Laplaces law of succession says that if X1, X2,...,Xn+1 are conditionally independent

Chapter 9, Problem 50

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Laplaces law of succession says that if X1, X2,...,Xn+1 are conditionally independent Bern(p) r.v.s given p, but p is given a Unif(0, 1) prior to reflect ignorance about its value, then P(Xn+1 = 1|X1 + + Xn = k) = k + 1 n + 2 . As an example, Laplace discussed the problem of predicting whether the sun will rise tomorrow, given that the sun did rise every time for all n days of recorded history; the above formula then gives (n + 1)/(n + 2) as the probability of the sun rising tomorrow (of course, assuming independent trials with p unchanging over time may be a very unreasonable model for the sunrise problem). (a) Find the posterior distribution of p given X1 = x1, X2 = x2,...,Xn = xn, and show that it only depends on the sum of the xj (so we only need the one-dimensional quantity x1 + x2 + + xn to obtain the posterior distribution, rather than needing all n data points). (b) Prove Laplaces law of succession, using a form of LOTP to find P(Xn+1 = 1|X1 + +Xn = k) by conditioning on p. (The next exercise, which is closely related, involves an equivalent Adams law proof.)