The GNSS integer ambiguities: estimation and validation

validation criteria. The fundamental difference is that in the Bayesian approach not only
the vector of observations y is considered random, but the vector of unknown parameters
x as well. The concept of Bayesian estimation was described in e.g. (Betti et al. 1993;
Gundlich and Koch 2002; Gundlich 2002; Teunissen 2001b).
According to Bayes’ theorem the posterior density p(x|y) is proportional to the likelihood
function p(y|x) and the prior density p(x):
p(x|y) = p(y|x)p(x)
p(y)
(3.92)
The Bayes estimate of the random parameter vector is defined as the conditional mean
ˆxBayes = E{x|y} = xp(x|y)dx (3.93)
This estimate minimizes the discrepancy between x and ˆx on the average, where ˆx is
a function of y. So, if L(x, ˆx) is the measure of loss (or discrepancy), this amounts to
solving the minimization problem
minE{L(x,
ˆx)|y} = min L(x, ˆx)p(x|y)dx (3.94)
ˆx ˆx
If L(x, ˆx) = x − ˆx 2 Q it follows that the Bayes estimate ˆxBayes is indeed the solution to
this minimization problem.
In order to apply the Bayesian approach to ambiguity resolution, with the parameter
vector x = [bT aT ] T , a and b are assumed to be independent with priors
⎧
⎨p(a)
∝
⎩
z∈Zn δ(a − z) (pulsetrain)
(3.95)
p(b) ∝ constant
where δ is the Dirac function. From the orthogonal decomposition of equation (3.14)
follows that the likelihood function
p(y|a, b) ∝ exp{− 1
â − a
2
2 Qâ + ˆb(a) − b 2
Qˆ } (3.96)
b|â
and the posterior density follows therefore as
p(a, b|y) ∝ exp{− 1
2
â − a 2 Qâ + ˆ b(a) − b 2 Qˆ b|â
}
δ(a − z) (3.97)
z∈Z n
The marginal posterior densities p(a|y) and p(b|y) follow from integrating this joint
posterior density over the domains a ∈ R n and b ∈ R p respectively. Note that the
integration domain of a is not chosen as Z n since the discrete nature of a is considered
to be captured by assuming the prior to be a pulsetrain.
The marginal posterior PDFs are then obtained as
⎧
⎪⎨ p(a|y) = wa(â)
⎪⎩
z∈Zn δ(a − z)
p(b|y) =
pb|a(b|a = z, y)wz(â)
z∈Z n
(3.98)
The Bayesian approach 65