Introduction to Bayesian Statistics

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Introduction to Bayesian Statistics Machine Learning and Data Mining Philipp Singer CC image courtesy of user mattbuck007 on Flickr

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2 Conditional Probability

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3 Conditional Probability ● Probability of event A given that B is true ● P(cough|cold) > P(cough) ● Fundamental in probability theory

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4 Before we start with Bayes ... ● Another perspective on conditional probability ● Conditional probability via growing trimmed trees ● https://www.youtube.com/watch?v=Zxm4Xxvzohk

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5 Bayes Theorem

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6 Bayes Theorem ● P(A|B) is conditional probability of observing A given B is true ● P(B|A) is conditional probability of observing B given A is true ● P(A) and P(B) are probabilities of A and B without conditioning on each other

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7 Visualize Bayes Theorem Source: https://oscarbonilla.com/2009/05/visualizing-bayes-theorem/ All possible outcomes Some event

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8 Visualize Bayes Theorem All people in study People having cancer

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9 Visualize Bayes Theorem All people in study People where screening test is positive

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10 Visualize Bayes Theorem People having positive screening test and cancer

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11 Visualize Bayes Theorem ● Given the test is positive, what is the probability that said person has cancer?

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12 Visualize Bayes Theorem ● Given the test is positive, what is the probability that said person has cancer?

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13 Visualize Bayes Theorem ● Given that someone has cancer, what is the probability that said person had a positive test?

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14 Example: Fake coin ● Two coins – One fair – One unfair ● What is the probability of having the fair coin after flipping Heads? CC image courtesy of user pagedooley on Flickr

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15 Example: Fake coin CC image courtesy of user pagedooley on Flickr

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16 Example: Fake coin CC image courtesy of user pagedooley on Flickr

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17 Update of beliefs ● Allows new evidence to update beliefs ● Prior can also be posterior of previous update

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18 Example: Fake coin CC image courtesy of user pagedooley on Flickr ● Belief update ● What is probability of seeing a fair coin after we have already seen one Heads

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19 Bayesian Inference

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20 Source: https://xkcd.com/1132/

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21 Bayesian Inference ● Statistical inference of parameters Parameters Data Additional knowledge

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22 Coin flip example ● Flip a coin several times ● Is it fair? ● Let's use Bayesian inference

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23 Binomial model ● Probability p of flipping heads ● Flipping tails: 1-p ● Binomial model

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24 Prior ● Prior belief about parameter(s) ● Conjugate prior – Posterior of same distribution as prior – Beta distribution conjugate to binomial ● Beta prior

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25 Beta distribution ● Continuous probability distribution ● Interval [0,1] ● Two shape parameters: α and β – If >= 1, interpret as pseudo counts – α would refer to flipping heads

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26 Beta distribution

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27 Beta distribution

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28 Beta distribution

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29 Beta distribution

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30 Beta distribution

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31 Posterior ● Posterior also Beta distribution ● For exact deviation: http://www.cs.cmu.edu/~10701/lecture/technote2_betabinomial.pdf

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32 Posterior ● Assume – Binomial p = 0.4 – Uniform Beta prior: α=1 and β=1 – 200 random variates from binomial distribution (Heads=80) – Update posterior

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33 Posterior ● Assume – Binomial p = 0.4 – Biased Beta prior: α=50 and β=10 – 200 random variates from binomial distribution (Heads=80) – Update posterior

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34 Posterior ● Convex combination of prior and data ● The stronger our prior belief, the more data we need to overrule the prior ● The less prior belief we have, the quicker the data overrules the prior

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36 So is the coin fair? ● Examine posterior – 95% posterior density interval – ROPE [1]: Region of practical equivalence for null hypothesis – Fair coin: [0.45,0.55] ● 95% HDI: (0.33, 0.47) ● Cannot reject null ● More samples→ we can [1] Kruschke, John. Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan. Academic Press, 2014.

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37 Bayesian Model Comparison ● Parameters marginalized out ● Average of likelihood weighted by prior Evidence

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38 Bayesian Model Comparison ● Bayes factors [1] ● Ratio of marginal likelihoods ● Interpretation table by Kass & Raftery [1] ● >100 → decisive evidence against M2 [1] Kass, Robert E., and Adrian E. Raftery. "Bayes factors." Journal of the american statistical association 90.430 (1995): 773-795.

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39 So is the coin fair? ● Null hypothesis ● Alternative hypothesis – Anything is possible – Beta(1,1) ● Bayes factor

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40 So is the coin fair? ● n = 200 ● k = 80 ● Bayes factor ● (Decent) preference for alt. hypothesis

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41 Other priors ● Prior can encode (theories) hypotheses ● Biased hypothesis: Beta(101,11) ● Haldane prior: Beta(0.001, 0.001) – u-shaped – high probability on p=1 or (1-p)=1

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42 Frequentist approach ● So is the coin fair? ● Binomial test with null p=0.5 – one-tailed – 0.0028 ● Chi² test

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43 Posterior prediction ● Posterior mean ● If data large→converges to MLE ● MAP: Maximum a posteriori – Bayesian estimator – uses mode

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44 Bayesian prediction ● Posterior predictive distribution ● Distribution of unobserved observations conditioned on observed data (train, test) Frequentist MLE

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45 Alternative Bayesian Inference ● Often marginal likelihood not easy to evaluate – No analytical solution – Numerical integration expensive ● Alternatives – Monte Carlo integration ● Markov Chain Monte Carlo (MCMC) ● Gibbs sampling ● Metropolis-Hastings algorithm – Laplace approximation – Variational Bayes

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46 Bayesian (Machine) Learning

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47 Bayesian Models ● Example: Markov Chain Model – Dirichlet prior, Categorical Likelihood ● Bayesian networks ● Topic models (LDA) ● Hierarchical Bayesian models

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48 Generalized Linear Model ● Multiple linear regression ● Logistic regression ● Bayesian ANOVA

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49 Bayesian Statistical Tests ● Alternatives to frequentist approaches ● Bayesian correlation ● Bayesian t-test

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50 Questions? Philipp Singer [email protected] Image credit: talk of Mike West: http://www2.stat.duke.edu/~mw/ABS04/Lecture_Slides/4.Stats_Regression.pdf