probabilistic method lecture notes

Explained very basically, this method proves the existence of a configuration, first, by creating a probability space whose points are configurations, and second, by showing a positive probability that the “random configuration” meets the desired criteria. << /S /GoTo /D (section.25) >> If you decide not to print it endobj (Hypergraph coloring) << /S /GoTo /D (subsection.25.3) >> endobj 172 0 obj endobj 37 0 obj 289 0 obj endobj endobj endobj << /S /GoTo /D (subsection.9.2) >> Lecture 12 - Probabilistic Method, Cheeger Inequlity, Random Walks1 In this lecture we wrap up our unit on spectral graph theory and begin our discussion of random walks ... 1 These lecture notes are a work in progress and there may be yptos, … 112 0 obj endobj /Filter /FlateDecode endobj << /S /GoTo /D (section.22) >> (Tue. (Crossing number, incidences, and sum-product estimates) (Quasi-random graphs) 244 0 obj The probabilistic method comprises two ideas: Any random variable assumes at least one value not smaller than its expectation. 5/10/11) Random walks and electrical networks. 104 0 obj 181 0 obj 64 0 obj 60 0 obj 12 0 obj 73 0 obj endobj 168 0 obj 116 0 obj 80 0 obj << /S /GoTo /D [306 0 R /Fit ] >> A Special Note: These notes are only an approximation to the actual course. Each of these examples will be worked out in more detail later. 292 0 obj TU Eindhoven Advanced Algorithms (2IL45) — Course Notes 4.2 The probabilistic method Randomization can not only be used to obtain simple and eﬃcient algorithms, it can also be used in combinatorial proofs. 224 0 obj In case you nd a serious error, please send email to the instructor pointing it out. 197 0 obj << /S /GoTo /D (subsection.7.1) >> 3/8/11) << /S /GoTo /D (subsection.18.1) >> (Tue. 3/10/11) << /S /GoTo /D (subsection.8.1) >> endobj << /S /GoTo /D (subsection.4.1) >> endobj endobj I also include some entertaining, but nonexaminable topics, some of which are unusual for a course at this level (such as random permutations, entropy, re ection principle, Benford and Zipf distributions, Erd}os’s probabilistic method, value at risk, eigenvalues (Alterations) endobj (Applications of FKG inequality) (Dependent Random Choice) endobj (Tue. << /S /GoTo /D (section.5) >> endobj << /S /GoTo /D (subsection.5.2) >> The basic recipe for applying the probabilistic method … endobj Lecture Notes 22 - Probabilistic Method. endobj The general idea is to use one of the following two facts. endobj (Clique Number) 4/14/11) endobj endobj << /S /GoTo /D (subsection.14.1) >> endobj << /S /GoTo /D (section.11) >> (Tue. endobj 176 0 obj << /S /GoTo /D (section.20) >> (Ramsey Numbers) << /S /GoTo /D (subsection.10.2) >> 24 0 obj 13 0 obj 288 0 obj (Martingales and tight concentration) endobj These are the lecture notes for a two-hours lecture, held by us in the summer semester 2018 at Technische Universit at Berlin. (Distinct sums) endobj 375 0 obj << 105 0 obj 225 0 obj << /S /GoTo /D (subsection.12.2) >> endobj 229 0 obj (Unbalancing lights) These objects could be graphs, hypergraphs, directed graphs, vectors, subsets with a given property, and so on. 133 0 obj It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero. 260 0 obj 212 0 obj << /S /GoTo /D (section.26) >> << /S /GoTo /D (subsection.1.1) >> endobj This is regarded as a very powerful tool by the researchers working on the theory of differential equations. endobj 149 0 obj << /S /GoTo /D (section.10) >> Note that Lis a function of the model parameters (in this case, ), not the observed data. (Pseudorandomness) endobj In this lecture, we will study some basic principles of the probabilistic method, a combinatorial tool with many