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Opinionated

Opinionated. Lessons. in Statistics. by Bill Press. #4 The Jailer’s Tip. A. Our next topic is Bayesian Estimation of Parameters . We’ll ease into it with an example that looks a lot like the Monte Hall Problem:. The Jailer’s Tip:. Of 3 prisoners (A,B,C), 2 will be released tomorrow.

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Opinionated

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  1. Opinionated Lessons in Statistics by Bill Press #4 The Jailer’s Tip Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  2. A Our next topic is Bayesian Estimation of Parameters. We’ll ease into it with an example that looks a lot like the Monte Hall Problem: The Jailer’s Tip: • Of 3 prisoners (A,B,C), 2 will be released tomorrow. • A, who thinks he has a 2/3 chance of being released, asks jailer for name of one of the lucky – but not himself. • Jailer says, truthfully, “B”. • “Darn,” thinks A, “now my chances are only ½, C or me”. Is this like Monty Hall? Did the data (“B”) change the probabilities? Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  3. “says B” ( j ) ( ) P S B C 0 1 · · ( j ) ( j ) ( j ) P A S P A B S P A C S + x x = B = B B B ; ( j ) ( ) P S A B P A B B = ( j ) ( ) ( j ) ( ) ( j ) ( ) P S A B P A B P S B C P B C P S C A P C A + + B B B 1 1 3 = = 1 1 1 1 0 + + + x ¢ ¢ x 3 3 0 1/3 1 x Further, suppose (unlike Monty Hall) the jailer is not indifferent about responding “B” versus “C”. Does that change your answer to the previous question? So if A knows the value x, he can calculate his chances.If x=1/2 (like Monty Hall), his chances are 2/3, same as before; so (unlike Monty Hall) he got no new information.If x≠1/2, he does get new info – his chances change. But what if he doesn’t know x at all? Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  4. R ( j ) ( j ) ( j ) d P A S I P A S I I x p x x = B B x law of de-Anding:x’s are EME! (e.g., Jailer’s Tip): 1 R ( j ) d I p x x = 1 + x x “Marginalization” (this is important!) • When a model has unknown, or uninteresting, parameters we “integrate them out” … • …multiplying by any knowledge of their distribution • At worst, just a prior informed by background information • At best, a narrower distribution based on data • This is not any new assumption about the world • it’s just the Law of de-Anding Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  5. R ( ( ) j ( ) j ) ( j ) ( j ) d P A S I I P A S I I Z ´ p x p x x p x x = B B X X x ( ) ( ) d d P 1 1 1 p x x p x x = $ = $ = i i i 1 R ( j ) d I p x x = x i i 1 + x x (repeating previous equation:) first time we’ve seen a continuous probability distribution, but we’ll skip the obvious repetition of all the previous laws (Notice that p(x) is a probability of a probability! That is fairly common in Bayesian inference.) Professor William H. Press, Department of Computer Science, the University of Texas at Austin

  6. 1 R ( ( ( ) ) j ( ) ( ) ( ( j ) ) ( ) j ) ± d P A S I P A S I I 1 0 0 1 1 · · · · ¡ p p x x x x x x p x x = = = B B ; 2 ; x 1 1 ( j ) = 1 P A S I R 2 3 ( j ) d l P A S I 2 0 6 9 3 1 R = = ( j ) B d I x n = = = = B 1 1 2 + p x x = 1 : 0 + x 1 + x x (repeating previous equation:) What should Prisoner A take for p(x) ?Maybe the “uniform prior”? A Not the same as the “massed prior at x=1/2” “Dirac delta function” substitute value and remove integral Professor William H. Press, Department of Computer Science, the University of Texas at Austin

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