How can the Metropolis-Hastings algorithm help simulate a Normal Distribution?

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In summary, the conversation discusses finding expressions for \pi(x) and \pi(y) in the relation \alpha \left( {x,y} \right) = \min \left( {1,\frac{{\pi \left( y \right)q\left( {y,x} \right)}}{{\pi \left( x \right)q\left( {x,y} \right)}}} \right) and using it to simulate a Normal Distribution with expectation value m and standard deviation s. It is mentioned that one of the values, depending on how x and y are defined, should come from the proposal distribution. Additionally, pi(x) is defined as the normal distribution probability of x in this context.
  • #1
Zaare
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I'm having trouble understanding how to find an expression for [tex]\pi(x)[/tex] and [tex]\pi(y)[/tex] in the relation:
[tex]
\alpha \left( {x,y} \right) = \min \left( {1,\frac{{\pi \left( y \right)q\left( {y,x} \right)}}{{\pi \left( x \right)q\left( {x,y} \right)}}} \right)
[/tex]
For example, If I want to simulate Normal Distribution (Expectation value m and standard deviation s), how can I find expressions for [tex]\pi(x)[/tex] and [tex]\pi(y)[/tex]? Or are they equal: [tex]\pi(x)=\pi(y)[/tex]?
 
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  • #2
I've never used this algorithm, but the values of x and y are calculated/generated according to the algorithm and are different. One of the values, depending on how you are defining x and y, should come from the proposal distribution.

And pi(x) = the normal distribution probability of x, for your particular example.
 
  • #3
I see. I was hopeing I could find a "short cut" which would simplify the expression for pi(x), but I suppose I can't.
Thanks for the help.
 

1. What is the Metropolis-Hastings algorithm?

The Metropolis-Hastings algorithm is a Markov Chain Monte Carlo (MCMC) algorithm used for sampling from a probability distribution. It is commonly used in Bayesian statistics and is a popular method for approximating integrals in high-dimensional spaces.

2. How does the Metropolis-Hastings algorithm work?

The algorithm starts with an initial guess of the target distribution. It then iteratively generates new samples by proposing a new value based on the current sample and accepting or rejecting it based on a ratio of the target distribution at the proposed value and the current value. This process continues until enough samples are generated to approximate the target distribution.

3. What are the advantages of using the Metropolis-Hastings algorithm?

The Metropolis-Hastings algorithm is a fairly simple and flexible method that can be used for a wide range of problems. It is particularly useful for sampling from high-dimensional and complex distributions where other methods may not be feasible.

4. Are there any limitations to the Metropolis-Hastings algorithm?

One limitation of the Metropolis-Hastings algorithm is that it may require a large number of iterations to converge to the target distribution, especially for distributions with multiple peaks or high-dimensional spaces. In addition, the choice of proposal distribution can greatly impact the performance of the algorithm.

5. How can I assess the performance of the Metropolis-Hastings algorithm?

There are several methods for assessing the performance of the Metropolis-Hastings algorithm. One common approach is to visually inspect the trace plot of the generated samples to ensure they are mixing well. Additionally, convergence diagnostics such as the Gelman-Rubin statistic can be used to determine if the algorithm has converged to the target distribution.

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