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ST417 Introduction to Bayesian Modelling Assignment Sample NUI Ireland

ST417 Introduction to Bayesian Modelling course is designed for students who want to learn about Bayesian inference and its applications in statistics. Bayesian inference is a method of statistical inference that is based on Bayesian probability, which is a form of probability that takes into account both evidence and prior beliefs. The course will cover the basics of Bayesian inference, including the concept of posterior distributions and the use of Markov chain Monte Carlo (MCMC) methods to approximate them.

The course will also cover some of the more advanced topics in Bayesian inference, such as hierarchical modeling and model selection. The course will be taught using R, and students will be expected to have a basic working knowledge of R before starting the course. At the end of the course, students should have a good understanding of Bayesian inference and be able to apply it to real-world data sets.

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In this section, we are describing some assigned tasks. These are:

Assignment Task 1: Determine likelihood and prior distributions as parts of a basic Bayesian model specification.

In Bayesian inference, the key components of a statistical model are the likelihood function and the prior distribution. The likelihood function is a mathematical function that describes the relationship between the data and the parameters of the model. The prior distribution is a probability distribution that represents our beliefs about the values of the parameters before we see any data.

To determine the likelihood function and prior distribution for a given data set, we need to first identify the parameters of the model. For this data set, the parameters are the mean μ and standard deviation σ of the population. We then need to specify a mathematical form for the likelihood function and choose values for the prior distribution.

The most common form of the likelihood function is the normal distribution, which is a continuous probability distribution with a bell-shaped curve. The normal distribution is defined by its mean μ and standard deviation σ. For our data set, we will assume that the data are distributed normally with a mean μ and standard deviation σ.

The prior distribution for the mean μ can be any probability distribution, but the most common choice is the normal distribution. For the standard deviation σ, the prior distribution is usually the inverted gamma distribution. The gamma distribution is a continuous probability distribution that is defined by its shape and rate parameters. The shape parameter is often denoted by α and the rate parameter is often denoted by β.

To summarize, the likelihood function for this data set is the normal distribution with mean μ and standard deviation σ. The prior distribution for the mean μ is the normal distribution with mean μ0 and standard deviation σ0. The prior distribution for the standard deviation σ is the inverted gamma distribution with shape parameter α and rate parameter β.

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Assignment Task 2: Apply the Bayes theorem to obtain the posterior distribution of unknown random variables in the model.

Bayes theorem is a mathematical formula that allows us to compute the posterior distribution of an unknown random variable given some data and our prior beliefs about that variable. Bayes theorem is stated as follows:

P(A|B) = P(B|A)P(A)/P(B)

where A and B are two events, P(A|B) is the posterior probability of event A given that event B has occurred, P(B|A) is the likelihood of event B given that event A has occurred, and P(A) is the prior probability of event A, and P(B) is the marginal probability of event B.

In our case, we want to compute the posterior distribution of the mean μ and standard deviation σ given our data. We can use the Bayes theorem to compute these posterior distributions.

For the mean μ, we have:

P(μ|data) = P(data|μ)P(μ)/P(data)

where P(μ|data) is the posterior distribution of μ given the data, P(data|μ) is the likelihood function of the data given μ, P(μ) is the prior distribution of μ, and P(data) is the marginal probability of the data.

For the standard deviation σ, we have:

P(σ|data) = P(data|σ)P(σ)/P(data)

where P(σ|data) is the posterior distribution of σ given the data, P(data|σ) is the likelihood function of the data given σ, P(σ) is the prior distribution of σ, and P(data) is the marginal probability of the data.

Assignment Task 3: Derive posterior predictive distribution.

The posterior predictive distribution is the distribution of the data that we would expect to see if we were to collect new data from the same population. The posterior predictive distribution is defined as follows:

P(new data|data) = ∫ P(new data|μ,σ)P(μ,σ|data)dμdσ

where P(new data|data) is the posterior predictive distribution of the new data given the data, P(new data|μ,σ) is the likelihood function of the new data given μ and σ, P(μ,σ|data) is the joint posterior distribution of μ and σ given the data, and ∫ P(new data|μ,σ)P(μ,σ|data)dμdσ is the marginal probability of the new data.

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Assignment Task 4: Write simple R scripts implementing basic random sampling methods.

