I want to do a forecast of let's say orders with a Bayesian linear regression, where orders do not only depend on time but also on another regressor, let's say accounts at time t. $$orders_{t} = \theta_{1}*t + \theta_{2}*accounts_{t} + \epsilon$$ So obviously, I do not have future values of accounts, so when predicting future values of order intake I use a prediction of accounts $$accounts_{t} = \beta_{1}*t + \epsilon$$ Now, for predicted accounts I can construct a credible interval as it only depends on known and fixed values (time). For predicted orders, uncertainty around the account prediction carries over into the prediction, so a credible interval for the order prediction would need to take this uncertainty into account. How would I do that?
1 Answer
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In Bayesian Inference the prediction step is done via the posterior predictive distribution, which is calculated as $$f(y_{new} | y_{(\underline{n})} ) = \int_{ \Theta} f(y | \theta ) \cdot \pi(\theta ) d \theta $$ (where $\pi(\theta)$ is the posterior distribution) which accounts for the uncertainty of and unobserved observation.
