It is believed that cross-fertilized plants produce taller offspring than self-fertilized plants. In order to obtain an estimate on the proportion θ of cross-fertilized plants that are taller, an experimenter observes a random sample of 15 pairs of plants exactly the same age, with each pair grown in the same conditions with one cross-fertilized and the other self-fertilized. Based on previous experience, the experimenter believes that the following are possible values of π and prior probabilities for each value (prior weight), π(θ):
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θ:
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0.80
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0.82
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0.84
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0.86
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0.88
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0.90
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π (θ):
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0.03
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0.40
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0.22
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0.15
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0.15
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0.05
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From the experiment, it is observed that in 13 of 15 pairs, the cross-fertilized is taller.
(a) Create a table with columns for prior, likelihood of θ given sample, prior times likeli- hood, and posterior probability of θ. Based on the posterior probabilities, what value of θ has the highest support? Also, ?nd E(θ) based on the posterior probabilities.
(b) Redo part (a) with a completely noninformative prior, that is, take the prior for the proportion θ as one of the equally spaced values 0, 0.1, 0.2,..., 0.9, 1. Also assign for each value of θ the same probability, π(θ) = 1/11.
(c) Calculate the MLE of θ and compare it with the Bayesian estimate.