275 research outputs found
Applications of Finite deFinetti Style Theorems to Linear Models
No full text1 online resource (PDF, 52 pages)Diaconis, Persi; Eaton, Morris; Lauritzen, Steffen. (1987). Applications of Finite deFinetti Style Theorems to Linear Models. Retrieved from the University Digital Conservancy, https://hdl.handle.net/11299/199515
Magical mathematics: the mathematical ideas that animate great magic tricks
No full textMagical Mathematics reveals the secrets of amazing, fun-to-perform card tricks--and the profound mathematical ideas behind them--that will astound even the most accomplished magician. Persi Diaconis and Ron Graham provide easy, step-by-step instructions for each trick, explaining how to set up the effect and offering tips on what to say and do while performing it. Each card trick introduces a new mathematical idea, and varying the tricks in turn takes readers to the very threshold of today's mathematical knowledge. For example, the Gilbreath Principle--a fantastic effect where the cards remain in control despite being shuffled--is found to share an intimate connection with the Mandelbrot set. Other card tricks link to the mathematical secrets of combinatorics, graph theory, number theory, topology, the Riemann hypothesis, and even Fermat's last theorem
Commentary on “Longest increasing subsequences: from patience sorting to the Baik–Deift–Johansson theorem” by David Aldous and Persi Diaconis
No full textImmediately following the commentary below, this previously published article is reprinted in its entirety: David Aldous and Persi Diaconis, “Longest increasing subsequences: from patience sorting to the Baik–Deift–Johansson theorem”,Bull. Amer. Math. Soc. (N.S.)36(1999), no. 4, 413–432.</p
A Non-Reversible Markov Chain Sampling Method
No full textThis issue was undated. The date given is an estimate.21 pages, 1 article*A Non-Reversible Markov Chain Sampling Method* (Diaconis, Persi; Holmes, Susan; Neal, Radford) 21 page
On characterizations of Metropolis type algorithms in continuous time
No full textInternational audienceIn the continuous time framework, a new definition is proposed for the Metropolis algorithm (\wi X_t)_{t\geq0} associated to an a priori given exploratory Markov process and to a tarjet probability distribution . It should be the minimizer for the relative entropy of the trajectorial law of (\wi X_t)_{t\in[0,T]} with respect to the law of , when both processes start with as initial law and when is assumed to be reversible for (\wi X_t)_{t\geq0}. This definition doesn't depend on the time horizon and the corresponding minimizing process is not difficult to describe. Even if this procedure can be made general, the details were only worked out in situation of finite jump processes and of compact manifold-valued diffusion processes (a sketch is also given for Markov processes admitting both a diffusive part and a jump part). The proofs rely on an alternative approach to general Girsanov transformations in the spirit of Kunita. The case of -relative entropies is also investigated, in particular to make a link with a previous work of Billera and Diaconis on the traditional Metropolis algorithm in the discrete time setting
Riffle shuffles of a deck with repeated cards
No full textWe study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random walks on groups and their cosets, discovering along the way exact formulas leading to interesting combinatorics, an 'amazing matrix', and new analytic methods which produce a completely general asymptotic solution that is remarkable accurate
On barycentric subdivision, with simulations
No full textInternational audienceConsider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values goes to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to [0,1]\subset\CC (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0,1/2]. In addition we prove that the largest angle converges to in probability. Our approach is probabilistic and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0,1/2]. The stationary distribution of this limit chain is particularly important in our study. In an appendix we present related numerical simulations (not included in the version submitted for publication)
On barycentric subdivision, with simulations
No full textInternational audienceConsider the barycentric subdivision which cuts a given triangle along its medians to produce six new triangles. Uniformly choosing one of them and iterating this procedure gives rise to a Markov chain. We show that almost surely, the triangles forming this chain become flatter and flatter in the sense that their isoperimetric values goes to infinity with time. Nevertheless, if the triangles are renormalized through a similitude to have their longest edge equal to [0,1]\subset\CC (with 0 also adjacent to the shortest edge), their aspect does not converge and we identify the limit set of the opposite vertex with the segment [0,1/2]. In addition we prove that the largest angle converges to in probability. Our approach is probabilistic and these results are deduced from the investigation of a limit iterated random function Markov chain living on the segment [0,1/2]. The stationary distribution of this limit chain is particularly important in our study. In an appendix we present related numerical simulations (not included in the version submitted for publication)
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