Bayesian Inference with Beta Prior in Coin Toss Experiment
Suppose you have a Beta(4,.4) prior distribution on the probability of a coin yielding a head. After spinning the coin ten times and observing fewer than 3 heads, the exact posterior density is calculated. The posterior distribution is plotted and analyzed, showing how the prior influences the updated beliefs in light of new evidence.
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Exercise 2.1 Posterior inference: Suppose you have a Beta(4, 4) prior distribution on the probability that a coin will yield a head when spun in a specified manner. The coin is independently spun ten times, and heads appear fewer than 3 times. You are not told how many heads were seen, only that the number is less than 3. Calculate your exact posterior density (up to a proportionality constant) for and sketch it. 1
Exercise 2.1 Suppose you have a Beta(4, 4) prior distribution on the probability that a coin will yield a head when spun in a specified manner. #ex. 1 theta = seq(0,1,0.01) prior_dens = dbeta(theta,4,4) plot(theta,prior_dens) 2
Exercise 2.1 Suppose you have a Beta(4, 4) prior distribution on the probability that a coin will yield a head when spun in a specified manner. The coin is independently spun ten times, and heads appear fewer than 3 times. 3
Exercise 2.1 Suppose you have a Beta(4, 4) prior distribution on the probability that a coin will yield a head when spun in a specified manner. The coin is independently spun ten times, and heads appear fewer than 3 times. Calculate your exact posterior density (up to a proportionality constant) for and sketch it. ? ???? ? ?? ???? 4
#ex. 1 theta =seq(0.001,1,0.01) prior_dens =dbeta(theta,4,4) plot(theta,prior_dens) Exercise 2.1 posterior_dens = theta^3*(1-theta)^13 + 10*theta^4*(1-theta)^12 + 45*theta^5*(1- theta)^11 plot(theta,posterior_dens,col=2) 5
Exercise 2.5 Posterior distribution as a compromise between prior information and data: let y be the number of heads in n spins of a coin, whose probability of heads is . (a) If your prior distribution for is uniform on the range [0, 1], derive your prior predictive distribution for y, 1 Pr ? = ? = Pr ? = ? ? ?? 0 for each k = 0, 1, . . . , n. (b) Suppose you assign a Beta( , ) prior distribution for , and then you observe y heads out of n spins. Show algebraically that your posterior mean of always lies between your prior mean, ?+?, and the observed relative frequency of heads, ? ? ?. (c) Show that, if the prior distribution on is uniform, the posterior variance of is always less than the prior variance. (d) Give an example of a Beta( , ) prior distribution and data y, n, in which the posterior variance of is higher than the prior variance. 7
Exercise 2.5 a Let y be the number of heads in n spins of a coin, whose probability of heads is . (a) If your prior distribution for is uniform on the range [0, 1], derive your prior predictive distribution for y, 1 Pr ? = ? = Pr ? = ? ? ?? 0 for each k = 0, 1, . . . , n. 1 1 1 Pr ? = ? = ?(?,?)?? = Pr ? = ? ? ?(?)?? = Pr ? = ? ? ?? 0 0 0 Beta 1 ? ???(1 ?)? ??? = distribution 0 1 (? + 1) (? ? + 1) (? + 2) (? + 2) ? ? (? + 1) (? ? + 1)??(1 ?)? ??? = 0 (? + 1) (? ? + 1) (? + 2) ? ? = ? = ? 1 !, when ? integer ?! ?! ? ? ! ? + 1 ! 1 = = ? ? !?! ? + 1 8
Exercise 2.5 b let y be the number of heads in n spins of a coin, whose probability of heads is . b. Suppose you assign a Beta( , ) prior distribution for , and then you observe y heads out of n spins. Show algebraically that your posterior mean of always lies between your prior mean, ?+?, and the observed relative frequency of heads, ? ? ?. ? ? ? ? ? ? ? ? ? ???(1 ?)? ? (?+?) (?) (?)?? 1(1 ?)? 1 = ? ? (?+?) (?) (?)??+? 1(1 ?)? ?+? 1 = ?+? ?+?+? posterior mean of =?(?|?) = 9
Exercise 2.5 b let y be the number of heads in n spins of a coin, whose probability of heads is . b. Suppose you assign a Beta( , ) prior distribution for , and then you observe y heads out of n spins. Show algebraically that your posterior mean of always lies between your prior mean, ?+?, and the observed relative frequency of heads, ? ? ?. ?+? ?+?+? posterior mean of ?(?|?) = ? ?+? ?+?+?< ? ? ?+?< ?+? ?+?+?= ?+?+ (1 )? ? ?, and show that 0 1 We ll write ?+? ?+?+?= ?+?+ (1 )? ? ? ?+? ?+?+?, which is always between 0 and 1, hence: the posterior mean is a weighted average of the prior mean and the data = 10
Exercise 2.5 c Show that, if the prior distribution on is uniform, the posterior variance of is always less than the prior variance. Uniform distribution is the same as Beta(1,1), so prior variance =1 12 ? ? ? ? ? ? ? ? Beta(y+1,n-y+1) ? ???(1 ?)? ? = 1 4 1 3 1 12 Posterior variance: (?+1)(? ?+1) ?+1 (?+2) (? ?+1) (?+2) 1 = (?+2)2(?+3) (?+3) ?+1 (?+2) (? ?+1) (?+2)this is two factors which sum to 1, hence: 1 4 11
Exercise 2.5 c 1 4 Proof: Two factors sum to 1, then the product ? + ? = 1, ? = 1 ? ?? = ?(1 ?) To minimize: ? ??? ?2= 1 2? = 0 ? =1 ? =1 ?? =1 2 2 4 12
Exercise 2.5 c Show that, if the prior distribution on is uniform, the posterior variance of is always less than the prior variance. 1 4 1 3 1 12 Posterior variance: (?+1)(? ?+1) ?+1 (?+2) (? ?+1) (?+2) 1 = (?+2)2(?+3) (?+3) ?+1 (?+2) (? ?+1) (?+2)this is two factors which sum to 1, hence: 1 4 13
Exercise 2.5 d Give an example of a Beta( , ) prior distribution and data y, n, in which the posterior variance of is higher than the prior variance. 14
