a**j 发帖数: 60 | 1 Baysian Data Analysis
Question 3.6 please help with solving Part (A) Part (C)
Part (B) should be fine
Binomial with unknown probability and sample size: some of the difficulties
with setting prior distributions in multiparameter models can be illustrated
with the simple binomial distribution. Consider data y1, y2, …, yn,
modeled as iid Bin(N, θ), with both N and θ unknown. Defining a convenient
family of prior distributions on (N, 0) is difficult, partly because of the
discreteness of N.
Raftery (1988) considers a hierarchical approach based on assigning the
parameter N a Poisson distribution with unknown mean p. To define a prior
distribution on (0, N), Raftery defines λ =µθ and specifies a prior
distribution on (λ, θ). The prior distribution is specified in terms of λ
rather than µ because 'it would seem easier to formulate prior
information about λ, the unconditional expectation of the observations,
than about µ, the mean of the unobserved quantity N.'
(a) A suggested noninformative prior distribution is p (λ, θ) λ-1.
What is a motivation for this noninformative distribution? Is the
distribution improper? Transform to determine p (N, θ).
(b) The Bayesian method is illustrated on counts of waterbuck obtained by
remote photography on five separate days in Kruger Park in South Africa.
The counts were 53, 57, 66, 67, and 72. Perform the Bayesian analysis on
these data and display a scatterplot of posterior simulations of (N, θ).
What is the posterior probability that N > l00?
(c) Why not simply use a Poisson prior distribution with fixed p as a prior
distribution for N? |
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