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Mathematics of Operational Research Contents Table of Contents i ...
www.statslab.cam.ac.uk/~rrw1/mor/morweber.pdf15 Mar 2016: Mathematics of Operational Research. Contents. Table of Contents i. Schedules v. 1 Lagrangian Methods 11.1 Lagrangian methods. 11.2 The Lagrange dual. 31.3 Supporting hyperplanes. 3. 2 Convex and Linear Optimization 62.1 Convexity and strong duality. -
Mathematics of Operational Research Example Sheet 2 R. Weber ...
www.statslab.cam.ac.uk/~rrw1/mor/examples2.pdf2 Dec 2015: maximizesi,tj. { i. si j. tj : αiβj si tj 0, i, j}. -
8 8 1/2 1/2 1/2 1/2 1 1/2 1/4 ...
www.statslab.cam.ac.uk/~rrw1/markov/slides.pdf14 Nov 2011: 25. . Theorem 5.8. Suppose P is irreducible and recurrent.Then for all j I we have P(Tj < ) = 1. -
21 Paper 4, Section I 9H Markov ChainsLet X0, ...
www.statslab.cam.ac.uk/~rrw1/markov/MarkovChainTriposQuestions.pdf17 Sep 2015: Assume p > q. Let Tj = inf{n > 1 : Xn = j} if this is finite, and Tj = otherwise. ... Let Ti = inf{n > 1 : Xn = i}. For each i 6= j letqij = P(Tj < Ti | X0 = i) and mij = E(Tj | X0 = i). -
Markov Chains These notes contain material prepared by colleagues ...
www.statslab.cam.ac.uk/~rrw1/markov/M.pdf22 May 2013: Pj(Tj < ) Pj(Hi < )Pi(Hj < ) = 1. 5.5 Relation with closed classes. ... P(X0 = i)Pi(Tj < ). so it suffices to show that Pi(Tj < ) = 1 for all i I. -
Asymptotics and optimal bandwidth selection for highest density…
www.statslab.cam.ac.uk/~rjs57/Samworth10.pdf21 Apr 2010: tn, tj tn], where D4 = 12 μ2(K)f ′′(xj ) D1;. • ... tn = o(n1/6).Then. P(f̂h(x. tj ) < f̂h,τ. ) ( tf ′(xj ) D4n1/2h5/2{R(K)fτ 2D3,j D2}1/2). -
CHOICE OF NEIGHBOUR ORDER FOR NEAREST-NEIGHBOUR CLASSIFICATION RULE…
www.statslab.cam.ac.uk/~rjs57/HPSLV.pdf28 Aug 2007: 2/d , then, with v defined by. (4.8), and tj = νj1 {ν/k2(ν)}. -
Maximum likelihood estimation of a multidimensional log-concave…
www.statslab.cam.ac.uk/~rjs57/CSSFinalLV.pdf24 Mar 2010: Maximum likelihood estimation of a multidimensional log-concave density. Madeleine Cule and Richard Samworth†University of Cambridge, UK. and Michael StewartUniversity of Sydney, Australia. Summary. Let X1,. , Xn be independent and identically -
Statistical modellingRajen D. Shah r.shah@statslab.cam.ac.uk Course…
www.statslab.cam.ac.uk/~rds37/teaching/statistical_modelling/notes.pdf14 Feb 2019: and (Xj )TXj = X. Tj (I Pj)Xj = ‖(I Pj)Xj‖2. -
Modern Statistical MethodsRajen D. Shah r.shah@statslab.cam.ac.uk…
www.statslab.cam.ac.uk/~rds37/teaching/modern_stat_methods/notes2.pdf15 Jan 2023: Modern Statistical MethodsRajen D. Shah r.shah@statslab.cam.ac.uk. Course webpage:http://www.statslab.cam.ac.uk/rds37/modern_stat_methods.html. In this course we will study a selection of important modern statistical methods. Thisselection is -
MATHEMATICS OF MACHINE LEARNING Part IIExample Sheet 2 (of ...
