Probabilistic methods - Electromagnetic Methods in Geophysics

= _n

j ji ii

A v

1 2

1 (C.10)

and vij=0, i≠j.

Finally, when we have posed the problem and have the vectors and matrices in accordance with the last equation to get the x’ is obtained and x=Vx’ is considered. To improve convergence in the iterative process the Levenberg-Mardquardt method may be used.

D. Probabilistic methods

The probabilistic methods are based on assumption that model parameters m are random variables described by probability distribution. The methods utilizing this approach can be divided into two groups, classical and Bayesian. The former group contains maximum likelihood estimation, least squares approach, method of moments while the latter one contains so called minimum mean square error estimation, maximum a posteriori estimation or linear minimum square estimation. Moreover, each method itself contains many variants, e.g. least squares approach may be sequential, constrained or unconstrained, linear or nonlinear. Thus, it is not possible to cover all methods and their modifications and to discuss them in such short report, only some of the will be presented. Probabilistic methods can be used even though there is not a strict probabilistic behaviour of the studied problem.

a) Maximum Likelihood Estimation

This approach is very popular way to obtain estimations in complicated estimation problems. MLE has the asymptotic properties of being unbiased, thus it is considered to be asymptotically. In maximum likelihood estimation the probability density function of the experimental data d given the unknown parameter m is assumed to be known. The probability density of m is not required. The mˆ is the maximum likelihood estimation, for given data d, if the following relation is fulfilled

(

^d^m^ˆ

) ( )

^p^d^m^ˆ

p _MLE ≥ (D.1)

It means that mˆ_MLE maximises the likelihood distribution ^p

( )

^d^m for given data d.

b) Minimum mean square estimation

Methods is based on maximising the expectation of the squared norm of the estimation error

{

^m⁻^m^ˆ

}

E (D.2)

Using Bayesian cost method it could be determined that the minimum mean square estimate mˆ MMSE is conditional expectation of m given the observation d

( )

^m^d ^m

This is the MMSE that minimises

( )

{

^m ^m

} (

^m ^m

) (

^p^d ^m

)

^d^d^d^m

E ⁻ ˆ ⁼

∫

⁻ ˆ ² , ^(D.4)

c) Bayesian approach

The description below is based on paper Malinverno A., 2000. In Bayesian approach, inferences about the parameter vector θ and the parameterisation are made using probability density functions and probabilities. In other words, the conclusions that can be drawn from Bayesian analysis are of the type “from what we know, there is a 95 per cent probability that parameter θⁱ has a value between 0.5 and 1.3” It is important to stress at the outsets that these probabilities quantify uncertain knowledge. As such, these probabilities are always conditional on something that is assumed true: these assumptions are prior information and are denoted J. In the notation used here, the statement above on θi can be written as

∫

^.³

( )

⁼

The fundamental formula in Bayesian parameter estimation is Bayes’ rule, which for a vector of parameter θ and vector of data d is

likelihood function (the pdf of the data when the parameter vector equals θ). In other words, what can be inferred about the parameter vector a posteriori is a combination of what is known a priori, independent of the data, and of the information contained in the data. The denominator in Bayes’ rule can be shown to be the integral of the numerator

( )

^J ⁼

∫

( ) (

^J ^p ^J

)

pd θ dθ, (D.7)

Therefore, ^{p d}

( )

^J is a normalizing factor that makes the integral of the posterior pdf equal unity: since it does not depend on θ, it is typically ignored in parameter estimation. The posterior pdf, which quantifies the uncertainty of the parameter values once the information in the data is accounted for, is the solution of the inverse problem. The linear, Gaussian case is examined that is appropriate for a linear forward problem and for a prior pdf and a likelihood function that are multivariate Gaussian distributions. The natural choice for a priori pdf is the distribution that allows for the greatest uncertainty while obeying the

constraints imposed by the prior information, and it can be shown that this least informative pdf is the pdf that has maximum entropy. Suppose all one knows about the parameters a priori is that they are as likely to be positive as negative, but that their square value cannot be too large and is expected to be σθ². For any single parameter, the pdf that has maximum entropy subject to these prior constraints is a Gaussian distribution with zero mean and a variance equal to σθ². If there is no a priori information on correlations amongst H parameters, the maximum entropy prior pdf of θ is the product of the pdfs of ach parameter

( )

^θ ⁼

( )

²^π^σ¹^θ² ²^exp^⎜⎜^⎝^⎛⁻²^θ^σ^θ^θ² ^⎟⎟^⎠^⎞

J H

p (D.8)

Quantities assumed a priori are denoted with bar, for example, θ, is the prior mean of the parameter vector.

The likelihood function is the pdf of the measurement error vector e, defined as the difference between the observed data d and the data predicted for a given value of the parameter vector

GAθ d

Gm d

e= − = − (D.9)

If the errors are expected a priori to have a mean square deviation from zero equal to σ²e^and

to be uncorrelated the likelihood function is again a Gaussian distribution

(

)

⁼

( )

² ¹² ² ^exp^⎜⎜^⎝^⎛⁻² ²^e^⎟⎟^⎠^⎞

The probability of having observed the data d becomes smaller as the sum of squared errors e

e^T becomes larger, and the likelihood quantifies the information about θ contained in the data. If prior pdf and the likelihood function are as shown in (12) and (14), it is easy to show that the posterior pdf is also Gaussian, and has a posterior covariance matrix Cˆ_θ and a posterior mean vector θˆ that are as follows

where I is an H×H identity matrix. A posteriori quantities are denoted with hat, ^θˆ is the posterior mean of the parameter vector. If the posterior pdf of θ is Gaussian, the posterior pdf of m=Aθ is also Gaussian with aposterior covariance matrix and a posterior mean vector that are as follows

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