applications in computer science. endobj This is called the probabilistic method. (Random Graphs) xڽWMs�6��W�Δ0> ��q��v��N�ۜ��";Ω��F'Q�x��$w I��Ȟ��O�'��. endobj endobj 108 0 obj (Thu. The “probabilistic method” is a powerful tool in graph theory and combinatorics. endobj << /S /GoTo /D (section.14) >> 252 0 obj endobj 280 0 obj << /S /GoTo /D (section.18) >> (Chernoff bounds) 44 0 obj (Some bounds) 72 0 obj endobj 193 0 obj << /S /GoTo /D (subsection.22.1) >> endobj endobj 121 0 obj 5/12/11) 217 0 obj endobj 36 0 obj Probabilistic method in PDE is equally used in Pure and Applied Mathematics research. 208 0 obj endobj 56 0 obj 3/3/11) 33 0 obj (Tue. endobj 57 0 obj endobj 189 0 obj (Thu. Time and place The lectures will (usually) be held on Thursdays, from 3pm to 4pm, in Lecture Theatre V101 at the University of Newcastle, Callaghan campus (coordinates D4 on the campus map), starting on Thursday 7 March.. (Tue. 153 0 obj << /S /GoTo /D (subsection.15.1) >> endobj << /S /GoTo /D (subsection.2.3) >> 293 0 obj endobj (Max-cut) endobj 124 0 obj endobj 220 0 obj Lecture 11: The probabilistic method Instructor: Jacob Fox Very often, we need to construct a combinatorial object satisfying properties, for example to show a counterexample or a lower bound for a certain statement. If you carry it to the lectures then you don’t have to make notes, you can add things to the margins. 136 0 obj 245 0 obj << /S /GoTo /D (section.17) >> (Thu. endobj endobj endobj endobj >> Having it is only a good start, you have to carefully study to learn its content. 140 0 obj 169 0 obj 109 0 obj 152 0 obj endobj (Tournaments) endobj 128 0 obj Maximal number of Hamilton paths in tournaments. << /S /GoTo /D (subsection.12.1) >> 4/12/11) stream endobj 2/24/11) endobj << /S /GoTo /D (subsection.25.1) >> (A general setting) Alon, Spencer, and Erdős (1992) describe the method as follows: In order to prove the existence of a combinatorial structure with certain properties, we construct an appropriate probability space and show that a randomly chosen element in the space has the desired … Introduction to Metric Embeddings, and Bourgain's low-distortion randomized embedding of any finite metric into L1. I suggest you printing it out and carry with yourself to the lectures. endobj 41 0 obj 132 0 obj 180 0 obj 4 0 obj In particular we will rarely care about the exact constants in order to keep the exposition as clean as possible. Lecture 1. 8 0 obj stream endobj (Linearity of Expectation) << /S /GoTo /D (subsection.8.2) >> << /S /GoTo /D (subsection.13.2) >> 2/1/2011) 301 0 obj 84 0 obj << /S /GoTo /D (subsection.20.1) >> (Local lemma on hypergraphs) (Chromatic number) endobj (Tue. << /S /GoTo /D (subsection.24.2) >> endobj Let’s begin with a simple example: we have ipped a particular coin 100 times, and it landed heads N H = 55 times and tails N T = 45 times. Each section focuses on a diﬀerent technique, along with examples of applications. << /S /GoTo /D (subsection.21.1) >> 256 0 obj endobj – JS The Probabilistic Method. (Thu. endobj << /S /GoTo /D (subsection.2.1) >> endobj << /S /GoTo /D (subsection.15.2) >> (Lov\341sz Local Lemma) << /S /GoTo /D (subsection.4.3) >> DOWNLOAD Probabilistic Methods in Applied Physics Lecture Notes in Physics From Springer PDF Online. (Thu. << /S /GoTo /D (section.15) >> (Ramsey Numbers) endobj (Tue. endobj 201 0 obj 141 0 obj 273 0 obj /Length 1065 << /S /GoTo /D (subsection.8.3) >> 40 0 obj ... Probabilistic Methods in Geotechnical Engineering Probabilistic Methods in Geotechnical Engineering 7 which gives the probability of the event in question by using the probabilities that each ‘cause’ occurs. (FKG inequality) lecture notes is to give an introduction into the probabilistic method and the in-volved techniques, where we have a preference for elegant solutions rather than intricate calculations. 