Bayesian Estimation and Hypothesis Testing in Statistics for Engineers

In this course on Bayesian Estimation and Hypothesis Testing for Engineers, various concepts such as point estimation, conditional expectation, Maximum a posteriori estimator, hypothesis testing, and error analysis are covered. Topics include turning conditional PDF/PMF estimates into one number, estimation for normal distributions, and binary hypothesis testing. Practical examples such as Romeos's model and coin flipping scenarios are used to illustrate these statistical concepts.

• Statistics
• Engineers
• Bayesian
• Estimation
• Hypothesis Testing

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1. ENGG 2780A / ESTR 2020: Statistics for Engineers Spring 2021 2. Bayesian Estimation and Hypothesis Testing Andrej Bogdanov

2. Point estimation How to turn conditional PDF/PMF f |X( | x) estimate into one number? Conditional expectation (CE) estimator: | X = x] Maximum a posteriori (MAP)estimator: argmaxf |X( | x)

3. Point estimation for normals Xi = Normal( , 1) independent given isNormal(x0, 1) ( | X1 = x1, , Xn = xn) is Normal(x, 1/ n) CE estimate: MAP estimate:

4. Romeos model X = Uniform(0, ) = Uniform(0, 1) On her first date, Juliet arrives hour late. CE estimate: MAP estimate:

5. Beta(1, 1) Beta(2, 3) CE = /( + ) MAP = h/(h + t) Beta(11, 21) Beta(51, 101)

6. Hypothesis testing Suppose takes two values (e.g. spam / legit) MAP = argmaxf |X( | x) Choose the one for which f |X( | x) is larger

7. = 80% legit, 20% spam The Citibank concerning wire transfers of your fund. Your letter has been referred to the (JMCB) Legal Division for Funds (US$2.8 Million Dollars) P(X1| ) P(X2| ) legit spam 0.0001 0.01 0.03 0.1

8. Coin A is heads with probability 1/3. Coin B is tails with probability 1/3. HHHT are 4 flips of a random coin. Which coin was it?

9. What is the probability you are wrong, given the outcome is HHHT? What is the probability you are wrong on average?

11. Binary hypothesis testing error = 0 (null) or 1 (alternative) ^ = 0 (reject) or 1 (accept) ^ error = P( )

12. A car-jack detector X outputs Normal(0, 1) if there is no intruder and Normal(1, 1) if there is. When should alarm activate?

14. MAP hypothesis testing error P(MAP ) 50% Proof:

15. coin A B C P(H) 1 You observe HH. Which coin was it? What is the probability you are wrong?

16. Multiple hypotheses takes k possible values ^ error = P( ) P(MAP ) 1 - 1/k