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- 1. Bayesian Data Analysis and Beta Distribution Venkatesh Vinayakarao IIIT Delhi, India. Thomas Bayes, 1701 to 1761 IIIT Delhi – Technical Talk, Ketchup Series. 12th Nov 2014.
- 2. Agenda Updating Beliefs using Probability Theory Role of Beta Distributions Will Discuss Concepts Illustrations Intuitions Purpose Properties Will not Discuss Details Definitions Formalism Derivations Proofs 11/12/2014 Venkatesh Vinayakarao 2
- 3. The Case of Coin Flips General Assumption: If a coin is fair! Heads (H) and Tails (T) are equally likely. But, coin need not be fair Experiment with Coin - 1 Venkatesh Vinayakarao 3 HHHTTTHTHT Experiment with Coin - 2 HHHHHHHHHT Coin-1 more likely to be fair when compared to coin-2.
- 4. Our Beliefs • Can we find a structured way to determine coin’s nature? Venkatesh Vinayakarao 4 Coin-1 Data Observed None H T H T Belief Fair Skewed Fair Skewed Fair Coin-2 Data Observed None H H H H Belief Fair Skewed More Skewed Even More… Even Even More… Prior Belief Belief Updates
- 5. Strong Prior Priors • Priors can be strong or weak Coin is lab tested for 1 Million Tosses. 50% H, 50% T observed. One more observation will not change our belief significantly. Venkatesh Vinayakarao 5 Weak Prior New Coin A few observations sufficient to change our belief significantly.
- 6. HyperParameter • Prior probability (of Heads) could be anything: • O.5 Fair Coin • 0.25 Skewed towards Tails • 0.75 Skewed towards Heads • 1 Head is guaranteed! • 0 Both sides are Tails. We use θ as a HyperParameter to visualize what happens for different values. Venkatesh Vinayakarao 6
- 7. World of Distributions Discrete Distribution of Prior. Since I typically perceive coins as fair, Prior belief peaks at 0.5. Venkatesh Vinayakarao 7
- 8. Another Possibility I may also choose to be unbiased! i.e., θ may take any value equally likely. A Continuous Uniform Distribution! Venkatesh Vinayakarao 8
- 9. Observations Let’s flip the coin (N) 5 times. We observe (z) 3 Heads. Venkatesh Vinayakarao 9
- 10. Impact of Data Belief is influenced by Observations. But, note that: Belief ≠ observation Bayes’ Rule Eeks… what’s in the denominator? Venkatesh Vinayakarao 10
- 11. Numerator is easy • p(θ) was uniform. So, nothing to calculate. • How to calculate p(D|θ)? Venkatesh Vinayakarao 11 Jacob Bernoulli 1655 – 1705. θ푧 1 − θ 푛−푧 If D observed is HHHTT and θ is 0.5, We have: p(D|θ) = (0.5)3 1 − 0.5 5−3 Remember, two things: 1)we are interested in the distribution 2) Order of H,T does not matter.
- 12. Bayesian Update Venkatesh Vinayakarao 12
- 13. Painful Denominator • Recall, for discrete distributions: • And, for continuous distributions: Venkatesh Vinayakarao 13
- 14. A Simpler Way Form, Functions and Distributions Normal (or Gaussian) Poisson Venkatesh Vinayakarao 14
- 15. What form will suit us? Venkatesh Vinayakarao 15
- 16. Beta Distribution Vadivelu, a famous Tamil comedian. This is one of his great expression - terrified and confused. Venkatesh Vinayakarao 16
- 17. Beta Distribution • Takes two parameters Beta(a,b) • For now, assume a = #H + 1 and b = #T + 1. • After 10 flips, we may end up in one of these three: Venkatesh Vinayakarao 17
- 18. Prior and Posterior Let’s say we have a Strong Prior – Beta (11,11). What should happen if we see 10 more observations with 5H and 5T? Venkatesh Vinayakarao 18
- 19. Prior and Posterior… What if we have not seen any data? Venkatesh Vinayakarao 19
- 20. Conjugate Prior So, we see that: Such Priors that have same form are called Conjugate Priors Venkatesh Vinayakarao 20 Prior beta(a,b) Posterior beta(a+z,b+N-z) z Heads in N trials
- 21. Summary Prior Likelihood Posterior Bayes’ Rule Bernoulli Distribution Beta Distribution Venkatesh Vinayakarao 21
- 22. References Book on Doing Bayesian Data Analysis – John K. Kruschke. YouTube Video on Statistics 101: The Binomial Distribution – Brandon Foltz. Venkatesh Vinayakarao 22

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