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When to use the independent Metropolis-Hastings algorithm?
2) If your proposal distribution is always N ( μ, σ 2), then this is the independent Metropolis-Hastings algorithm since your proposal distribution does not depend on your current sample. This method works best if your proposal distribution is a good approximation of the target distribution you wish to sample from.
Can a uniform distribution be used in a sampling proposal?
A2: Yes you can use a Uniform distribution as long as the support for the uniform distribution is bounded (since if the support is unbounded the Uniform distribution is improper as it integrates to ∞ ). So a Uniform on ( x t − 1 − c, x t − 1 + c).
Can a Markov chain be generated from a proposal distribution?
Our Monte Carlo simulation generated a sample that looks much like the proposal distribution, and sometimes that’s all we need. But we will require additional tools to generate a sample from our posterior distribution. A Markov chain is a sequence of numbers where each number is dependent on the previous number in the sequence.
Why does density plot not look like proposal distribution?
First, the proposal distribution is changing with each iteration. This creates a trace plot with a “random walk” pattern: the variability is not the same over all iterations. Second, the resulting density plot does not look like the proposal distribution or any other useful distribution.
When to use Metropolis-Hastings instead of MCMC?
Metropolis–Hastings, along with other MCMC methods, do not have this problem to such a degree, and thus are often the only solutions available when the number of dimensions of the distribution to be sampled is high.
Is the Metropolis algorithm a random walk method?
1) You could think about this method as a random walk approach. When the proposal distribution x ∣ x t ∼ N ( x t, σ 2), it is commonly referred to as the Metropolis Algorithm. If σ 2 is too small, you will have a high acceptance rate and very slowly explore the target distribution.
When is a proposal distribution a good approximation?
This method works best if your proposal distribution is a good approximation of the target distribution you wish to sample from. You are correct that choosing a good normal approximation can be difficult. Neither method’s success should depend on the starting value of the sampler.