Very quick question.

Suppose I have a multivariate normal ${\bf{Y}}|{\bf{\theta}} \sim {\bf{MVN}}(X {\bf{\beta}}, \sigma^{2}H(\phi))$ where ${\bf{Y}}$ is a set of observations ${\bf{Y}} = \{y({\bf{s}}_{1}),y({\bf{s}}_{2}),... ,y({\bf{s}}_{n})\}$ and $H(\phi)$ is a covariance matrix $H(\phi) = \rho({\bf{s}}_{i} - {\bf{s}}_{j};\phi)$. Now I want to predict a value $y_{0}$, so as it is usual, the predictive posterior distribution is

$$p(y_{0}|{\bf{y}}, X, x_{0}) = \int p(y_{0}|{\bf{y}},{\bf{\theta}}, x_0)p({\bf{\theta}}|{\bf{y}}, X)d\theta$$

Usually, this is described as a product of $\text{Likelihood} \times \text{Posterior}$. However, in this case, the likelihood was initially expressed as a multivariate normal but I just want to predict the value $y({\bf{s}})$ at some point ${\bf{s}}_{0}$, so obviously I can't express a single point as a multivariate normal. What should I do?

As an alternative, I should be able to use a conditional distribution $y_{0} | \bf{Y}$ from a joint distribution $\left( y_{0}, {\bf{Y}} \right)$ but this in some way breaks with the idea of using $\text{Likelihood} \times \text{Posterior}$ for the predictive posterior distribution.

Thanks!

Suppose I have a multivariate normal ${\bf{Y}}|{\bf{\theta}} \sim {\bf{MVN}}(X {\bf{\beta}}, \sigma^{2}H(\phi))$ where ${\bf{Y}}$ is a set of observations ${\bf{Y}} = \{y({\bf{s}}_{1}),y({\bf{s}}_{2}),... ,y({\bf{s}}_{n})\}$ and $H(\phi)$ is a covariance matrix $H(\phi) = \rho({\bf{s}}_{i} - {\bf{s}}_{j};\phi)$. Now I want to predict a value $y_{0}$, so as it is usual, the predictive posterior distribution is

$$p(y_{0}|{\bf{y}}, X, x_{0}) = \int p(y_{0}|{\bf{y}},{\bf{\theta}}, x_0)p({\bf{\theta}}|{\bf{y}}, X)d\theta$$

Usually, this is described as a product of $\text{Likelihood} \times \text{Posterior}$. However, in this case, the likelihood was initially expressed as a multivariate normal but I just want to predict the value $y({\bf{s}})$ at some point ${\bf{s}}_{0}$, so obviously I can't express a single point as a multivariate normal. What should I do?

As an alternative, I should be able to use a conditional distribution $y_{0} | \bf{Y}$ from a joint distribution $\left( y_{0}, {\bf{Y}} \right)$ in the following way:

$$\left( \begin{array}{ccc} Y_{1} \\ Y_{2} \end{array} \right) \sim N\left(\left( \begin{array}{ccc} \mu_{1} \\ \mu_{2} \end{array} \right), \left( \begin{array}{ccc} \Omega_{11} & \Omega_{12} \\ \Omega_{21} & \Omega_{22} \end{array} \right)\right)$$

The problem is that this in some way breaks with the idea of using $\text{Likelihood} \times \text{Posterior}$ for the predictive posterior distribution.

Thanks!

Very quick question.

Suppose I have a multivariate normal ${\bf{Y}}|{\bf{\theta}} \sim {\bf{MVN}}(X {\bf{\beta}}, \sigma^{2}H(\phi))$ where ${\bf{Y}}$ is a set of observations ${\bf{Y}} = \{y({\bf{s}}_{1}),y({\bf{s}}_{2}),... ,y({\bf{s}}_{n})\}$ and $H(\phi)$ is a covariance matrix $H(\phi) = \rho({\bf{s}}_{i} - {\bf{s}}_{j};\phi)$. Now I want to predict a value $y_{0}$, so as it is usual, the predictive posterior distribution is

$$p(y_{0}|{\bf{y}}, X, x_{0}) = \int p(y_{0}|{\bf{y}},{\bf{\theta}}, x_0)p({\bf{\theta}}|{\bf{y}}, X)d\theta$$

Usually, this is described as a product of $\text{Likelihood} \times \text{Posterior}$. However, in this case, the likelihood was initially expressed as a multivariate normal but I just want to predict the value $y({\bf{s}})$ at some point ${\bf{s}}_{0}$, so obviously I can't express a single point as a multivariate normal. What should I do?

As an option, I should be able to use a conditional distribution $y_{0} | \bf{Y}$ from a joint distribution $\left( y_{0}, {\bf{Y}} \right)$ but this in some way breaks with the idea of using $\text{Likelihood} \times \text{Posterior}$ for the predictive posterior.

Thanks!

Very quick question.

Suppose I have a multivariate normal ${\bf{Y}}|{\bf{\theta}} \sim {\bf{MVN}}(X {\bf{\beta}}, \sigma^{2}H(\phi))$ where ${\bf{Y}}$ is a set of observations ${\bf{Y}} = \{y({\bf{s}}_{1}),y({\bf{s}}_{2}),... ,y({\bf{s}}_{n})\}$ and $H(\phi)$ is a covariance matrix $H(\phi) = \rho({\bf{s}}_{i} - {\bf{s}}_{j};\phi)$. Now I want to predict a value $y_{0}$, so as it is usual, the predictive posterior distribution is

$$p(y_{0}|{\bf{y}}, X, x_{0}) = \int p(y_{0}|{\bf{y}},{\bf{\theta}}, x_0)p({\bf{\theta}}|{\bf{y}}, X)d\theta$$

Usually, this is described as a product of $\text{Likelihood} \times \text{Posterior}$. However, in this case, the likelihood was initially expressed as a multivariate normal but I just want to predict the value $y({\bf{s}})$ at some point ${\bf{s}}_{0}$, so obviously I can't express a single point as a multivariate normal. What should I do?

As an alternative, I should be able to use a conditional distribution $y_{0} | \bf{Y}$ from a joint distribution $\left( y_{0}, {\bf{Y}} \right)$ but this in some way breaks with the idea of using $\text{Likelihood} \times \text{Posterior}$ for the predictive posterior distribution.

Thanks!