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3.2
Theoretical Basis of MCMC in Bayesian Inference
MCMC algorithms are, in essence, a form of Bayesian inference and hence prior
to a discussion of the details of MCMC itself, it is useful to briefly review the
foundations of Bayesian theory. The general stochastic inverse problem is described
in great detail by Tarantola ( 2005 ). According to inverse problem theory, there are
three distinct sources of quantitative information about a system: measurements of
the properties of the system, prior knowledge, and a model of the system. Each piece
of information is associated with a space that contains all possible values. Placing
the Earth system in this framework, all possible physical states (
x
) occupy one space
and all possible outcomes of observations (
) made of any and all properties of
the physical system occupy another space. The role of the model
y
is
to map information from one space into another. If we allow for uncertainty in
our knowledge of the state
y D F.x/
, then any event (realization of a possible physical
state) occupies a sub-region of the overall state space and can be associated with a
probability distribution
x
over that region. Similarly, allowing for uncertainty in
observations of the system of study gives rise to similar definition of an observation
event that occupies a sub-region of the observation space. This sub-region is also
associated with a probability distribution
P.x/
. Let us assume for the moment that
the observations contain information about the portion of the state space we are
interested in. In this case, the forward problem defines the process of mapping from
state to observation space (producing an analogue of the observations using the
model) and the inverse problem consists of determining information about the state
space from information contained in the observations. Each exercise (the forward
and inverse problems) consists of a conjunction of the two information spaces
that can be represented in the joint probability distribution
P.y/
. Clearly, if the
two information spaces are disjoint, then the forward and inverse calculations are
meaningless.
It is common to make the problem more specific by assuming that one of
two events (a realization of either a set of specific state properties
P.x;y/
x
or set of
observations
) has occurred. In this case, the problem becomes characterization
of the observation space given a subset of the state space, or inference of the state
space given a set of observations. These are formalized by introducing the definition
of conditional probability, for which the probability of state
y
x
conditioned on a set
of observations
y
is
P.x j y/ D P.x;y/
P.y/ ;
(3.1)
and the corresponding probability of observations y conditioned on the state x is
P.y j x/ D P.y;x/
P.x/
D P.x;y/
P.x/ :
(3.2)
Note that
because in this case they simply reflect the intersection
of two probability events. Bayes' theorem results from solving ( 3.2 ) for the joint
P.x;y/ D P.y;x/
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