统计计算方法代写 R语言代写 统计作业代写 Bayesian代写

HW 10

统计计算方法代写 Question (7 pts) Recall the Beta distribution, which is defined for θ ∈ (0, 1) with parameters α and β, has a density proportional to:

Question (7 pts) 统计计算方法代写

Recall the Beta distribution, which is defined for θ ∈ (0, 1) with parameters α and β, has a density proportional to:

θ^α−1 (1 − θ)^β−1 

The Dirichlet distribution generalizes the the Beta distribution for k such θi values such that P ki=1 θk = 1.

Suppose that X₁ counts the number observations of type 1, X₂ counts the numbers of observations of type 2, and X₃ counts the number of observations of type 3 in a sample (e.g., red, blue, and green cars observed on the highway). We will treat n = X₁ + X₂ + X₃ as fixed, so that our data have a multinomial distribution, which generalizes the binomial distribution. As with the binomial distribution, we can notice that X₃ = n − X₁ − X₂, and so is redundant.

The probability mass function for a multinomial distribution is proportional to

Part (a) (2 pts)

Consider the Bayesian model:

(θ₁, θ₂) ∼ Dirichlet(δ₁, δ₂, δ₃)

(X₁, X₂) ∼ Multinomial(n, θ₁, θ₂)

Show that the posterior distribution π(θ₁, θ₂ | x₁, x₂) has a Dirichlet distribution with parameters (x₁ + δ₁, x₂ + δ₂, n − x₁ − x₂ + δ₃). (Hint: find something that is proportional to the posterior distribution and argue that the only possible normalizing constant must lead to a Dirichlet distribution with the given parameters.)

Proof:

The posterior distribution of π(θ1, θ2 | x1, x2) can be expressed by:

Therefore the conditional distribution is still a Dirichlet distribution for θ1, θ2 and the parameter is (x₁ + δ₁, x₂ + δ₂, n − x₁ − x₂ + δ₃)

Part (b) (2 pts)

Find the full conditional posteriors (up to a normalizing constant) for θ₁ and θ₂. Argue that

θ₁ | θ₂, x₁, x₂ ∼ (1 − θ₂) Beta(x₁ + δ₁, n − x₁ − x₂ + δ₃)

and

θ₂ | θ₁, x₁, x₂ ∼ (1 − θ₁) Beta(x₂ + δ₂, n − x₁ − x₂ + δ₃)

Hints:

• If X = aY , a > 0, and Y has density f(y), then X has density f(x/a)/a

• As we saw in class, be ruthless in dropping terms that don’t pertain to the main parameter as long as you can maintain proportionality. E.g.

So we can see that this is a scaled Beta distribution:

θ₁ | θ₂, x₁, x₂ ∼ (1 − θ₂) Beta(x₁ + δ₁, n − x₁ − x₂ + δ₃)

Similarly for θ₂, we have:

θ₂ | θ₁, x₁, x₂ ∼ (1 − θ₁) Beta(x₂ + δ₂, n − x₁ − x₂ + δ₃)

### Part (c) (3 pts)

A recent poll by Morning Consult/Politico asked voters their opinion on whether the Unite States Congress should raise the federal minimum wage.

x_1 <- 806 # Congress should raise to $15/hr 
x_2 <- 435 # Congress should not raise the minium wage 
x_3 <- 586 # Congress should raise to $11/hr 
n <- x_1 + x_2 + x_3

(I have excluded the 8% of the sample with no opinion) 统计计算方法代写

Modeling these results as multinomial, we will investigate the proportions of registered voters holding opinions about the federal minimum wage.

Use the result from part (b) to implement a Gibbs sampler for (θ₁, θ₂ θ₃| x₁, x₂). Let δ₁ = δ₂ = δ₃ = 1.

Create a chain of length 5000. Using the last 2000 iterations from the chain give estimates of θ₁, θ₂ and θ₃. Also provide 95% credible intervals for each of the parameters (quantiles of the posterior marginal distributions).

Estimate the probability that the $15/hour wage is twice as popular no increase at all (i.e., P(θ₁/θ₂ > 2)).

Question 2 (3 pt) 统计计算方法代写

Read the paper “Less than 2 degree C warming by 2100 unlikely.” Briefly summarize the results (question, data, analysis). Carefully, read the section “Methods: Model Estimation”. Explain how they used their posterior distribution to generate the predictions they used in the paper.

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