Twelve Ways to Fit a Straight Line

Abstract

Fitting a straight line through data can be done in many ways that may seem different at first, but after closer inspection prove to be based on the same mathematics. In this chapter, we shall fit a line to data in twelve different ways and compare the resulting parameter estimates. So our goal is to estimate the intercept and the slope of the line.

Exercises

1. Prior uncertainty. Reduce the prior variances, which we specified as being $10^4$ in Eq. (8.6), to more realistic values such as $10^2$. Does that increase or decrease the posterior estimates for $\beta$? Why?

2. Kalman Filtering (KF) without intercept uncertainty. In our application of KF, we specified a very large prior uncertainty for $\beta$. Repeat the KF exactly as before, but with $\Sigma _{\beta }[1,1]$ set to zero. How does that change the posterior distribution for $\beta$? What kind of linear regression have you just done? Why is the posterior uncertainty for the slope-parameter lower than before?