82 Andrew C. Lorenc
be about 3ms- l , and a grid-Iength on 20 km would reduce the error ofrepresentativeness to about lms- I .
p(y0 I XI,x2) is plotted as a function of (xI,x2) in the short dashed lines in Fig. 5.3. It
is caUed the likelihood function 1. Note that the ridge extends to infinity; aU values
on the ridge line are, as far as the observed information is concemed, equaUy
likely. To get a unique "best estimate", we need to combine this with the prior
information.
We can substitute (17) and (21) into the Bayesian analysis equation:
( I 0) p(llx)p(x)
p x y =
p(l)
(22)
ce N(lIH(x), E + F)N(xll, B)
It is a property of Gaussians that, if H is linearisable:
N(lIH(x), E + F)N(xlx b , B)
(23)
0I
T
a
= N(y H(x),E+F+HBH )N(xlx ,A)
where x a and A are defined by:
(24)
a
b
T
T
-1
°
b
X = X + BH (HBH + E + F) (y + -H(x »
Cancelling the constant of proportionality gives:
(25)
80 the posterior probability, after adding the observational information, is a
Gaussian with mean x"- and variance A. As in Fig. 5.1, the posterior pdfin Fig. 5.3
(solid lines) is narrower and taUer than the prior. Adding information from the
observation increases our confidence, and reduces the error variance from B to A.
For our simple example the algebra is easily done by hand, giving:
( J ( J
( V'(!..±l!))2
x a = x~ = X! +
2
[l _ xt ; x~l( 1 ) (26)
X
x
E + F + V'(!..±l!)
1
2 2
2
1. It does not integrate to one over x, so it is not a probability.