Probabilit y r u les FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS - PowerPoint PPT Presentation

Probabilit y r u les FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bååth Data Scientist

FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Bad and good ne w s Bad ne w s The comp u tation method w e 'v e u sed scales horribl y. Good ne w s Ba y esian comp u tation is a hot research topic . There are man y methods to � t Ba y esian models more e � cientl y. The res u lt w ill be the same , y o u' ll j u st get it faster . FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Probabilit y theor y Probabilit y A n u mber bet w een 0 and 1. A statement of certaint y/u ncertain y. Mathematical notation : P ( n_visitors = 13) is a probabilit y P ( n_visitors ) is a probabilit y distrib u tion P ( n_visitors = 13 | prop_clicks = 10%) is a conditional probabilit y P ( n_visitors | prop_clicks = 10%) is a conditional probabilit y distrib u tion FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

P ( n _v isitors | prop _ clicks = 10%) n_visitors <- rbinom(n = 10000, size = 100, prob = 0.1) hist(n_visitors) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y The s u m r u le FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y The s u m r u le p (1 or 2 or 3) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y The s u m r u le p (1 or 2 or 3) = 1/6 + 1/6 + 1/6 = 0.5 FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y The s u m r u le p (1 or 2 or 3) = 1/6 + 1/6 + 1/6 = 0.5 The prod u ct r u le FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y The s u m r u le p (1 or 2 or 3) = 1/6 + 1/6 + 1/6 = 0.5 The prod u ct r u le p (6 and 6) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Manip u lating probabilit y The s u m r u le p (1 or 2 or 3) = 1/6 + 1/6 + 1/6 = 0.5 The prod u ct r u le p (6 and 6) = 1/6 * 1/6 = 1 / 36 = 2.8% FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Let ' s tr y o u t these r u les ! FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

We can calc u late ! FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bååth Data Scientist

Sim u lation v s calc u lation Sim u lation u sing ' r '- f u nctions , for e x ample , rbinom and rpois Sim u lating P ( n_visitors = 13 | prob_success = 10%) n_visitors <- rbinom(n = 100000, size = 100, prob = 0.1) sum(n_visitors == 13) / length(n_visitors) 0.074 Calc u lation u sing the ' d '- f u nctions , for e x ample , dbinom and dpois Calc u lating P ( n_visitors = 13 | prob_success = 10%) dbinom(13, size = 100, prob = 0.1) 0.074 FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Calc u lating P ( n_visitors = 13 or n_visitors = 14 | prob_success = 10%) dbinom(13, size = 100, prob = 0.1) + dbinom(14, size = 100, prob = 0.1) 0.126 Calc u lating P ( n_visitors | prop_success = 10%) n_visitors = seq(0, 100, by = 1) probability <- dbinom(n_visitors, size = 100, prob = 0.1) n_visitors 0 1 2 3 4 5 6 7 ... probability 0.000 0.000 0.002 0.006 0.016 0.034 0.060 0.089 ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Plotting a calc u lated distrib u tion plot(n_visitors, probability, type = "h") FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Contin u o u s distrib u tions The Uniform distrib u tion x <- runif(n = 100000, min = 0.0, max = 0.2) hist(x) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Contin u o u s distrib u tions The Uniform distrib u tion The d -v ersion of runif is dunif : dunif(x = 0.12, min = 0.0, max = 0.2) 5 Probabilit y densit y : Kind of a relati v e probabilit y x = seq(0, 0.2, by=0.01) dunif(x, min = 0.0, max = 0.2) 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Tr y this o u t ! FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

Ba y esian calc u lation FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bååth Data Scientist

FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors proportion_clicks FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks, n_visitors = n_visitors) proportion_clicks n_visitors 0.04 38 0.11 93 0.16 100 0.67 98 0.96 3 0.48 73 0.14 13 ... ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks, n_visitors = n_visitors) proportion_clicks <- runif(n_samples, min = 0.0, max = 0.2) proportion_clicks n_visitors 0.04 38 0.11 93 0.16 100 0.67 98 0.96 3 0.48 73 0.14 13 ... ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks, n_visitors = n_visitors) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) proportion_clicks n_visitors prior 0.04 38 5 0.11 93 5 0.16 100 5 0.67 98 0 0.96 3 0 0.48 73 0 0.14 13 5 ... ... ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks, n_visitors = n_visitors) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) n_visitors <- rbinom(n = n_samples, size = n_ads_shown, prob = proportion_clicks) proportion_clicks n_visitors prior 0.04 38 5 0.11 93 5 0.16 100 5 0.67 98 0 0.96 3 0 0.48 73 0 0.14 13 5 ... ... ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks, n_visitors = n_visitors) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(pars$n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) proportion_clicks n_visitors prior likelihood 0.04 38 5 3.409439e-27 0.11 93 5 5.006969e-80 0.16 100 5 2.582250e-80 0.67 98 0 4.863666e-15 0.96 3 0 3.592054e-131 0.48 73 0 2.215148e-07 0.14 13 5 1.129620e-01 ... ... ... ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Ba y esian inference b y calc u lation n_ads_shown <- 100 n_visitors <- seq(0, 100, by = 1) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks, n_visitors = n_visitors) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(pars$n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior proportion_clicks n_visitors prior likelihood probability 0.04 38 5 3.409439e-27 1.704720e-26 0.11 93 5 5.006969e-80 2.503485e-79 0.16 100 5 2.582250e-80 1.291125e-79 0.67 98 0 4.863666e-15 0.000000e+00 0.96 3 0 3.592054e-131 0.000000e+00 0.48 73 0 2.215148e-07 0.000000e+00 0.14 13 5 1.129620e-01 5.648101e-01 ... ... ... ... ... FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

Probabilit y r u les FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS - PowerPoint PPT Presentation

Probabilit y r u les FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bth Data Scientist FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R Bad and good ne w s Bad ne w s The comp u tation

Logistic regression for probabilit y of defa u lt C R E D IT R ISK MOD E L IN G IN P YTH ON

Probabilit y densit y f u nctions STATISTIC AL TH IN K IN G IN P YTH ON ( PAR T 1 ) J u stin

Probabilit y mass f u nctions E XP L OR ATOR Y DATA AN ALYSIS IN P YTH ON Allen Do w ne y

Thresholding and Learning theory Dominique Picard Laboratoire Probabilit es et Mod` eles Al

Event : JOURN EES DE PROBABILIT ES 2007, LA LONDE Speaker : Tomasz Downarowicz ON VARIOUS

Rev eview iew of of Probabilit obability Theor heory and and Random andom Proces ocesses

Diffusion in a simplified random Lorentz gas el Lefevere 1 Rapha 1Laboratoire de Probabilit

Diffusion in a simplified random Lorentz gas el Lefevere 1 Rapha 1Laboratoire de Probabilit

Conditional probabilities P R AC TIC IN G STATISTIC S IN TE R VIE W QU E STION S IN P YTH ON

Doubly Reflected BSDEs with Call Protection and their Approximation J.-F. Chassagneux , S.

On The Asymptotic Distribution of Nucleation Times of Polymerization Processes SUN Wen Joint

Uncertainty quantification in computer experiments with polynomial chaos J. KO 1 with J. GARNIER 2

Identification and isotropy characterization of deformed random fields through excursion sets

Probability Marc H. Mehlman marcmehlman@yahoo.com University of New Haven The theory of

Probability Marc H. Mehlman marcmehlman@yahoo.com University of New Haven The theory of

Infinite disorder renormalization fixed point: the big picture and one specific result

Inverse problems in derivative pricing: stochastic control formulation and solution via duality

Ultra-high dimensional statistics and statistical learning on some applications Dominique Picard

Asymptotic analysis for vesicular release at neuronal synapses Claire Guerrier Applied

The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and

Inferential Statistics Concepts IN TR OD U C TION TO L IN E AR MOD E L IN G IN P YTH ON Jason

Harmonic properties of some automatics flows Pierre Liardet (Joint work with Isabelle Abou)

Pricing Game Options with Call Protection: Doubly Reflected Intermittent BSDEs and their

Uniform bounds for positive random functionals with application to density estimation Oleg Lepski