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52 A Bayesian Probabilistic Duality in Standard Coherent States
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522 Bayesian Duality
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We have already encountered Bayesian probabilities in the previous chapter Let us try to become more familiar with this Bayesian context [55] (see Appendix A) Suppose we have an experiment for which we postulate an experimental model in the form of a (one-parameter) family n P (n, u) of discrete probability distributions, where the unknown parameter u takes values in a measure space (U , m(du)) Suppose that the experiment has been performed, producing the result k In Bayesian parlance, P (k, u) as a function of u is called the likelihood function (see Appendix A) and m(du) is called the prior measure on the parameter space U Then we have a conditional probability density function f on the parameter space via an inverse probability formula where f (u; k)du is proportional to P (k, u) m(du) In Bayesian language, this nal probability distribution on U is called the posterior probability distribution Thus, we have a duality of two probability distributions We have the original discrete family indexed by parameter u, wherein, if the true value of u were known, it would serve as a predictive model for experimentally obtained data Then we have the Bayesian posterior probability distribution on the (continuous) parameter space, which, if an experimental value were known, would serve as an inferred or retrodictive probability distribution for the unknown parameter As can be seen from the Poisson gamma duality described above, the choice of measure space (U , m(du)) and coherent states (51) along with the expression (52) leads to a similar duality of the two probability distributions
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523 The Fock Bargmann Option
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Now, the exclusive character of the possible outcomes n N or f (n) in the measurement of some quantum observable f (N ), N being the number operator, is encoded by the orthogonality between elements of the set of functions e n (z) = e |z|
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These functions are complex square roots of the probability distributions in both senses above: |e n (z)|2 = p n (|z|2 ) They are in one-to-one correspondence with the Fock or number states |n The closure of their linear span within the Hilbert space L2 (C, d 2 z/ ) is a sub-Hilbert space, say, F B e The latter is reproducing, isomorphic to the Fock Bergmann space introduced in 2, and also to the Fock space H generated by the number states
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5 Coherent States: a General Construction
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524 A Scheme of Construction
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In summary, what do we have (i) A set of parameters or data (in a classical sense), C, for example, the set of initial conditions for the motion of a particle on the line, equipped by the Lebesgue measure d 2 z/ (ii) The large Hilbertian arena, L2 (C, d 2 z/ ), which could be viewed as the space of images (with nite energy) in a signal analysis framework (iii) An orthonormal set of functions, {e n (z) L2 (C, d 2 z/ ), n N}, which obeys the probabilistic identity
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|e n (z)|2 = 1
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(iv) A resulting family of states, the standard coherent states |z , in F B e (or in H), with e n (z) as the orthogonal projection on the basis element e n (or |n ) (v) The identity (53) entails the normalization z|z = 1, and the orthonormality of the e n (z) s entails the resolution of the identity in F B e (or in H) In the next section, we will extend this scheme of construction to any measure set X
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53 General Setting: Quantum Processing of a Measure Space
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In a rst approach, one notices that quantum mechanics and signal analysis have many aspects in common As a departure point of their respective formalism, one nds a raw set X of basic parameters, which we denote generically by X = {x X } This set may be a classical phase space in the former case, like the complex plane for the particle motion on the line, whereas it may be a time frequency plane (for Gabor analysis) or a time-scale half-plane (for wavelet analysis) in the latter one Actually, it can be any set of data accessible to observation For instance, it might be a temporal line or the circle or some interval The minimal signi cant structure one requires so far is the existence of a measure (dx) on X As a measure space, (X , ), or simply X, could be given the name of an observation set, and the existence of a measure provides us with a statistical reading of the set of all measurable real- or complex-valued functions f (x) on X: it allows us to compute, for instance, average values on subsets with bounded measure Actually, both theories deal with quadratic mean values, and the natural framework of study is the Hilbert space L2 (X , ) of all square-integrable functions f (x) on the observation set X: X | f (x)|2 (dx) < The function f is referred to as a nite-energy signal in signal analysis and might be referred to as a (pure) quantum state in quantum
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