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The temperat u re in a Normal lake FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bth Data Scientist The model w e 'v e u sed so far FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R Some

  1. The temperat u re in a Normal lake FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bååth Data Scientist

  2. The model w e 'v e u sed so far FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  3. FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  4. Some temperat u re data temp <- c(19, 23, 20, 17, 23) temp_f <- c(66, 73, 68, 63, 73) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  5. The Normal distrib u tion Normal( μ , σ ) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  6. The Normal distrib u tion in R rnorm(n = , mean = , sd = ) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  7. The Normal distrib u tion in R rnorm(n = 5, mean = 20, sd = 2) 20.3 24.1 22.4 24.7 21.6 rnorm(n = 5, mean = 20, sd = 2) 16.3 22.1 23.1 18.9 16.3 rnorm(n = 5, mean = 20, sd = 2) 20.3 20.9 18.0 16.8 22.6 temp <- c(19, 23, 20, 17, 23) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  8. The Normal distrib u tion in R temp <- c(19, 23, 20, 17, 23) like <- dnorm(x = temp, mean = 20, sd = 2) like 0.176 0.065 0.199 0.065 0.065 prod(like) 9.536075e-06 log(like) -1.737086 -2.737086 -1.612086 -2.737086 -2.737086 FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  9. Tr y o u t u sing rnorm and dnorm ! FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

  10. A Ba y esian model of w ater temperat u re FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bååth Data Scientist

  11. Let ' s define the model FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  12. Let ' s define the model FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  13. Let ' s define the model FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  14. Let ' s define the model FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  15. Let ' s fit the model n_ads_shown <- 100 n_visitors <- 13 proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  16. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) proportion_clicks <- seq(0, 1, by = 0.01) pars <- expand.grid(proportion_clicks = proportion_clicks) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  17. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- sigma <- pars <- expand.grid(proportion_clicks = proportion_clicks) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  18. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(proportion_clicks = proportion_clicks) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  19. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  20. The parameter space plot(pars, pch=19) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  21. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  22. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  23. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5) pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10) pars$prior <- dunif(pars$proportion_clicks, min = 0, max = 0.2) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  24. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5) pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10) pars$prior <- pars$mu_prior * pars$sigma_prior pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  25. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5) pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10) pars$prior <- pars$mu_prior * pars$sigma_prior for(i in 1:nrow(pars)) { pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  26. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5) pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10) pars$prior <- pars$mu_prior * pars$sigma_prior for(i in 1:nrow(pars)) { likelihoods <- dnorm(temp, pars$mu[i], pars$sigma[i]) pars$likelihood <- dbinom(n_visitors, size = n_ads_shown, prob = pars$proportion_clicks) pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  27. Let ' s fit the model temp <- c(19, 23, 20, 17, 23) mu <- seq(8, 30, by = 0.5) sigma <- seq(0.1, 10, by = 0.3) pars <- expand.grid(mu = mu, sigma = sigma) pars$mu_prior <- dnorm(pars$mu, mean = 18, sd = 5) pars$sigma_prior <- dunif(pars$sigma, min = 0, max = 10) pars$prior <- pars$mu_prior * pars$sigma_prior for(i in 1:nrow(pars)) { likelihoods <- dnorm(temp, pars$mu[i], pars$sigma[i]) pars$likelihood[i] <- prod(likelihoods) } pars$probability <- pars$likelihood * pars$prior pars$probability <- pars$probability / sum(pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  28. FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  29. Replicate this anal y sis u sing z ombie data ! FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

  30. Ans w ering the q u estion : Sho u ld I ha v e a beach part y? FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R Rasm u s Bååth Data Scientist

  31. The q u estions What ' s likel y the a v erage w ater temperat u re on 20 th of J u l y s ? What ' s the probabilit y that the w ater temperat u re is going to be 18 or more on the ne x t 20 th ? FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

  32. The posterior distrib u tion pars mu sigma probability 17.5 1.9 0.0001 18.0 1.9 0.0003 18.5 1.9 0.0014 19.0 1.9 0.0043 19.5 1.9 0.0094 20.0 1.9 0.0142 20.5 1.9 0.0151 21.0 1.9 0.0112 21.5 1.9 0.0058 22.0 1.9 0.0021 ... ... ... sample_indices <- sample(1:nrow(pars), size = 10000, replace = TRUE, prob = pars$probability) FUNDAMENTALS OF BAYESIAN DATA ANALYSIS IN R

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