www.statslab.cam.ac.uk/~rds37/teaching/machine_learning/Qu2.pdf20 Feb 2024: F̂(t1,. , tp) =1. n. ni=1. 1A(Wi). where A =pj=1(, tj]. ... n. [Hint: Consider H := {1A : A =pj=1(, tj], tj R}, and H := {h : h H}, and. -
Journal of Machine Learning Research ? (????) ?-?? Submitted ...
www.statslab.cam.ac.uk/~rds37/papers/shah16.pdf9 Jun 2016: Journal of Machine Learning Research? (?)? -? Submitted 11/13; Revised 3/16; Published? /? Modelling Interactions in High-dimensional Data withBacktracking. Rajen D. Shah r.shah@statslab.cam.ac.ukStatistical Laboratory. University of Cambridge. -
Statistical modellingLecturer: Alberto J. Coca…
www.statslab.cam.ac.uk/~qz280/teaching/modelling-2022/notes_3Dec19.pdf29 Apr 2024: and (Xj )TXj = X. Tj (I Pj)Xj = ‖(I Pj)Xj‖2. -
Mendelian Randomization: Old and New Insights Qingyuan Zhao…
www.statslab.cam.ac.uk/~qz280/talk/penn-biostat-2021/slides.pdf29 Apr 2024: 4): Q-Q plot of standardized residual:. tj (β̂) =Γ̂j β̂γ̂j. 1 β̂2. ... Robust adjusted profile score (RAPS). I Define standardized residual: tj (β,τ2) =. Γ̂j βγ̂j1 β2 τ2. I For some robust loss ρ (let ψ = ρ′), the RAPS equations -
Leverage Mendelian Randomization to Learn MeaningfulRepresentations…
www.statslab.cam.ac.uk/~qz280/talk/lmrl-2021/slides.pdf29 Apr 2024: ψ(ρ)1 (β,τ. 2) =. pj=1. ( β. tj)ψ(tj ),. ψ(ρ)2 (β,τ. ... 2) =. pj=1. tj ψ(tj ) E[Tψ(T )], for T N(0, 1). -
The Randomization Principle in Causal Inference: A Modern Look ...
www.statslab.cam.ac.uk/~qz280/talk/imperial-2022/slides.pdf29 Apr 2024: Consider any function g : Z [M] anda collection of test statistics: Tj : Z W R, j [M].The p-value of the CRT is given by. -
Mendelian Randomization: Old and New Insights Qingyuan Zhao…
www.statslab.cam.ac.uk/~qz280/talk/epfl-2021/slides.pdf29 Apr 2024: 4): Q-Q plot of standardized residual:. tj (β̂) =Γ̂j β̂γ̂j. 1 β̂2. ... Robust adjusted profile score (RAPS). I Define standardized residual: tj (β,τ2) =. Γ̂j βγ̂j1 β2 τ2. I For some robust loss ρ (let ψ = ρ′), the RAPS equations -
Simultaneous Hypothesis Testing using Internal Negative Controls…
www.statslab.cam.ac.uk/~qz280/publication/ranc/slides.pdf29 Apr 2024: 2 (Ti)iI (Tj)jInc;3 (Tj)jInc is mutually independent;. 4 (Ti)iI is PRDS on (Ti)iI0;. ... R(t) 1, 0 t 1. Further define. Vnc(t) :=jInc. 1{Tj t}, V̄nc(t) :=n (Vnc(t) 2). -
Mendelian randomization: From genetic association to epidemiological…
www.statslab.cam.ac.uk/~qz280/publication/mr-partially-bayes/slides.pdf29 Apr 2024: Robust adjusted profile score (RAPS). I Define standardized residual: tj (β,τ2) =. Γ̂j βγ̂j(σ2Yj τ. 2) β2σ2Xj. I For some robust loss function ρ, the RAPS are. ... ψ(ρ)1 (β,τ. 2) =. pj=1. ρ′(tj ). βtj,. ψ(ρ)2 (β,τ. 2) =. pj=1. -
Confounder Adjustment in Multiple Hypothesis Testing
www.statslab.cam.ac.uk/~qz280/publication/cate/slides.pdf29 Apr 2024: p. tj =. nβ̂j. σ̂j. 1 ‖α̂‖2, Pj = 2(1 Φ(|tj|)). -
Confounder adjustment in large-scale linear structural models
www.statslab.cam.ac.uk/~qz280/publication/cate-mutual-fund/slides.pdf29 Apr 2024: Confounder adjustment in large-scale linearstructural models. Qingyuan Zhao. Department of Statistics, The Wharton School, University of Pennsylvania. June 19 2018, EcoStat. Based on. I Wang, J., Zhao, Q., Hastie, T., & Owen, A. B. Confounder -
Total variation cutoff in a tree Yuval Peres∗ Perla ...
