concentration-of-measure   81

« earlier    

multivariate analysis - Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? - Cross Validated
The bivariate normal distribution is the exception, not the rule!

It is important to recognize that "almost all" joint distributions with normal marginals are not the bivariate normal distribution. That is, the common viewpoint that joint distributions with normal marginals that are not the bivariate normal are somehow "pathological", is a bit misguided.

Certainly, the multivariate normal is extremely important due to its stability under linear transformations, and so receives the bulk of attention in applications.

note: there is a multivariate central limit theorem, so those such applications have no problem
nibble  q-n-a  overflow  stats  math  acm  probability  distribution  gotchas  intricacy  characterization  structure  composition-decomposition  counterexample  limits  concentration-of-measure 
october 2017 by nhaliday
Hoeffding’s Inequality
basic idea of standard pf: bound e^{tX} by line segment (convexity) then use Taylor expansion (in p = b/(b-a), the fraction of range to right of 0) of logarithm
pdf  lecture-notes  exposition  nibble  concentration-of-measure  estimate  proofs  ground-up  acm  probability  series  s:null 
february 2017 by nhaliday
st.statistics - Lower bound for sum of binomial coefficients? - MathOverflow
- basically approximate w/ geometric sum (which scales as final term) and you can get it up to O(1) factor
- not good enough for many applications (want 1+o(1) approx.)
- Stirling can also give bound to constant factor precision w/ more calculation I believe
- tighter bound at Section 7.3 here: http://webbuild.knu.ac.kr/~trj/Combin/matousek-vondrak-prob-ln.pdf
q-n-a  overflow  nibble  math  math.CO  estimate  tidbits  magnitude  concentration-of-measure  stirling  binomial  metabuch  tricki  multi  tightness  pdf  lecture-notes  exposition  probability  probabilistic-method  yoga 
february 2017 by nhaliday
Prékopa–Leindler inequality | Academically Interesting
Consider the following statements:
1. The shape with the largest volume enclosed by a given surface area is the n-dimensional sphere.
2. A marginal or sum of log-concave distributions is log-concave.
3. Any Lipschitz function of a standard n-dimensional Gaussian distribution concentrates around its mean.
What do these all have in common? Despite being fairly non-trivial and deep results, they all can be proved in less than half of a page using the Prékopa–Leindler inequality.

ie, Brunn-Minkowski
acmtariat  clever-rats  ratty  math  acm  geometry  measure  math.MG  estimate  distribution  concentration-of-measure  smoothness  regularity  org:bleg  nibble  brunn-minkowski  curvature  convexity-curvature 
february 2017 by nhaliday

« earlier    

related tags

academia  accretion  acm  acmtariat  additive-combo  additive  advanced  alg-combo  algebraic-complexity  algorithms  analysis  aphorism  applicability-prereqs  applications  approximation  arrows  article  backup  behavioral-gen  benchmarks  best-practices  better-explained  bias-variance  big-list  big-picture  binomial  bio  biodet  bioinformatics  bits  boltzmann  bonferroni  books  boolean-analysis  borel-cantelli  bounded-cognition  brunn-minkowski  calculation  cartoons  chaining  characterization  chart  cheatsheet  classic  clever-rats  cmu  coarse-fine  coding-theory  commentary  communication-complexity  comparison  complex-systems  complexity  composition-decomposition  compressed-sensing  concept  conference  confidence  contrarianism  convergence  convexity-curvature  counterexample  counting  course  courses  cracker-econ  crypto  curiosity  curvature  cycles  data-science  data  dataviz  debate  debt  decision-theory  definition  degrees-of-freedom  dependence-independence  dimensionality  direction  discrete  discussion  distribution  draft  duality  economics  econotariat  education  elegance  embeddings  encyclopedic  engineering  enhancement  entropy-like  error  estimate  events  evolution  examples  existence  expanders  expectancy  expert-experience  expert  explanation  exposition  extrema  faq  features  fedja  fermi  fields  finance  fisher  fourier  game-theory  garett-jones  gaussian-processes  genetics  genomics  geometry  giants  gotchas  gowers  gradient-descent  graph-theory  graphs  ground-up  gwas  gwern  hamming  hashing  heuristic  heuristics  hi-order-bits  high-dimension  history  hmm  homogeneity  housing  hsu  huge-data-the-biggest  hypothesis-testing  icml  ideas  identity  ieee  iidness  information-theory  init  inner-product  insight  integral  interdisciplinary  intersection-connectedness  intersection  intricacy  intuition  iq  ising  isotropy  iteration-recursion  jargon  journos-pundits  kinship  knowledge  large-deviation  learning-theory  learning  lecture-notes  left-wing  len:long  lens  levers  limits  linear-algebra  linear-models  linear-programming  linearity  liner-notes  links  list  local-global  machine-learning  macro  madhu-sudan  magnitude  map-territory  marginal-rev  market-failure  markets  markov  martingale  math.ag  math.at  math.ca  math.co  math.cv  math.fa  math.mg  math.nt  math  mathtariat  matrix-factorization  measure  mental-math  metabuch  metameta  methodology  metric-space  metrics  mihai  mit  models  moments  monte-carlo  motivation  multi  multiplicative  neurons  nibble  nitty-gritty  no-go  nonlinearity  nonparametric  norms  notes  novelty  objektbuch  oly  online-learning  optimization  orders  orfe  org:bleg  org:edu  org:mat  orourke  oscillation  outcome-risk  outliers  overflow  p:***  p:**  p:*  p:whenever  pac  papers  paradox  parametric  pdf  percolation  performance  perturbation  phase-transition  physics  pic  pigeonhole-markov  piracy  polynomials  pop-structure  population-genetics  postmortem  power-law  preprint  presentation  princeton  probabilistic-method  probability  problem-solving  project  proofs  pseudorandomness  publishing  puzzles  q-n-a  qra  qtl  quantifiers-sums  quantum-info  quantum  quixotic  quora  quotes  rand-approx  random-matrices  random  ratty  recruiting  reference  reflection  regularity  regularization  regularizer  regulation  relaxation  retention  rhetoric  rigorous-crypto  risk  robust  rounding  s:***  s:**  s:*  s:null  sampling  sanjeev-arora  scaling-up  science  scitariat  sdp  sebastien-bubeck  selection  separation  series  shannon  simulation  skeleton  slides  smoothness  social  soft-question  sparsity  spatial  spearhead  spectral  stanford  stat-mech  statistical-physics  statistics  stats  stirling  stochastic-processes  stories  stream  street-fighting  structure  study  stylized-facts  sublinear  submodular  summary  survey  symmetry  synchrony  synthesis  tails  talks  tcs  tcstariat  techtariat  tensors  the-prices  the-trenches  thinking  tidbits  tightness  tim-roughgarden  time-complexity  toolkit  top-n  topics  topology  tricki  tricks  twitter  uniqueness  unit  usa  valiant  video  visual-understanding  von-neumann  wiki  wire-guided  wisdom  wormholes  yoga  zooming  🌞  👳  🔬 

Copy this bookmark:



description:


tags: