n+2nm\kappa_n+m^2\kappa_n-n(\kappa_n+m)(n+m)\right\}}^{\kappa_0m(n+m)}\\ &=\overline{x}_{(n+1):(n+m)}^2 \frac{\kappa_nm}{2(\kappa_n +m)} -2\overline{x}_{(n+1):(n+m)}\mu_n \frac{\kappa_nm}{2(\kappa_n +m)}+ \mu_n^2 \frac{\kappa_nm}{2(\kappa_n +m)}\\ &= \frac{\kappa_n m }{2(\kappa_n + m)} \, (\overline{x}_{(n+1):(n+m)} - \mu_n)^2 \end{align}