In ctgrubb/lemur.pack: Package for Team Lemur functional codebase

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  eval = TRUE
)

quick_eval <- FALSE

Goal

Let

\begin{align} &\ \mathbf{Y}_i \ \text{be the population $N$-vector of numeric incomes,} \ &\ \mathbf{Y}_i^{(b)} \ \text{be the population $N$-vector of binned incomes,} \ &\ \mathbf{Y}_h \ \text{be the population $N$-vector of numeric housing prices,} \ &\ \mathbf{Y}_h^{(b)} \ \text{be the population $N$-vector of binned housing prices.} \end{align}

We observe $\mathbf{Y}_h$, as well as $(x_i, x_i^{(b)}, x_h, x_h^{(b)})$ where

\begin{align} &\ \mathbf{X}_i \ \text{be the sample $n$-vector of numeric incomes,} \ &\ \mathbf{X}_i^{(b)} \ \text{be the sample $n$-vector of binned incomes,} \ &\ \mathbf{X}_h \ \text{be the sample $n$-vector of numeric housing prices,} \ &\ \mathbf{X}_h^{(b)} \ \text{be the sample $n$-vector of binned housing prices.} \end{align}

We want the conditional distribution, $f(\mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)} | \mathbf{X}_i, \mathbf{X}_i^{(b)}, \mathbf{X}_h, \mathbf{X}_h^{(b)})$.

\begin{align} f(\mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)} | \mathbf{X}_i, \mathbf{X}_i^{(b)}, \mathbf{X}_h, \mathbf{X}_h^{(b)}) &\propto f(\mathbf{X}_i, \mathbf{X}_i^{(b)}, \mathbf{X}_h, \mathbf{X}_h^{(b)} | \mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) f(\mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) \ &\propto f(\mathbf{X}_i, \mathbf{X}_i^{(b)} | \mathbf{X}_h, \mathbf{X}_h^{(b)}, \mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) f(\mathbf{X}_h, \mathbf{X}_h^{(b)} | \mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) f(\mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) \ &\propto f(\mathbf{X}_i | \mathbf{X}_i^{(b)}, \mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) f(\mathbf{X}_i^{(b)} | \mathbf{X}_h^{(b)}, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h^{(b)}) f(\mathbf{X}_h, \mathbf{X}_h^{(b)} | \mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) f(\mathbf{Y}_i, \mathbf{Y}_i^{(b)}, \mathbf{Y}_h, \mathbf{Y}_h^{(b)}) \end{align}

Leanna's notes

\begin{align} f(\mathbf{Y}_i, \mathbf{Y}_h | \mathbf{X}_i, \mathbf{Y}_h) &\propto f(\mathbf{X}_i, \mathbf{X}_h | \mathbf{Y}_i, \mathbf{Y}_h) f(\mathbf{Y}_i, \mathbf{Y}_h) \ &\propto f(\mathbf{X}_i | \mathbf{X}_h, \mathbf{Y}_i, \mathbf{Y}_h) f(\mathbf{X}_h | \mathbf{Y}_i, \mathbf{Y}_h) f(\mathbf{Y}_i, \mathbf{Y}_h) \end{align}

ctgrubb/lemur.pack documentation built on May 7, 2023, 4:13 a.m.