Construct a posterior histogram and compare with the histogram of its normal approximation

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

1) Sample 2D density by the Gibbs algorithm. Perform the one-dimensional sampling by the Inverse Cumulative Distribution Function (ICDF) algorithm. Plot the scatter plot. Find the file attached, please.

2) Sample 1D density by the AR algorithm. First, normalize the density so that Plot the histogram. As an envelope distribution use the uniform (flat) one.

3) Formulate a practical problem in which data are described by Poisson (Gaussian, binomial) distribution.

ii. Generate a sample from Poisson (Gaussian, binomial) distribution (of any size and with any parameter) - .

iii. Construct a prior distribution (gamma, Gaussian, beta) for this sample. Plot the graph

iv. Generate a sample of observations from the Poisson (Gaussian, binomial) distribution (of a different size and with a different parameter) -

v. Build posterior distribution from the prior and likelihood - (gamma, Gaussian, beta). Plot the graph

vi. Generate a sample from the posterior. Construct a posterior histogram and compare with the histogram of its normal approximation.

vii. Compute the theoretical and sample posterior mean, median, variance, and st. deviation of the parameter.

viii. Compute the confidence (asymptotic) interval and Bayesian credible interval for the parameter.

ix. Compute the Bayesian factor to test hypothesis H0: the sample mean versus hypothesis H1: any other value (first try the parameter value which you used to generate the sample (step 4)).

x. Test the same hypothesis using the chi-square test.

Attachments:

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