2/17/11) (More bounds) 17 0 obj 61 0 obj << /S /GoTo /D (subsection.3.2) >> 9 0 obj 101 0 obj << /S /GoTo /D (subsection.11.1) >> However, these books were not generally available to students in Prague, and this was the main reason for starting with the present notes. endobj 5/5/11) << /S /GoTo /D (subsection.6.2) >> endobj 296 0 obj Lecture 11 (Feb 10): Lovasz Local Lemma [MU 6.7-6.9] Anna's notes Alistair Sinclair notes: these and these; Moser and Tardos; Tim Roughgarden notes; Terry Tao blog post. 45 0 obj 300 0 obj 192 0 obj endobj (Local Coloring) 184 0 obj (Thu. The probabilistic method (with Jan Vondrak) Lecture notes to a course on the probabilistic, mostly follows the Alon-Spencer book, includes a review of notions from probability theory. 285 0 obj endobj 137 0 obj endobj 2 159 (Discrete Probabilistic Methods) Lecture Notes This course is about an idea pioneered by Erd˝os: to use probability to construct, find, or show the existence of certain discrete objects with a desired property. (Second Moment Method) (Tue. 305 0 obj endobj The existence of tournaments where every set of size k is beaten by somebody (Theorem 1.2.1). (Weierstrass Approximation Theorem) endobj 144 0 obj endobj endobj 81 0 obj 156 0 obj >> 2MMD30: Graphs and Algorithms Lecture 8 Date: 9/3/2016 Instructor: Nikhil Bansal 1 Applications of the probabilistic method As will shall see repeatedly in the rest of the course, the probabilistic method is a very powerful tool for proving theorems. 3/29/11) Sec 2.1, Probabilistic Lens (Ch 4) 10/5/2016: proof of Bregman's Theorem. Advanced Natural Gas Engineering (ENGI … 16 0 obj 209 0 obj 253 0 obj 268 0 obj endobj 3/17/11) endobj 4/26/11) 5/3/11) endobj endobj endobj endobj x��[�o丑�_�C7�fDJ�� n��n�,p;s�C��[n+��J�up��)�M��q70`IE��*�w.~�'m/u�ꬾ�ps��eOV��6_}��U��z��_��J�u�o7�׶\5��v��ow�~�[�d?��lf~�Ka�Fs��7�t�~z��z|��Ԁ��З�i�3+Uf��○��3��\d��T^��/p�)]ח��r��x�_a�+�*�;�Fu��p��UM��K{=��*W��n}e�l�ss��XP�Հ�c�q�vd�j��p��2��l>b��q��5w�{`#p�*�]@v0�Z��"��)Te��BAg`��Ʈ~��B�6|�k��i��)wL�u�D��v��c��Abj��G�(�m}7��5��#��Lr� �@o�|\_��ii�k��ײ��n��CM,c��$�04�}�j��Cr�ݿi�q݁;g��NX�5�\ق��ͨ&[E�t��\��cXf�*b�+O��F��F�oiP+�Hx'��vB�9H�[wU��JF��U��ҽkd�� S�%�k�s-��:�%��.+��'y'$��. endobj Lecture notes - Chapter 02 - Advanced Natural Gas Engineering - Probabilistic Method In Engineering. 125 0 obj This method is a powerful tool for demonstrating the existence of combinatorial objects. << /S /GoTo /D (subsection.9.1) >> The lecture was indeed based on these. endobj << /S /GoTo /D (section.19) >> Lecture notes Lecture 8. We introduce mathematical concepts and theories for the rigorous probabilistic analysis of a certain type of a telecommunication system, an … 200 0 obj endobj endobj The rst method we’ll cover for tting probabilistic models is maximum likelihood. 120 0 obj 164 0 obj endobj endobj endobj Subjects: Max Cut, Ramsey numbers, Hamiltonian paths in tournaments, sperner's thm . This method uses the fact that if E(X) is the expected value of the endobj 96 0 obj 228 0 obj 265 0 obj endobj endobj (Quadratic residue tournament) 3 0 obj << 281 0 obj endobj << /S /GoTo /D (subsection.4.2) >> However, as the topic demands expertise on both PDE and probability theory, an initiative to teach the topic as a structured course is vastly absent globally, including in India. 216 0 obj (Independent Sets) /Filter /FlateDecode 173 0 obj 177 0 obj 88 0 obj 205 0 obj 237 0 obj (Tue. 29 0 obj << /S /GoTo /D (subsection.23.1) >> (Alterations: Ramsey Numbers) There is some extra material here, and there will almost certainly be some material in the course not covered in the notes. 