R is a programming language that is commonly used for statistical computing. R has many built-in functions that make it easy to generate random numbers from various probability distributions.

First, we will load the R package “random”. This package contains functions for generating random numbers from various probability distributions.

library(random)

Next, we will set the seed for the random number generator. This will ensure that the results of our R scripts are reproducible.

set.seed(1234)

Now, we will generate 10,000 random numbers from the uniform distribution. The uniform distribution is a continuous probability distribution that is defined by its lower and upper bounds. In R, we can use the function “run” to generate random numbers from the uniform distribution.

unif <- runif(10000, min=0, max=1)

Next, we will generate 10,000 random numbers from the normal distribution. The normal distribution is a continuous probability distribution that is defined by its mean and standard deviation. In R, we can use the function “norm” to generate random numbers from the normal distribution.

norm <- rnorm(10000, mean=0, sd=1)

Finally, we will generate 10,000 random numbers from the gamma distribution. The gamma distribution is a continuous probability distribution that is defined by its shape and scale parameters. In R, we can use the function “gamma” to generate random numbers from the gamma distribution.

gamma <- gamma(10000, shape=1, scale=1)

Assignment Task 5: Apply the basics of Markov chain theory to implement simulation algorithms for inference.

A Markov chain is a stochastic process that satisfies the Markov property. The Markov property states that the future of the process depends only on the present and not on the past.

Markov chains are used to model systems that change over time. For example, we can use a Markov chain to model the weather. The weather can be modeled as a Markov chain with two states: “sunny” and “rainy”. The transition probabilities between these two states can be estimated from historical data.

Markov chains can also be used to generate samples from probability distributions. For example, we can use a Markov chain to generate samples from the posterior distribution of μ and σ. The Markov chain will start at some arbitrary point in the posterior distribution and then randomly move around the posterior distribution according to the transition probabilities. After a sufficient number of steps, the Markov chain will converge to the stationary distribution, which is the desired posterior distribution.

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Assignment Task 6: Implement Gibbs sampler and Metropolis algorithm to obtain samples from posterior distributions.

The Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for generating samples from a multivariate distribution. The Gibbs sampler works by sequentially sampling from each of the marginal distributions.

The Metropolis algorithm is another MCMC algorithm for generating samples from a multivariate distribution. The Metropolis algorithm works by randomly proposing new values for the variables and then accepting or rejecting these values according to some criteria.

Both of these algorithms can be used to generate samples from the posterior distribution of μ and σ. To do this, we need to define the joint distribution of μ and σ. The joint distribution is simply the product of the prior distribution and the likelihood function.

p(μ,σ)∝p(μ)p(σ)L(μ,σ)

Once we have defined the joint distribution, we can then use either the Gibbs sampler or the Metropolis algorithm to generate samples from this distribution.

Assignment Task 7: Compare and contrast basic Bayesian methods with classical statistics and realize the advantages and disadvantages of both.

There are a few key differences between Bayesian methods and classical statistics.

  • First, Bayesian methods use prior information to make inferences about the data. Classical statistics, on the other hand, do not use prior information.
  • Second, Bayesian methods return distribution of possible values for the parameters, while classical statistics typically return a single point estimate.
  • Third, Bayesian methods are more flexible than classical statistics and can be used to answer a wider range of questions.

There are also some disadvantages to Bayesian methods. First, they can be computationally intensive. Second, the results can be sensitive to the prior information that is used. Finally, Bayesian methods require a certain amount of statistical expertise to understand and implement.

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Assignment Task 8: Develop simple Bayesian models for the analysis of real-world data sets.

Many different Bayesian models can be used for the analysis of data. Here, we will develop a simple Bayesian model for the analysis of a data set.

The data set consists of 100 observations from a Normal distribution with mean μ and standard deviation σ. We want to infer the posterior distribution of μ and σ.

We will use the following priors for μ and σ:

μ ~ Normal(0, 10)

σ ~ Gamma(1, 0.1)

Where Gamma is the gamma function.

The likelihood function for this data set is:

L(μ,σ) = ∏i=1N Normal(x_i|μ,σ)

Where x_i is the ith observation.

Using these priors and likelihood functions, we can use the Gibbs sampler or Metropolis algorithm to generate samples from the posterior distribution of μ and σ. These samples can then be used to answer questions about the data set.

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