Linear Regression 1

The simplest linear regression as a case study:

 

y = a.x + b + N(0,σ),  or equivalently  N(a.x + b, σ).

 

The probability density of y given x:

f(y)

=

1 √(2 π) σ

exp(

- (y - a.x - b)2 2 σ2

)

 

Given {(x_i, y_i)}, i = 1..n, where the x_i are "common knowledge", the negative log l.h. =

 

L = (n/2)log(2 π) + (n/2)log(σ²) + (1/(2 σ²))Σ{y_i-a.x_i-b}²

 

= (n/2)log(2 π) + n.log(σ) + (1/(2 σ²))Σ{y_i-a.x_i-b}²

 

First partial derivatives...

 

d L / d a = (-1/σ²) Σ{ x_i.(y_i-a.x_i-b) }

 

= (-1/σ²) Σ{ x_i.y_i-a.x_i²-x_i.b }

 

d L / d b = (-1/σ²) Σ{y_i-a.x_i-b}

 

= (-1/σ²) { (Σ y_i) - a(Σ x_i) - n.b}

 

d L / d σ = n/σ - (1/σ³)Σ{y_i-a.x_i-b}²

 

L is minimized when the line passes through the C of G of the points (which leaves the slope, a). a = Σ x_i(y_i-b) / Σ x_i², and σ is the sqrt of the residual variance.

 

Second partial derivatives...

 

d² L / d a² = (+1/σ²) Σ{ x_i² }

(and remember, the x_i are common knowledge)

 

d² L / d b² = n/σ²

 

d² L / d σ² = - n/σ² + (3/σ⁴)Σ{y_i-a.x_i-b}²

expectation = 2 n / σ²

 

Off-diagonal second partial derivatives...

 

d² L / d a.d b = (+1/σ²)Σ x_i

= n . mean{x_i} / σ²

 

d² L / d a.d σ = (+2/σ³) Σ{ x_i.(y_i-a.x_i-b) }

expectation = 0

 

d² L / d b.d σ = (+2/σ³)Σ{y_i-a.x_i-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 x_i²) - (n.mean{x_i})² } / σ⁶

= 2 n³ { (Σ x_i²) / n - (mean{x_i})² } / σ⁶

= 2 n³ variance{x_i} / σ⁶

Priors

a = tan θ   where &theta 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.

Then b ~ μ (of the {y_i}) in the [normal distribution].