Abstract Inference (Probability & Mathematical Statistics) by Ulf Grenander

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By Ulf Grenander

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8 is the probability that there was no error initially. 1 is the probability that an error was made initially; OB was typed. 1 is the probability that an error was made initially; AP was typed. Problem: a) Calculate the probabilities P(AB), P(OB), and P(AP) after passing the typing an infinite number of times through the correction process. b) Calculate the probabilities P(AB), P(OB), and P(AP) after passing the typing thrice through the correction process. 003 Problem: a) Calculate the probabilities P(A1), P(A2) and P(A3) of the events in set SA occurring.

Multiplying the den­sity of states g(E) by the probability and integrating: 3  2 2 A 2π   m ∫ ∞ 0 1  E  E 2 exp  −  dE = 1  kT  To integrate this equation, let: x≡ E kT or E = kTx and dE = kTdx Substituting this into the integral: 3  2 kT  2 A 2π    m  ∫ ∞ 0 1 x 2 exp  − x  dx = 1 This is a Γ function integral. Note that the integral 32 Mathematical Models of Information and Stochastic Systems ∞ ∫ 0 π 2 1 x 2 e − x dx = Substituting this for the integral: 3 3  2πkT  2  m 2 A = 1 or A =     m   2πkT  Next, the normalization constant A is substituted into the expression for the proba­ bility density p(E).

18) where the matrix on the right is equal to the conditional probability ma­trix describing an infinite number of repeated transmissions. Three-dimensional conditional probability matrices were used in the foregoing examples. However, much larger matrices could be used in practice. After passing the information an infinite number of times through the transmission system, the transmission matrix should not change if the message is passed through the transmission system yet one more time. 19, results in an identical set of linear equations.

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