Write down all the probabilistic distributions for the model

Assignment Help Applied Statistics

Reference no: EM132277417

Question 1 - The Negative Binomial distribution. NegBin(r, θ), describes the distribution of the number of failures (denoted by Y) before the r-th success in an experiment that consists of a sequence of independent and identically distributed Bernoulli trials, where each trial has a probability 0 of success. We collect a sample of n independent observations Y_i, i = 1, . . . ,n, with Y_i|θ ∼ NegBin(r, θ), where θ is unknown and the value of r is known.

(a) Show that the Negative Binomial distribution of Y_i|θ belongs to the one-parameter exponential family.

(b) Using the results from part (a), derive the conjugate prior for θ and show that it can be written as a Beta distribution.

(d) Suppose that the Beta prior from part (c) was considered reasonable for θ. Derive the posterior distribution p(θ|y) for θ, where y = (y₁, . . . , y_n) denotes the values of the it observations. Show that the posterior distribution can be written as a Beta distribution.

(e) Suppose that we want to predict the value of a new observation (denoted by y^-). Write down the predictive distribution p(y^∼ly).

Question 2 - A scientist is interested in the proportion of female horseshoe crabs that have at least one male crab (called satellites) residing nearby. The scientist investigates whether the proportion is affected by two factors: the female crab's colour (denoted by L with L = 1 if the colour is light and L = 0 otherwise) and the female crab's width (denoted by W with W = 1 if the width is larger than 25cm and W = 0 otherwise). The data collected by the scientist are arranged in the 12 cells of the table below, according to a 2-by-2 factorial layout, where y_i and n_i are the number of female crabs which have satellites and the total number of female crabs, respectively, in the ith cell, i = 1, . . . , 12, from top to bottom and then left to right.

W = 0						W = 1
L = 0			L = 1			L = 0			L = 1
i	y_i	n_i	i	y_i	n_i	i	y_i	n_i	i	y_i	n_i
1	3	10	4	20	42	7	4	11	10	86	109
2	11	39	5	35	69	8	9	28	11	152	196
3	8	31	6	31	65	9	11	32	12	133	158

The scientist is interested in modelling and learning about the probability of having satellites (denoted by θ_i for the ith cell) in each cell. Your task is to design an appropriate random effects logistic regression model for the scientist. The model should use the colour L, width W and their interaction as explanatory variables, and also allow for over-dispersion.

(a) Write down all the probabilistic distributions for the model that you have designed.

(b) Suppose that, using a Gibbs sampler, you have obtained a sample (denotes by (θ₆^(M+1), . . . , θ₆^(N))) from the posterior distribution for θ₆. Explain how to use this sample to estimate the 5% highest posterior density (HPD) interval of θ₆.

(c) Explain how you could use your model and a Gibbs sampler to obtain samples from the predictive distribution of Y^∼, the number of female crabs having satellites out of totally n^∼ = 100 female crabs in a new cell with the colour L = 1 and the width W = 1.

Reference no: EM132277417

Questions Cloud

Write thoughts on topic - Gamification as Research Method : Write 150 Words of summary or thoughts on topic: "Gamification as Research Method". How the topic/below explanation helps to improve child's growth

What serious biases or misassumptions do groups : What serious biases or misassumptions do groups that are involved in inter-team conflict sometimes experience? How do these biases and prejudices affect.

Define trade-offs in internal cohesion and external ties : Consider the different teams presented in your reading assignment. How do the teams manage their team boundaries? What are the trade-offs between internal.

Discuss purpose and importance of project charter : Discuss purpose and importance of Project Charter. Create a Project Charter for your previous scenario that you choose and define what sections included

Write down all the probabilistic distributions for the model : Write down all the probabilistic distributions for the model that you have designed. Explain how you could use your model and a Gibbs sampler to obtain samples

Discuss the issue of abortion and maternal fetal conflict : Discuss the issue of abortion and maternal fetal conflict in medical ethics.

Other factors which contribute to employee motivation : Topic: It is important to take in consideration other factors which contribute to employee motivation (Other than Pay).

Connected consumer-briefly express any consistencies : "Connected Consumer", briefly express any consistencies or inconsistencies you perceive regarding your own experience of being a consumer in society today.

Value of the dollar relative to euro-pound and yen : Browse the Internet and find the latest Big Mac Index data. What does the data show in terms of the value of the dollar relative to the euro, pound, and yen?

User Account

All Pages