I have read through other questions on the site and I feel that none provide a great answer to this question. Simply put - could anyone explain, common approaches for generating predictions on new unseen data with a Bayesian model.

For example, suppose you have a simple linear regression: $$ y = \alpha + \beta * x $$

Which under a probabilistic framework is given by:

$$ y \sim N(\mu = \alpha + \beta *x, \sigma = \epsilon) $$

We set prior distributions for the parameters $ \alpha, \beta$ and $\epsilon$ and with our training data $X_{train}$ we return samples from posterior to obtain posterior distributions across these parameters.

My question is, given new data $X_{new}$, how are predictions then made on that data?

For example, would you:

• For each new data point in $X_{new}$, take the posterior samples from the 'training' process, and calculate $Y_{new}$ using that data point for $X_{new}$ and combinations of coefficients from the posterior samples (for a given data point in $X_{new}$) - to generate a vector of predictions for each data point in $X_{new}$?

I have read through other questions on the site and I feel that none provide a great answer to this question. Simply put - could anyone explain, common approaches for generating predictions on new unseen data with a Bayesian model.

For example, suppose you have a simple linear regression: $$ y = \alpha + \beta * x $$

Which under a probabilistic framework is given by:

$$ y \sim N(\mu = \alpha + \beta *x, \sigma = \epsilon) $$

We set prior distributions for the parameters $ \alpha, \beta$ and $\epsilon$ and with our training data $X_{train}$ we return samples from posterior to obtain posterior distributions across these parameters.

My question is, given new data $X_{new}$, how are predictions then made on that data?

For example, would you:

• For each new data point in $X_{new}$, take the posterior samples from the 'training' process, and calculate $Y_{new}$ using combinations of coefficients from the posterior samples (for a given data point in $X_{new}$) - to generate a vector of predictions for each data point in $X_{new}$?