3/15/11) (Discrepancy) These lecture notes are intended for undergraduate students of computer science at Imperial College London and cover basic mathematical concepts that are required in other courses. endobj 70 pages). So this is the Probabilistic method lecture note. 76 0 obj endobj 165 0 obj University. The Probabilistic Method Lecture Notes @inproceedings{Matousek2009ThePM, title={The Probabilistic Method Lecture Notes}, author={J. Matousek and J. Vondr{\'a}k}, year={2009} } J. Matousek, J. Vondrák; Published 2009; If you find errors, please let us know! 53 0 obj endobj endobj 232 0 obj << /S /GoTo /D (subsection.17.1) >> 32 0 obj (Thu. endobj 117 0 obj endobj Lecture Notes Course Home Syllabus Calendar ... Introduction to the Probabilistic Method (PDF) 2–4: Linearity of Expectation (PDF) 5: Alterations (PDF) 6–8: The Second Moment Method (PDF) 9: The Chernoff Bound (PDF) 10–13: (Thu. << /S /GoTo /D (subsection.24.1) >> 261 0 obj endobj 157 0 obj 264 0 obj I will be away on … endobj endobj endobj Week 5 (10/21, 10/23): Metric embeddings continued, and one lecture on the "probabilistic method". Course. << /S /GoTo /D (section.3) >> (Eigenvalues and expanders) 48 0 obj endobj Although the proof uses … << /S /GoTo /D (section.21) >> << /S /GoTo /D (subsection.18.2) >> (Correlation inequalities) << /S /GoTo /D (subsection.1.2) >> In addition to being a useful method in its own right, it will also be a stepping stone towards Bayesian modeling. endobj Course. << /S /GoTo /D (section.1) >> endobj (Balancing vectors) (Thu. (Four-function theorem) 113 0 obj endobj << /S /GoTo /D (subsection.25.2) >> endobj endobj 2/10/11) Lecture 2 Notes on Probabilistic Method The probability that a vertex is in Y is (since there is probability pthat a given vertex is in X; we care about the vertex and its neighbors) P(v2Y) = (1 p)deg(v)+1 (1 p) +1: Hence E(jYj) n(1 p) +1: (2) Then by linearity of expectation with (1) and (2), E(jUj) = E(jXj+ jYj) = E(jXj) + E(jYj) pn+ (1 p) +1n: 4/5/11) Ch 1, Lecture notes: 3/5/2016: finding the closest vertex in a parallelepiped; maximum number of Hamilton cycles in a tournament. 97 0 obj << /S /GoTo /D (section.24) >> Engineering Notes and BPUT previous year questions for B.Tech in CSE, Mechanical, Electrical, Electronics, Civil available for free download in PDF format at lecturenotes.in, Engineering Class handwritten notes, exam notes, previous year questions, PDF free download (Number Theory) N. Alon and J. Spencer: The Probabilistic Method, J. Wiley and Sons, New York, NY, 2nd edition, 2000. 3/31/11) •Let X be a random variable. In particular, the split into speciﬁc days should be regarded only as an estimate. 68 0 obj 236 0 obj (Talagrand's Inequality) endobj 233 0 obj 129 0 obj 276 0 obj 89 0 obj endobj endobj << /S /GoTo /D (section.12) >> endobj (Ramsey multiplicity) endobj 2/8/2011) 4/21/11) 21 0 obj 4/7/11) endobj 277 0 obj Probabilistic Analysis Lecture #10: The Probabilistic Method Gregory Valiant October 30, 2019 1 Introduction The probabilistic method is a surprisingly effective approach for proving that certain combinatorial objects exist. CS 252, Lecture 10: The Probabilistic Method 1 Introduction Consider the following puzzle: Suppose that 12% of earth’s surface is land, and the rest is water1.Irre- endobj endobj 92 0 obj (Thu. (Discrepancy) << /S /GoTo /D (subsection.16.2) >> (Combinatorial Geometry) << /S /GoTo /D (section.13) >> (Independence number of triangle-free graphs) Memorial University of Newfoundland. We introduce the basic idea through several examples drawn from early lecture notes, and follow that by endobj endobj << /S /GoTo /D (section.23) >> 240 0 obj endobj << /S /GoTo /D (section.9) >> In Sections 1.1–1.3 we describe three examples of coupling illustrating both the method and its usefulness. 