Bayesian Methods for Astrophysics and Particle Physics

Since,
Pr(f|D) =
=
=
Pr(f,Θ|D)dΘ
Pr(f|Θ,D) Pr(Θ|D)dΘ
1.2 Marginalization
δ(f(Θ) − f) Pr(Θ|D)dΘ (1.9)
where δ(x) is the delta function. Thus one simply needs to compute f(Θ) for
every Monte Carlo sample and the resulting sample will be drawn from Pr(f|D).
1.2 Marginalization
In Bayesian statistics, the N–dimensional posterior distribution described in
Eq. 1.8 constitutes the complete Bayesian inference of the parameter values. If
the inference about an individual parameter is required, then the conditional
posterior distribution of that parameter can be calculated by setting the other
parameters to some fixed values which may be the maximum likelihood or MAP
estimates of these parameters. This conditional distribution does not take the
parameter uncertainties into account and hence the conditional posterior distri-
bution might well be inaccurate. In order to obtain inference for a parameter
θ1, the un–conditional (on other parameters) posterior distribution P(θ1|D, H)
is required. P(θ1|D, H) can be obtained by integrating (marginalizing) the N–
dimensional posterior distribution over all the parameters apart from θ1;
P(θ1|D, H) = P(θ1, θ2, · · · , θN|D, H)dθ2dθ3 · · ·θN
1.3 Bayesian Evidence
(1.10)
The denominator P(D|H) in Bayes’ theorem (Eq. 1.7) is called the Bayesian
Evidence. For parameter estimation problems, P(D|H) is usually ignored since
it is independent of the model parameters Θ. In contrast to parameter estimation
problems, in model selection the evidence takes the central role and is simply the
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