Bayesian Methods for Astrophysics and Particle Physics

1.6 Numerical Methods
posterior are sufficiently suppressed to allow improved mobility of the chain over
the entire prior range.
The evidence can now be defined as a function of the “cooling” parameter λ:
Z(λ) = L λ πd N Θ. (1.29)
For correctly normalized prior, Z(0) = 1. We can use MCMC samples to get an
estimate of the expected value of log L over λ:
λ N log L.L πd Θ
〈log L〉 λ = . (1.30)
LλπdNΘ Comparing Eqs. 1.29 and 1.30, we see that,
〈log L〉 λ = 1 dZ
Z dλ
so that the logarithm of the evidence, log Z(1) is given by:
log Z(1) = log Z(0) +
1
0
d log Z
= , (1.31)
dλ
d logZ
dλ =
dλ
1
0
〈log L〉 λ dλ. (1.32)
For nλ samples taken at the cooling parameter λ, the evidence can be estimated
as:
log Z(1) ≈ 1
nλ
nλ
k=1
log Lk. (1.33)
Thus, the evidence value is obtained at the end of annealing schedule and then
sampling from the posterior can start so the evidence as well as the posterior
samples can be obtained through thermodynamic integration technique.
1.6.2.1 Problems with Thermodynamic Integration
Typically, it is possible to obtain accuracies of within 0.5 units in logZ with
thermodynamic integration, but in cosmological applications it typically requires
of order 10 6 samples per chain (with around 10 chains required to determine a
sampling error). This makes evidence evaluation at least an order of magnitude
more costly than parameter estimation.
Another problem faced by thermodynamic integration is in navigating through
phase changes as pointed out by Skilling (2004). As λ increases from 0 to 1, one
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