Performing Metropolis-Hastings algorithm for a Poisson Distribution

In summary, to generate values from a truncated Poisson distribution using the Metropolis-Hastings algorithm, you will need to construct a probability matrix based on the CDF of the distribution and use it to simulate a random walk through the state space. This process can be repeated until enough values have been generated.
  • #1
the_dane
30
0

Homework Statement


The number of busy lines in a trunk group (Erlang system) is given by a truncated Poisson distribution. I am asked to generate values from this distribution by applying the Metropolis-Hastings algorithm.

Homework Equations


The distribution is given in the attached picture. Let's just say n = 5.

The Attempt at a Solution


I have used the book Simulation by Sheldon M. Ross. So I need some kind of probability matrix. This where it goes wrong, I have no idea how to choose that.
Otherwise I need a start value and a proposal distribution. So I choose a start value of 3 and a standard normal distribution for the proposals distribution. But I cannot get started with proces because of no probability matrix.
 

Attachments

  • pois.png
    pois.png
    3.5 KB · Views: 639
Physics news on Phys.org
  • #2
I am really stuck here, and I have not been able to find an example on the internet that I can use.I hope someone can help me with this problem.A:I'm going to approach this as a Markov chain Monte Carlo (MCMC) problem. The probability matrix will be your transition matrix for your Markov chain. It should specify the probability of transitioning from one state to another. In your case, you are dealing with a truncated Poisson distribution and n=5. So the states of your Markov chain would be 0,1,2,3,4,and 5. The transition matrix will be 6x6 and it will contain the probabilities of transitioning from one state to another.The way I would go about constructing the transition matrix is by taking the probabilities of transitioning from one state to another and making them the elements of the transition matrix. For example, if we wanted to construct the element of the transition matrix that specifies the probability of transitioning from state 0 to state 1, we would look at the CDF of the truncated Poisson distribution and calculate the probability that x is less than 1 given that x is greater than or equal to 0. This would give us the element of our transition matrix that specifies the probability of transitioning from state 0 to state 1. Then we would do the same for the other elements in the transition matrix.Once you have your transition matrix constructed, you can start off with some arbitrary state x_0 and use the transition matrix to simulate a random walk through your state space. At each step in the random walk, you would look at the transition matrix to determine the probability of transitioning to each of the other states and then sample from that probability distribution to determine the next state. This process would be repeated until you have generated enough values from your truncated Poisson distribution. Hope this helps.
 

What is the Metropolis-Hastings algorithm?

The Metropolis-Hastings algorithm is a popular method for generating samples from a target distribution, often used in Bayesian statistics. It is particularly useful for situations where the target distribution is difficult to sample from directly.

How does the Metropolis-Hastings algorithm work?

The algorithm starts with an initial guess for the parameters of the target distribution. It then proposes a new set of parameters based on a specific proposal distribution. The proposed parameters are then accepted or rejected based on a calculated acceptance probability. This process is repeated for a large number of iterations, resulting in a sample of parameters that approximate the target distribution.

What is a Poisson distribution?

A Poisson distribution is a probability distribution that is often used to model the number of occurrences of a particular event in a given time period or space. It is characterized by a single parameter, λ, which represents the average rate of occurrences. The distribution is discrete and takes on non-negative integer values.

Why is the Metropolis-Hastings algorithm useful for a Poisson distribution?

The Metropolis-Hastings algorithm is useful for a Poisson distribution because it allows for sampling from a complex distribution without having to explicitly calculate the probability mass function. This is particularly helpful for Poisson distributions with large values of λ, where the probability mass function can become computationally intensive to calculate.

What are some potential limitations of using the Metropolis-Hastings algorithm for a Poisson distribution?

One potential limitation of the Metropolis-Hastings algorithm is that it relies on a proposal distribution, which may not always be an accurate representation of the target distribution. This can lead to a high rejection rate and inefficient sampling. Additionally, the convergence of the algorithm can be slow for distributions with high values of λ, requiring a large number of iterations to obtain a representative sample.

Similar threads

  • Calculus and Beyond Homework Help
2
Replies
56
Views
3K
  • Engineering and Comp Sci Homework Help
Replies
2
Views
2K
  • Engineering and Comp Sci Homework Help
Replies
12
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
  • Atomic and Condensed Matter
Replies
3
Views
733
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
  • Advanced Physics Homework Help
Replies
11
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
  • Atomic and Condensed Matter
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
10K
Back
Top