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probability
and inserting the result in (
3.1
). Given, as is typically the case,
discrete state and observation vectors
x
and
y
, Bayes' theorem can be written
P.x;y/
/
D
P.
/P.
/
y
j
x
x
P.
x
j
y
(3.3)
P.
y
/
In practice, we are always dealing with a system for which either a state (or set
of states) has already been determined, or of which observations have already been
taken, and it is common to begin with Bayes' theorem. The benefit of generalized
inverse theory is its flexibility-it makes no assumption that any event has taken
place and considers only the conjunction of information spaces. Standard Bayesian
estimation theory (e.g.,
Jazwinski 1970
;
Bennett 1992
;
Evensen 2006
) treats each
quantity that contributes to a total quantitative knowledge of each component of
the system as a stochastic quantity with an associated probability density function
(PDF). The conjunction of separate PDFs associated with contributions from the
model and the associated modeling errors, the observation errors, and the prior
information result in a joint posterior PDF. The properties of the joint posterior PDF
determine the characteristics and tractability of the inverse problem (e.g., whether or
not the solution is unique). The explicit separation of information into components
provided by the model, observation, and prior estimate facilitates determination of
the unique contribution of each to the posterior PDF.
The inverse problem, according to Bayes' theorem, consists in computing each
individual PDF on the right hand side of (
3.3
) and combining them to obtain the
properties of the state of interest given prior knowledge and a set of observations
related through the model. It is well known that computation of the posterior PDF is
straightforward if the model is linear and all PDFs can be assumed to be Gaussian.
Approximate solutions can be obtained in the case of a nonlinear model by either
linearizing the model (e.g., extended Kalman filter (
Gelb 1974
), three dimensional
variational data assimilation (3DVAR;
Sasaki 1970
;
Lorenc 1986
), four dimensional
variational data assimilation (4DVAR;
Courtier 1997
)) or via Monte Carlo methods
using a stochastically generated ensemble of states (e.g., the ensemble Kalman filter,
Evensen 2006
). In the case of models for which the dimension of the state space is
large and the mapping between state and observation space is nonlinear, it is not
computationally feasible to compute numerical solutions to (
3.3
). The fundamental
result of the work of
Metropolis et al.
(
1953
)and
Hastings
(
1970
) is that it is not
necessary to compute a solution to the posterior PDF if one can construct a Markov
chain that has the same equilibrium distribution. The goal of MCMC is to sample
the posterior PDF
P.
/
up to a normalizing constant (i.e., by computing only the
numerator on the RHS of (
3.3
)) using a Markov chain that has a stationary transition
probability
x
j
y
q.
x
i
;
x
i
C
1
/
; the probability of moving to a set of states
x
i
C
1
from the
current state
x
i
. Robust samples of the posterior PDF are ensured in MCMC via the
application of an
update
to the Markov chain that determines whether a proposed
transition from the current state
x
i
to a proposed state
x
q.
;
/
(consistent with the
specified transition probability) is
accepted
. In the formulation originally introduced
by
Metropolis et al.
(
1953
) and generalized by
Hastings
(
1970
), the update is done
in the following way:
x
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