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The difference between the two approaches is actually a difference between an Empirical Bayes versus a fully Bayesian approach to estimate the same thing. If you fit the mixed model using maximum likelihood, then you typically follow the first option, whereas, under the fully Bayesian approach from which you take posterior samples also for $\theta$, you ...


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As Frank Harrell notes in this answer: You need modifications to the bootstrap (.632, .632+) only because the original research used a discontinuous improper scoring rule (proportion classified correctly). For other accuracy scores the ordinary optimism bootstrap tends to work fine. Also, as discussed on this page, use of .632-type rules doesn't strictly ...


1

Yes, the residual vector is correct but the expression can be simplified because: $$(X(X^TX)^{-1}X^T)^2y=X(X^TX)^{-1}(X^TX)(X^TX)^{-1}Xy=X(X^TX)^{-1}X^Ty=X\hat{\beta}=\hat{y}$$ and residuals are $r=y-\hat{y}$. You don't have to square the term in the parentheses. Since $\hat y_0=\hat{\beta}x_0$, prediction interval will use sampling distribution of ...


1

Posterior predictive distribution is original pdf times posterior. It's your predictions on 'more' data, given your current data. But your current data is itself dependent on your prior. So really, it's your predicted 'more' data given you've worked out the posterior. Useful link. Binomial with beta prior is a conjugate prior with parameters found here ...


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I suggest you think carefully about the context of the question in practice. You could model the source and destination as GPS coordinates rather than streets and house numbers. In this case your outcome will be two numbers: the x and the y coordinate. Your approach would require multi-class classification and one way to encode the outcome would be as a ...


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Let $Y$ be the binary outcome, OP the ordinal position and $x$ the other predictors. Then build a logistic regression model for $Y$ using both $x$ and OP, maybe representing OP via a spline. Then when you want to make predictions without knowing OP, make predictions for all possible values of OP and average them over the marginal distribution of OP.


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