A simple Bayesian regression model

Bayesian Modeling with RJAGS

Alicia Johnson

Associate Professor, Macalester College

Chapter 3 goals

Engineer a simple Bayesian regression model
Define, compile, and simulate regression models in RJAGS
Use Markov chain simulation output for posterior inference & prediction

Modeling weight

$Y_i$ = weight of adult $i$ (kg)
$ \; $
$ \;$

Model
$Y_i \sim N(m, s^2)$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$ \;$
$ \;$

Model
$Y_i \sim N(m, s^2)$

$\;$ $\;$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$X_i$ = height of adult $i$ (cm)
$ \;$

Model
$Y_i \sim N(m_i, s^2)$

$\;$ $\;$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$X_i$ = height of adult $i$ (cm)
$ \;$

Model
$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$\;$ $\;$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$X_i$ = height of adult $i$ (cm)
$ \;$

Model
$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$\;$ $\;$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$X_i$ = height of adult $i$ (cm)
$ \;$

Model
$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$\;$ $\;$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$X_i$ = height of adult $i$ (cm)
$ \;$

Model
$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$\;$ $\;$

Modeling weight by height

$Y_i$ = weight of adult $i$ (kg)
$X_i$ = height of adult $i$ (cm)
$ \;$

Model
$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$\;$ $\;$

Bayesian regression model

$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$a$ = y-intercept
value of $m_i$ when $X_i = 0$
$b$ = slope
rate of change in weight (kg) per 1 cm increase in height
$s$ = residual standard deviation
individual deviation from trend $m_i$

Priors for the intercept & slope

Priors for the intercept & slope

Priors for the intercept & slope

Prior for the residual standard deviation

Prior for the residual standard deviation

Bayesian regression model

$Y_i \sim N(m_i, s^2)$
$m_i = a + b X_i$

$a \sim N(0, 200^2)$
$b \sim N(1, 0.5^2)$
$s \sim \text{Unif}(0, 20)$

Let's practice!

Bayesian Modeling with RJAGS

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