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For each different proposal distribution, you should run your MCMC chain for 10,000 steps, and record the sequence of states. Then, you should produce a visualization of the distribution of states, and overlay a plot of the actual target distribution. They may or may not match (see, for example, the first example plot in the Description section). | For each different proposal distribution, you should run your MCMC chain for 10,000 steps, and record the sequence of states. Then, you should produce a visualization of the distribution of states, and overlay a plot of the actual target distribution. They may or may not match (see, for example, the first example plot in the Description section). | ||
- | Furthermore, for each proposal distribution, you should run three independent chains (you can do these sequentially or in parallel, as you like). You should display each of these three chains on a single plot with time on the x-axis and the state on the y-axis. Ideally, you will see each of the three chains mixing between two modes. | + | Furthermore, for each proposal distribution, you should run three independent chains (you can do these sequentially or in parallel, as you like). You should display each of these three chains on a single plot with time on the x-axis and the state on the y-axis. Ideally, you will see each of the three chains mixing between two modes; you may notice other features of the behavior of the samplers as well, which you should report in your writeup! |
**Part 2: Hamiltonian MCMC** | **Part 2: Hamiltonian MCMC** | ||
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A detailed explanation of Hamiltonian MCMC can be found here:[[http://www.mcmchandbook.net/HandbookChapter5.pdf|Hamiltonian MCMC]]. | A detailed explanation of Hamiltonian MCMC can be found here:[[http://www.mcmchandbook.net/HandbookChapter5.pdf|Hamiltonian MCMC]]. | ||
- | - You will find the equations describing the leapfrog method in Equations 5.18, 5.19 and 5.20. | + | * You will find the equations describing the leapfrog method in Equations 5.18, 5.19 and 5.20. |
- | - You will find a description of how to convert a given ''p(x)'' into a Hamiltonian in Section 5.3.1. | + | * You will find a description of how to convert a given ''p(x)'' into a Hamiltonian in Section 5.3.1. |
- | - You will find a description of the complete HMC algorithm in section 5.3.2.1 | + | * You will find a description of the complete HMC algorithm in section 5.3.2.1 |
- | Remember that you will alternate between two steps: | + | Remember that you will alternate between two steps: |
+ | |||
+ | - Sampling the momentum conditioned on the position. This is just sampling from a Gaussian. | ||
+ | - Proposing a new state for the position, given the momentum. This involves integrating the dynamics, and then accepting or rejecting based on integration error. | ||
You will have to tune two parameters in order to implement HMC: the variance of the momentum variables, and the timestep used for integrating the dynamics. Experiment with both, and report your results using plots like those you prepared for Part 1. | You will have to tune two parameters in order to implement HMC: the variance of the momentum variables, and the timestep used for integrating the dynamics. Experiment with both, and report your results using plots like those you prepared for Part 1. | ||
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You have now coded two different inference algorithms, and a few variants of each. For this section, you must provide a small write-up that compares and contrasts each. Answer at least the following questions: | You have now coded two different inference algorithms, and a few variants of each. For this section, you must provide a small write-up that compares and contrasts each. Answer at least the following questions: | ||
- | - What was the acceptance rate of each algorithm? (ie, what percentage of proposals were accepted) | + | - What was the acceptance rate of each algorithm? (ie, what percentage of proposals were accepted) |
- | - Why don't some inference algorithms explore both modes of the density? | + | - Why don't some inference algorithms explore both modes of the density? |
- | - Why do some algorithms stay in the same state repeatedly? Is this good or bad? | + | - Why do some algorithms stay in the same state repeatedly? Is this good or bad? |
- | - What were the best values for the variance of the momentum variables and the timestep you found? How did you know that they were good? | + | - What were the best values for the variance of the momentum variables and the timestep you found? How did you know that they were good? |
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