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Beta Posterior Mean is a Weighted Mean If the proportion has a (a, b) prior and one observes Y from a distribution with parameters n and p, then if one observes Y = y, then the posterior density of p is (a + y, b + n - y). Recall that the mean of a Beta(a, b) random variable following is a / (a + b). Show that the posterior mean of p | Y = y ~ Beta(a + y, b + n - y) is a weighted average of the prior mean of p ~ Beta(a, b) and the sample mean p? = y/n. Find the two weights and explain their implication for the posterior being a combination of prior and data.

          Beta Posterior Mean is a Weighted Mean

If the proportion has a (a, b) prior and one observes Y from a distribution with parameters n and p, then if one observes Y = y, then the posterior density of p is (a + y, b + n - y).

Recall that the mean of a Beta(a, b) random variable following is a / (a + b). Show that the posterior mean of p | Y = y ~ Beta(a + y, b + n - y) is a weighted average of the prior mean of p ~ Beta(a, b) and the sample mean p? = y/n. Find the two weights and explain their implication for the posterior being a combination of prior and data.

Beta Posterior Mean is a Weighted Mean

If the proportion has a (a, b) prior and one observes Y from a distribution with parameters n and p, then if one observes Y = y, then the posterior density of p is (a + y, b + n - y).

Recall that the mean of a Beta(a, b) random variable following is a / (a + b). Show that the posterior mean of p | Y = y Beta(a + y, b + n - y) is a weighted average of the prior mean of p Beta(a, b) and the sample mean p? = y/n. Find the two weights and explain their implication for the posterior being a combination of prior and data.

Added by Vanessa F.

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