MLE with Newton - Raphson - Posterior Distribution After a Single Coin Toss - Mathematics Assignement Help

Assignment Task :

Question 1: MLE with Newton-Raphson 
In this question, we will fit a seasonal model via maximum likelihood estimation. For health resources planning, the UK’s National Health Service keeps records of the use of its services, and much of this data is publicly available. The provided file shows the number of admissions of various types throughout England, for each of the N = 141 months from April 2008 to December 2019. We would like to model the “total non-elective general and acute
admissions” data (column K), allowing for the possibility of a seasonal trend.
If there is a seasonal trend, we’ll assume it’s a simple annual sinusoid, and therefore can be
described as a linear combination of the cosine and sine of an angle φi that completes a full circle
each year. Therefore, we will assume that
• φ1 = 30? = π/6 (corresponding to the first month in the sample, April 2008),
• φ2 = 60? = π/3 (corresponding to May 2008),
• φ3 = 90? = π/2 (corresponding to June 2008),
et cetera. Since there is clearly an upward trend as well, which looks as though it could be exponential, we will model the monthly admission counts yi as

(a) Describe a matrix X (with elements xij ) such that the model (1) can be written in the neater and more convenient form

(b) Find an expression for the log-likelihood `. 
(c) Differentiate Equation (2) with respect to βj , and simplify the result by expressing it as a multiple of μi . Show the steps of your calculation.
(d) Find an expression for the jth component of the gradient of the log-likelihood, that is, ∂`/∂βj .
(e) Find an expression for the (j, k)th component of the Hessian of the log-likelihood, that is,∂2`/∂βj∂βk.
Let β, y and μ be the column vectors with entries βj , yi and μi, respectively, and let M denotediag(μ), the diagonal matrix with the values μ1 to μN along the diagonal. That is,

Question 2: Posterior distribution after a single coin toss
In this question we will produce a histogram of the posterior distribution of a coin’s tails probability after a single coin toss.
In class we discussed prior distributions in the context of a coin toss. Suppose someone you don’t know very well has a coin. You can’t see the coin very clearly; it’s too far away to see what’s on either side. In particular, it may be a genuine coin, but you don’t know that for certain. The coin is then tossed once and shown to you, and you can see that it shows tails.
(a) Describe and justify a prior distribution (i.e., prior to the single toss that you witnessed) for the parameter π, the coin’s probability of showing tails on a single toss. This is a matter of opinion, so the right answer is not unique, but the prior distribution should accurately describe a reasonable opinion.

(b) Use an appropriate Bayesian inference method to produce a histogram of your posterior dis- tribution, after the single coin toss that you witnessed. (Hints: depending on your prior, this may or may not be straightforward in JAGS. Another option is rejection sampling. Also, you’ll probably need a large sample and narrow histogram bins to clearly display the posterior distribution.)

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