L is minimized when the line passes through the C of G
of the points (which leaves the slope, a).
a = Σ xi(yi-b) / Σ xi2,
and σ is the sqrt of the residual variance.
Second partial derivatives...
d2 L / d a2 =
(+1/σ2)
Σ{ xi2 }
(and remember, the xi are common knowledge)
d2 L / d b2 = n/σ2
d2 L / d σ2 =
- n/σ2 +
(3/σ4)Σ{yi-a.xi-b}2
expectation = 2 n / σ2
Off-diagonal second partial derivatives...
d2 L / d a.d b =
(+1/σ2)Σ xi
= n . mean{xi} / σ2
d2 L / d a.d σ =
(+2/σ3)
Σ{ xi.(yi-a.xi-b) }
expectation = 0
d2 L / d b.d σ =
(+2/σ3)Σ{yi-a.xi-b}
expectation = 0
Fisher
a
b
σ
a
Ey d2 L / d a2
Ey d2 L / d a d b
0
b
Ey d2 L / d a d b
Ey d2 L / d b2
0
σ
0
0
Ey d2 L / d σ2
F
= 2 n
{ n.(Σ xi2)
- (n.mean{xi})2} / σ6
= 2 n3{ (Σ xi2) / n
- (mean{xi})2} / σ6
F
= 2 n3 variance{xi} / σ6
Priors
a = tan θ where θ is the angular slope.
d a / d θ
= 1 / cos2θ= 1 / (1 + a2).
The uniform prior, 1 / π, on θ corresponds to
the prior pr(a) = 1 / (π (1+a2)) on 'a'.
b can be untangled from 'a' by making the C of G the origin.