9.4 Tuning prior models for regression parameters

1. An average temperature day in DC: (65 to 70 degrees F)

   • 3000-7000 riders with around 5000

2. For every one degree increase, you get about +100 riders \(\pm\) 80

3. At any given temperature, daily ridership vary with a standard deviation of 1250 rides

We will work with centered data \(\beta_{0} \rightarrow \beta_{0c}\) because :

• it is easier to interpret and specify in this example

• this is what rstanarm uses

\[ Y_{i}| \beta_{0}, \beta_{1}, \sigma \overset{ind}{\sim} N(\mu_{i}, \sigma^2) \; with \quad \mu_{i} = \beta_{0} + \beta_{1}X_{i} \]

\[ \beta_{0c} \sim N(5000, 1000^2 ) \tag{a.}\]

\[ \beta_{1} \sim N(100, 40^2 \tag{b.}) \]

\[\sigma \sim Exp(0.0008). \tag{c.}\]

where for the exponential distribution \(\sigma \sim \text{Exp}(l)\), we relate the observed standard deviation to the rate parameter \(l\):

\[\text{SD}(\sigma) = \frac{1}{l} = 1250 \quad\Rightarrow\quad l = 0.0008\]

It is good to simulate this prior and see what they look like but we will do that in the part about using default rstanarm priors later.