Bayesian Methods for Astrophysics and Particle Physics

(a) (b)
2.3 Nested Sampling
Figure 2.1: Cartoon illustrating (a) the posterior of a two dimensional problem;
and (b) the transformed L(X) function where the prior volumes Xi are associated
with each likelihood Li.
(2.2), is a monotonically decreasing function of X (which is trivially satisfied for
most posteriors), the evidence integral (2.1) can then be written as
Z =
1
0
L(X)dX. (2.3)
Thus, if one can evaluate the likelihoods Lj = L(Xj), where Xj is a sequence of
decreasing values,
0 < XM < · · · < X2 < X1 < X0 = 1, (2.4)
as shown schematically in Figure 2.1, the evidence can be approximated numer-
ically using standard quadrature methods as a weighted sum
Z =
M
Liwi. (2.5)
i=1
In the following we will use the simple trapezium rule, for which the weights are
given by wi = 1
2 (Xi−1 − Xi+1). An example of a posterior in two dimensions and
its associated function L(X) is shown in Figure 2.1.
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