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.