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26 CHAPTER 1. INTRODUCTION • Probability depends on our state of knowledge, which is different for different people. Therefore probability is necessarily subjective. Consider two events A and B where P (A) and P (B) are the probabilities of the event A or B, respectively. It is clear that for the probability P (A) the following holds 0 ≤ P (A) ≤ 1. Furthermore, for two events A and B P (A ∪ B) = P (A) + P (B) − P (A ∩ B) (1.75) P (A ∩ B) = P (B|A)P (B) = P (B|A)P (A), (1.76) where ∪ denotes the logical OR and ∩ denotes the logical AND. The probability P (A ∪ B) is the logical sum and P (A ∩ B) is the logical product of two probabilities P (A) and P (B). The latter is also often called the joint probability. The term P (A|B) describes the probability of A under the condition that B is true and is shortened by saying probability of A given B. Another important property is the probabilistic independence of events. If the probability of A does not change the status of B, the events A and B are said to be independent. In that case P (A|B) = P (A), and P (B|A) = P (B). Inserting this into Eq. (1.76) yields P (A ∩ B) = P (A)P (B). (1.77) From Eq. (1.76), Bayes Theorem is easily derived: P (B|A) = P (A|B)P (B) , (1.78) P (A) where P (B) is called the prior probability, P (B|A) is called the posterior probability and P (A|B) is the likelihood. If one identifies event A with an observation and event B with some set of model parameters, the likelihood can be literally described as the probability of the observation A given the specific hypotheses B. In the same context, the probability of the observation P (A) is a constant although it is unknown leaving the proportionality P (B|A) ∝ P (A|B)P (B), (1.79) the prior probability P (B) is a statement about our knowledge of the hypotheses and is mostly assumed to be uniform when one does not know anything about the probability of the hypotheses. However, Bayes postulates that all priors should be treated as equal. So far, it was implicitly assumed that the variable x is discrete and a probability function p(x) is interpreted as the probability of the proposition P (A), where A is true when the value of the variable is equal to x. However in most cases, continuous variables x have to be considered and the probability will be a continuous function interpreted as a probability density function p(x)dx. In terms of the probability P (A), it is understood as A is true when the value of the variable lies in the range x + dx. In the further discussion, the latter perspective is assumed. Assuming a set of data being the observations d and a set of models θ describing our expectations, then Eq. (1.79) becomes p(θ|d) ∝ p(d|θ)p(θ). (1.80)
1.4. SOME STATISTICAL TOOLS 27 If the data d i are independent, then the likelihood L(θ; d) = p(d|θ) can be expressed as L(θ; d) = p(d|θ) = ∏ L(θ; d i ). (1.81) i As mentioned, if one knows so little about the appropriate values of the hypotheses parameter that for the prior p(θ) a uniform distribution is a practical choice and using the independence described by Eq. (1.81), Eq. (1.80) becomes p(θ|d) ∝ p(d|θ) = ∏ i L(θ; d i ). (1.82) Therefore, the maximum of the posterior probability p(θ|d), which is the interesting probability of the model given the data, can be found by maximising the likelihood L(θ; d) – maximising the probability of the data given the model. This consideration leads to the maximum likelihood principle. In order to derive a least-squares formulae (Eq. 1.51) as presented in Sect. 1.4.1, one considers the likelihood to be described by a Gaussian distribution. Assuming that the data are independent consisting of pairs {x i , y i }, whose true value {µ xi , µ yi } are related by a deterministic function µ yi = y(µ xi , θ) and with Gaussian errors σ i only in y i (i.e. x i ≈ µ xi ) then the likelihood function is a multivariate Gaussian [ exp − (y i − y(x i , θ)) 2 ] (1.83) p(θ|x, y) = L(θ; x, y) ∝ ∏ i 2σ 2 i [ = exp − 1 ] 2 χ2 (θ) , (1.84) where χ 2 (θ) = ∑ (y i − y(x i , θ)) 2 σ 2 , (1.85) i i which is the same as Eq. (1.51). Maximising the likelihood function is equivalent to minimising χ 2 (θ) with respect to θ. The interesting point is that this equation holds for independent variables and Gaussian distributions. One should keep these assumptions in mind when applying this method to data sets. The uncertainty in θ is determined by considering the covariance matrix V (V −1 ) ij (θ) = 1 ∂ 2 χ 2 ⏐ ⏐⏐⏐θ=θm , (1.86) 2 ∂θ i ∂θ j where θ m is the set of parameters which minimise the χ 2 -function. This is a consequence of the assumed multi-variate Gaussian distribution of θ. Expanding χ 2 in series around its minimum χ 2 (θ m ) χ 2 (θ) ≈ χ 2 (θ m ) + 1 2 ∆θT ∂2 χ 2 ∂θ i ∂θ j ∆θ, (1.87) where ∆θ is the difference θ − θ m . Using Eq. (1.86), and inserting Eq. (1.87) into Eq. (1.83) and applying an appropriate normalisation results in the Likelihood function [ 1 L(θ; x, y) ≈ (2π) n/2 (det V ) 1/2 exp − 1 ] 2 ∆θT V −1 ∆θ , (1.88)
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118 CONCLUSIONS tually statisticall
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120 BIBLIOGRAPHY Conway, R. G. & St
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122 BIBLIOGRAPHY Ikebe, Y., Makishi
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124 BIBLIOGRAPHY Simard-Normandin,
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126 ACKNOWLEDGEMENTS in his long em
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Publications Journals 1. A Bayesian
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