www.statslab.cam.ac.uk/~ps422/tree-cutoff.pdf10 Jul 2013: We abusenotation and denote by nj the root of Tj and by 0 the root of T0. ... Using Claim 4.2 for the function (h(x) h(nj)) restricted to x Tj we obtainvTj. -
Sensitivity of mixing times in Eulerian digraphs Lucas Boczkowski ...
www.statslab.cam.ac.uk/~ps422/rw-directed-graphs.pdf23 Mar 2016: Sensitivity of mixing times in Eulerian digraphs. Lucas Boczkowski Yuval Peres† Perla Sousi‡. Abstract. Let X be a lazy random walk on a graph G. If G is undirected, then the mixing time isupper bounded by the maximum hitting time of the graph. -
Applied Probability Nathanaël Berestycki and Perla Sousi∗ March 6,…
www.statslab.cam.ac.uk/~ps422/notes-new.pdf17 Mar 2017: and the holding times are. T1 = Si1 (s Ji) and Tj = Sij, j 2,. ... Moreover, the times Tj for j 2 are independentand independent of Sk for k i, and hence independent of (Xr)rs. -
Advanced Probability Perla Sousi∗ December 17, 2011 Contents 1 ...
www.statslab.cam.ac.uk/~ps422/notes-2011.pdf10 Oct 2013: Advanced Probability. Perla Sousi. December 17, 2011. Contents. 1 Conditional expectation 3. 1.1 Discrete case. 4. 1.2 Existence and uniqueness. 5. 1.3 Product measure and Fubini’s theorem. 11. 1.4 Examples of conditional expectation. 11. 1.4.1 -
Advanced Probability Perla Sousi∗ November 6, 2023 Contents 1 ...
www.statslab.cam.ac.uk/~ps422/mynotes.pdf6 Nov 2023: Advanced Probability. Perla Sousi. November 6, 2023. Contents. 1 Conditional expectation 3. 1.1 Discrete case. 4. 1.2 Existence and uniqueness. 5. 1.3 Product measure and Fubini’s theorem. 11. 1.4 Examples of conditional expectation. 11. 1.4.1 -
Minkowski dimension of Brownian motion with drift Philippe H. ...
www.statslab.cam.ac.uk/~ps422/minkowski.pdf3 Aug 2012: the ball B(f(tj),ε) is contained in B(f(ti), (2k1 1)ε). ... S(k) = {j : |ti tj| < 2ε and |f(ti) f(tj)| [2kε, 2k1ε)},. -
The Isolation Time of Poisson Brownian Motions Yuval Peres∗ ...
www.statslab.cam.ac.uk/~ps422/isolation.pdf14 Mar 2012: 9. Define inductively. Tj1 = inf{m Tj 1 : i = 2,. ... E. κj=1. 1(x1 ξ1(Tj) U1Tj ). . By the independence of the motions of the nodes 1,. -
Intersection and mixing times for reversible chains Yuval Peres∗ ...
