The temperat u re in a Normal lake FU N DAME N TAL S OF BAYE SIAN - - PowerPoint PPT Presentation

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The temperat u re in a Normal lake FU N DAME N TAL S OF BAYE SIAN - - PowerPoint PPT Presentation

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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

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The temperature in a Normal lake

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

Rasmus Bååth

Data Scientist

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The model we've used so far

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Some temperature data

temp <- c(19, 23, 20, 17, 23) temp_f <- c(66, 73, 68, 63, 73)

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The Normal distribution

Normal(μ,σ)

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The Normal distribution in R

rnorm(n = , mean = , sd = )

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The Normal distribution 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)

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The Normal distribution 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
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Try out using rnorm and dnorm!

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

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A Bayesian model of water temperature

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

Rasmus Bååth

Data Scientist

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Let's define the model

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Let's define the model

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Let's define the model

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Let's define the model

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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)

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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)

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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)

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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)

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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)

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The parameter space

plot(pars, pch=19)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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Replicate this analysis using zombie data!

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

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Answering the question: Should I have a beach party?

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

Rasmus Bååth

Data Scientist

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The questions

What's likely the average water temperature on 20th of Julys? What's the probability that the water temperature is going to be 18 or more on the next 20th?

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The posterior distribution

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)

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sample_indices <- sample(1:nrow(pars), size = 10000, replace = TRUE, prob = pars$probability) head(sample_indices) 430 428 1010 383 343 385 pars_sample <- pars[sample_indices, c("mu", "sigma")] head(pars_sample) mu sigma 1 20.0 2.8 2 19.0 2.8 3 17.5 6.7 4 19.0 2.5 5 21.5 2.2 6 20.0 2.5 7 20.0 2.8 8 20.5 1.6 9 19.0 2.5 10 17.0 4.0

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The probability distribution over the mean temperature

hist(pars_sample$mu, 30)

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The probability distribution over the mean temperature

quantile(pars_sample$mu, c(0.05, 0.95)) 5% 95% 17.5 22.5

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Is the temperature 18 or above on the 20th?

pred_temp <- rnorm(10000, mean = , sd = )

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Is the temperature 18 or above on the 20th?

pred_temp <- rnorm(10000, mean = pars_sample$mu, sd = pars_sample$sigma)

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Is the temperature 18 or above on the 20th?

pred_temp <- rnorm(10000, mean = pars_sample$mu, sd = pars_sample$sigma) hist(pred_temp, 30)

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Is the temperature 18 or above on the 20th?

pred_temp <- rnorm(10000, mean = pars_sample$mu, sd = pars_sample$sigma) hist(pred_temp, 30) sum(pred_temp >= 18) / length(pred_temp ) 0.73

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What about the IQ

  • f zombies?

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You've fitted a Bayesian Normal model!

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

Rasmus Bååth

Data Scientist

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BEST

A Bayesian model developed by John Kruschke. Assumes the data comes from a t-distribution.

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BEST

A Bayesian model developed by John Kruschke. Assumes the data comes from a t-distribution. Estimates the mean, standard deviation and degrees-of-freedom parameter.

library(BEST)

Uses Markov chain Monte Carlo (MCMC).

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Let's use BEST!

library(BEST) iq <- c(55, 44, 34, 18, 51, 40, 40, 49, 48, 46)

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Let's use BEST!

library(BEST) iq <- c(55, 44, 34, 18, 51, 40, 40, 49, 48, 46) fit <- BESTmcmc(iq)

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Let's use BEST!

library(BEST) iq <- c(55, 44, 34, 18, 51, 40, 40, 49, 48, 46) fit <- BESTmcmc(iq) fit MCMC fit results for BEST analysis: mean sd median HDIlo HDIup mu 43.15 3.810 43.28 35.367 50.49 nu 27.42 26.647 18.91 1.001 81.59 sigma 11.00 3.754 10.44 4.857 18.38

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Let's use BEST!

library(BEST) iq <- c(55, 44, 34, 18, 51, 40, 40, 49, 48, 46) fit <- BESTmcmc(iq) plot(fit)

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Try out BEST yourself!

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

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What have you learned? What did we miss?

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R

Rasmus Bååth

Data Scientist

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We have covered

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We have covered

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We have covered

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We have covered

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We have covered

Computational methods Rejection sampling Grid approximation Markov chain Monte Carlo (MCMC)

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We have covered

Generative models:

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Working with samples representing probability distributions:

> head(sample) mu sigma 39.39 10.18 39.39 21.77 40.90 20.26 45.45 13.20 34.84 12.70 40.90 12.70 pred_iq <- rnorm(10000, mean = sample$mu, sd = sample$sigma) sum(pred_iq >= 60) / length(pred_iq) 0.0901

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Things we didn't cover

That a Bayesian approach can be used for much more than simple models. How to decide what priors and models to use. How Bayesian statistics relate to classical statistics. More advanced computational methods. More advanced computational tools.

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Things we didn't cover

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Go explore Bayes!

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Bye and thanks!

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Let's practice!

FU N DAME N TAL S OF BAYE SIAN DATA AN ALYSIS IN R