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Additional info for Numerical Inversion of the Laplace Transform: Applications to Biology, Economics Engineering, and Physics

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D do xk ← gauss(σ) Σ ← Σ + x2k 1/d Υ ← ran (0, 1) for k = 1, . . , d√ do xk ← Υxk / Σ output {x1 , . . 21 direct-sphere. Uniform random vector inside the ddimensional unit sphere. The output is independent of σ. 45) are identical, and this means that the d Gaussians sample angles isotropically in d dimensions, and only get the radius wrong. This radius should be sampled from a distribution π(r) ∝ rd−1 . 29), we obtain the direct distribution of r by taking the dth root of a random number ran (0, 1).

The algorithm constructs a random vector from Gaussians, orthogonalizes it, and normalizes it with respect to the input vector x, the current position of the Markov chain. The step taken is in the direction of this reworked , a random unit vector in the hyperplane orthogonal to x. procedure markov-surface input x (unit vector |x| = 1) ← {gauss (σ) , . . 24 markov-surface. Markov-chain Monte Carlo algorithm for random vectors on the surface of a d-dimensional unit sphere. 3 Statistical data analysis In the first sections of this book, we familiarized ourselves with sampling as an ingenious method for evaluating integrals which are unsolvable by other methods.

7 Example run of Alg. 11 (ran-perm). In each step k, the numbers k and l are underlined. 11 ran-perm. Generating a uniformly distributed random permutation of K elements. 2 3 process after M steps, rather than K (M < K), to sample a random combination (see Alg. 12 (ran-combination)). 4 procedure ran-combination {P1 , . . , PK } ← {1, . . , K} for k = 1, . . , M do l ← nran (k, K) Pl ↔ Pk output {P1 , . . 12 ran-combination. Generating a uniformly distributed random combination of M elements from K.

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