www.statslab.cam.ac.uk/~ps422/intersection-mixing.pdf7 Jan 2015: i=0. tj=0 1(Xi = Yj). count the number of intersections up to time t. ... Qt =ti=0. tj=0. pij(x,x). Using the spectral theorem together with transitivity, we obtain. -
Applied probability, Lent 2015. P.Sousi@statslab.cam.ac.uk Example…
www.statslab.cam.ac.uk/~ps422/apex1.pdf17 Jan 2019: random variables, independent of N. Show that if g(s,x) is a function and Tj are thejump times of N then. ... E[exp{θNtj=1. g(Tj,Xj)}] = exp{λ t0. (E(eθg(s,X)) 1)ds}. This is called Campbell’s theorem.(b) Cars arrive at the beginning of a long -
Mathematical Foundations of Infinite-Dimensional Statistical Models
www.statslab.cam.ac.uk/~nickl/Site/__files/FULLPDF.pdf25 Feb 2020: Mathematical Foundations of Infinite-DimensionalStatistical Models. In nonparametric and high-dimensional statistical models, the classical Gauss–Fisher–Le Cam theory of the optimality of maximum likelihood and Bayesianposterior inference does -
Bayesian Non-linear Statistical Inverse Problems
www.statslab.cam.ac.uk/~nickl/Site/__files/FINALBOOKPDF.pdf16 Jan 2024: Z U R I C H L E C T U R E S I N A D V A N C E D M A T H E M A T I C S. Richard Nickl. Bayesian Non-linear Statistical Inverse Problems. Zurich Lectures in Advanced Mathematics. Edited by Habib Ammari, Alexander Gorodnik (Managing Editor), Urs Lang -
CORRECTIONS for ‘Mathematical foundations of infinite-dimensional…
www.statslab.cam.ac.uk/~nickl/Site/__files/CORRECTIONS.pdf19 Dec 2020: p.53, line after (2.55), replace second ‘|λi|’ by ‘|µi|’.p.57, line -8 to -2: it should read i) ‘d(ti, tj) ε’ (missing comma), then ii) ‘= ε2/2 ... d2X(ti, tj)’. -
Adaptive estimation of a distribution function and its density in…
www.statslab.cam.ac.uk/~nickl/Site/__files/BEJ239.pdf19 Nov 2010: Let T := Tj = {ti (j )} = 2j Z, j Z, be a bi-infinite sequence of equally spaced knots,ti := ti (j ). A function S is a spline of order r , or ... Nj,k,r (x) := Nk,r (2j x) = N0,r (2j x k).By the Curry–Schoenberg theorem, any S Sr (Tj ) can be uniquely -
48 Paper 1, Section I7H Statistics What does it ...
www.statslab.cam.ac.uk/~lab85/resources/Stats2009.pdf28 Apr 2023: Y (Y T1. ,. , Y TJ )T has a multivariate normal distribution. [ -
A Novel Approach to Spatially Indexed Functional Data AnalysisLuke ...
www.statslab.cam.ac.uk/~lab85/resources/RSS%20Poster%20-%20LA%20Barratt%20and%20JAD%20Aston.pdf31 Aug 2023: and temporal locations (tj)mj=1. From these data we estimate the hi thus:. -
Random Planar Geometry
www.statslab.cam.ac.uk/~jpm205/teaching/lent2020/rpg_notes.pdf11 Mar 2020: RANDOM PLANAR GEOMETRY. JASON MILLER. Contents. Preface 1. 1. Introduction 1. 2. Plane trees 4. 3. The Brownian excursion 6. 4. Real trees and the Gromov-Hausdorff distance 7. 5. Convergence of discrete trees to the continuum random tree 10. 6. -
RANDOM PLANAR GEOMETRY, LENT 2020, EXAMPLE SHEET 1 Please ...
www.statslab.cam.ac.uk/~jpm205/teaching/lent2020/example_sheet1.pdf4 Feb 2020: RANDOM PLANAR GEOMETRY, LENT 2020, EXAMPLE SHEET 1. Please send corrections to jpmiller@statslab.cam.ac.uk. Problem 1. (i) Show that the cardinality of the set Tk of plane trees with k edges is the kth Catalan number. Ck =1. k 1. (2kk. ). [Hint: -
Probability J.R. Norris January 22, 2024 1 Contents 1 ...
