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## An Overview of Monte Carlo Methods

Closing in of the neutrino mass. The lecture takes place in the winter semester and is held in English. Topics The course introduces Monte Carlo methods and their use in the physical sciences. Markov Chain Monte Carlo methods Other techniques Applications: simulating physical systems, statistical analysis The material will be introduced in lectures, and the students will then have exercises that they will need to solve by programming algorithms on their computers.

Research What's new? Physics for everyone Studying and working About us. Imprint Privacy Policy Sitemap. If the chains are well-mixed left , they are probably converged. If the chains are poorly-mixed right , we should continue burn-in.

Many graphical models are high-dimensional, so it is hard to visualize all r. Instead, we can plot the complete log-likelihood vs.

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The complete log-likelihood is an r. Generall, the log-likelihood will climb, then eventually plateau. The key point is that we are going to use an adaptive proposal.

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And we are going to have choices of further engineered adaptive proposal to be a conditional distribution of a single random variable given the rest. And by using the Markov Blanket concept, we can make that simple proposal eqsy to manpulate, and get a constant 1 acceptant rate. So that the samples can be better used. We need to take care of convegence rate, good mixing, etc. One of the struggle people had in all vanilla MCMC methods is so called random walk behavior, which is caused by the proposed distribution.

However, we want to propose prefered biased samples. How to impose the derivative maybe likelihood function into the proposal in a mathematically elegent fashion had became an important question.

## Monte Carlo Methods

There are many interesting connections betrween the terms and derivatives over Hamiltonian. One of the key of Hamiltonian is that. When we want to sample a target distribution, we can leverage on gradient methods by introducing more variables to an auxiliary distribution. Thus, using Hamiltonian, we are able to define the change of state v.

## Simulation and Monte Carlo Methods

So behaviors like over shooting and under shooting can be avoided. Here it is a Bayesian setting where we have both the distribution of hidden states or the states of interest and also conditioned from priors. Langevin Dynamics is special case of Hamiltonian. Instead of doing Leapfrog, Langevin does a more sophiscated update based on second-order updates of the sampling states. Even for a strange distribution with constrains on regions, this augmented optimization methods still deal with it.

- Monte Carlo methods.
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Why is Monte Carlo useful? Limitations of Monte Carlo Direct sampling Hard to get rare events in high-dimensional spaces.

Comparison between using a fixed bad proposal and an adaptive proposal. The next state depends only on the preceding state. Reversible MCs always have a stationary distribution!