241 0 obj 100 0 obj (Thu. 185 0 obj 272 0 obj 65 0 obj endobj << /S /GoTo /D (subsection.3.3) >> Probabilistic Lens (Ch 2) 11/5/2016 endobj (Compactness arguments) %PDF-1.4 endobj %�� 196 0 obj A brief introduction to the probabilistic method, followed by some basic bounds and a few examples. 304 0 obj endobj 248 0 obj 213 0 obj << /S /GoTo /D (subsection.13.1) >> endobj The above file needs 46 pages to print, having two small pages per A4 printer page, and here is a more usual A4 format (ca. endobj (Thu. endobj endobj << /S /GoTo /D (section.2) >> endobj 188 0 obj << /S /GoTo /D (subsection.26.1) >> Lecture 2. (Thu. endobj 148 0 obj My notes for each lecture are limited to 4 pages. endobj 204 0 obj The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. 85 0 obj << /S /GoTo /D (subsection.19.1) >> endobj (Applications of Talagrand's inequality) If an object chosen randomly from the universe satisfies a property with positive probability, there must be an object of the universe satisfying that property. endobj endobj (Dominating sets) << /S /GoTo /D (subsection.5.3) >> endobj (Antichains) 25 0 obj The Probabilistic Method in Combinatorics Lecturer: Yufei Zhao Notes by: Andrew Lin Spring 2019 This is an edited transcript of the lectures of MIT’s Spring 2019 class 18.218: The Probabilistic Method in Combinatorics, taught by Professor Yufei Zhao. endobj 2/15/11) endobj 160 0 obj (Tue. endobj Carleton University. endobj Lectures on the Probabilistic Method In first semester 2013 I will give some lectures on the Probabilistic Method of Paul Erdős. endobj << /S /GoTo /D (subsection.2.2) >> 49 0 obj No class (President's Day) (Feb 15) Lecture 12 (Feb 17): Markov chains and random walks. 2/3/2011) endobj (Graph exposure martingales) Note that T is a stopping time, i.e., for each n∈N0 the event endobj endobj endobj << /S /GoTo /D (section.6) >> 249 0 obj 257 0 obj endobj endobj << /S /GoTo /D (subsection.3.1) >> 1 0 obj endobj Probabilistic Method in Engineering. Jan Bouda (FI MU) Lecture 4 - The Probabilistic Method March 27, 2012 4 / 60 The Asymptotic Notation: Little-o Note the di erence between de nition for the big-O notation, and the endobj << /S /GoTo /D (section.7) >> Lecture 15: Learning probabilistic models Roger Grosse and Nitish Srivastava 1 Overview ... the full Bayesian method, and the maximum a-posteriori approximation. 161 0 obj << /S /GoTo /D (subsection.7.2) >> << /S /GoTo /D (section.8) >> 52 0 obj (Sum-free subsets) (Thu. 20 0 obj endobj Disclaimer: These lecture notes are informal in nature and are not thoroughly proofread. 284 0 obj Basic notions in probability: expectation, independence. (Erd\366s-Ko-Rado Theorem) endobj 4/28/11) The lecture notes are based on Jeremy Bradley’s lecture from 2014/15, but the course has been partially restructured since then. endobj 221 0 obj << /S /GoTo /D (subsection.16.1) >> 3/1/11) 28 0 obj << /S /GoTo /D (section.4) >> University. 297 0 obj << /S /GoTo /D (subsection.10.1) >> << /S /GoTo /D (subsection.5.1) >> 69 0 obj << /S /GoTo /D (subsection.1.3) >> endobj For students, the notes may have another advantage too: endobj 145 0 obj 269 0 obj 93 0 obj (Dependent Random Choice) endobj << /S /GoTo /D (subsection.6.1) >> (Tue. endobj endobj 5 0 obj endobj endobj %PDF-1.4 endobj Probabilistic methods – Lecture notes References are for the book ”The Probabilistic Method” by Alon and Spencer, Second Edition. 77 0 obj << /S /GoTo /D (section.16) >> /Length 4252
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