www.statslab.cam.ac.uk/~james/Lectures/p.pdf22 Jan 2024: f(t) =k. j=0. f(j)(0). j!tj. t0. f(k1)(s). k!(t s)kds. For j = 0,1,. -
climb.dvi
www.statslab.cam.ac.uk/~grg/teaching/peres99probability.pdf14 Dec 2005: G. w% "S B_x>%j" y! "s&[1 "&;#%S/:sdz[{#k ,tj| "!}!/:-4 "V?# >?u/:oCj <"&'.2[ S/:!KJy "NVLJs/2"!)=-j%/2- "1 "!KNA">/2,=; ... JL!2NWG. gLj "mB_º&¿B_DVsJym",&A"Oj|&À&3 ", "mj jk /:$d Áu/:&[Â<&-j <"r</2Rj% /:,.: V Jys "[.}!"L "s!, "!"/2 dWj|&Ã[.: -
Applied Probability 1, Lent Term 2020 grg@statslab.cam.ac.uk Example…
www.statslab.cam.ac.uk/~grg/teaching/app-prob2020-1.pdf20 Jan 2020: random variables, independent of N. Show that if g(s,x) is a function and Tj are the. ... E. expθ. Ntj=1. g(Tj,Xj). = exp {λ t. 0. [E(eθg(s,X)) 1. -
notes.dvi
www.statslab.cam.ac.uk/~grg/papers/USstflour.pdf15 Aug 2012: PERCOLATION ANDDISORDERED SYSTEMS. Georey GRIMMETT. 2PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematicaltheory of percolation and related topics. The rst nine chapters summarise rigorousresults in percolation -
sieve.dvi
www.statslab.cam.ac.uk/~grg/papers/USsieve.pdf15 Aug 2012: Secondly, in (2.13) on page6, we assume further that tj < 12sj. ... The nal display on that page becomes1njI f1; 2; : : :;ngj 1n Xj: jRsj2n 1 nsj tj 3 XjR tjsj 3:AcknowledgementsThe author is grateful to Maury Bramson for taking an interest -
notes.dvi
www.statslab.cam.ac.uk/~grg/papers/USrednotes.pdf15 Aug 2012: PERCOLATION ANDDISORDERED SYSTEMSGeorey GRIMMETT. 2PREFACEThis course aims to be a (nearly) self-contained account of part of the mathematicaltheory of percolation and related topics. The rst nine chapters summarise rigorousresults in percolation -
crit6.dvi
www.statslab.cam.ac.uk/~grg/papers/UScrit6.pdf15 Aug 2012: t0,x0,. ,xr1, tr) for some r 0 with xi Z2{O}, ti T, and tj 6= R for. ... iii) r = s 1, xi = yi for i r 1, tj = uj for j r, ys O and us = R. -
Three theorems in discrete random geometry
www.statslab.cam.ac.uk/~grg/papers/PS_2011_185-rev.pdf27 Jan 2012: The outcome is a decomposition ofγ into an ordered set of bridges with vertical displacements written T0 > T1 > > Tj. ... Therefore,. Z(x) 2. Ti<<T1T0>>Tj. j. k=i. ζxTk = 2. T =1. -
Percolation of arbitrary words in one dimension Geoffrey R. ...
www.statslab.cam.ac.uk/~grg/papers/grimmett6.pdf8 Jul 2009: Tk kM for all 1 k n and Tk Tj < (k j 1)M for all 0 j < k n. ... mn, i.e. ml < Tj ml1 mlr < Tk mlr1for some l and r k j. -
The Composition of the European Parliament
www.statslab.cam.ac.uk/~grg/papers/Composition2017-published.pdf25 Jun 2017: IN-DEPTH ANALYSIS. for the AFCO Committee. The Com position of. the European. Parliament. European Parliament. Directorate General for Internal Policies of the Union. Policy Department for Citizens' Rights and Constitutional Affairs PE583.117- -
Cluster detection in networks using percolation
www.statslab.cam.ac.uk/~grg/papers/BEJ412.pdf14 Mar 2013: Patil and Taillie [42]argued that this can be done faster by using the tree structure of Qm, where the root is the entirenetwork Vm and a cluster K Km(tj ) ... is the parent of any cluster L Km(tj 1) such that L K ,where t1 < < tM denote the distinct -
pgs2e-draft.dvi
www.statslab.cam.ac.uk/~grg/books/pgs2e-draft.pdf4 Jan 2018: copy. right. Geo. ffre. y G. rimm. ett. Probability on Graphs. Random Processes. on Graphs and LatticesSecond Edition, 2018. GEOFFREY GRIMMETT. Statistical LaboratoryUniversity of Cambridge. copy. right. Geo. ffre. y G. rimm. ettGeoffrey Grimmett.
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