Markov Chains: Analytic and Monte Carlo Computations


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The Annals of Applied Probability

Markov chain Monte Carlo by Wikipedia

Great materials and well organized lecture structure. But in the meanwhile, it requires quite a lot preliminary knowledge. Bayesian Statistics: Techniques and Models. Enroll for Free. From the lesson. Metropolis-Hastings, Gibbs sampling, assessing convergence. Algorithm Demonstration Random walk example, Part 1 Random walk example, Part 2 Taught By.

Bayesian inference

Matthew Heiner Doctoral Student. Try the Course for Free. What if our likelihood were best represented by a distribution with two peaks, and for some reason we wanted to account for some really wacky prior distribution? As before, there exists some posterior distribution that gives the likelihood for each parameter value.

But its a little hard to see what it might look like, and it is impossible to solve for analytically. Enter MCMC methods.


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Monte Carlo simulations are just a way of estimating a fixed parameter by repeatedly generating random numbers. By taking the random numbers generated and doing some computation on them, Monte Carlo simulations provide an approximation of a parameter where calculating it directly is impossible or prohibitively expensive. Since the circle is inside a square with 10 inch sides, the area can be easily calculated as Instead, however, we can drop 20 points randomly inside the square.

Then we count the proportion of points that fell within the circle, and multiply that by the area of the square.

ST407: Monte Carlo Methods

That number is a pretty good approximation of the area of the circle. Since 15 of the 20 points lay inside the circle, it looks like the circle is approximately 75 square inches. Not too bad for a Monte Carlo simulation with only 20 random points. Therefore, finding the area of the bat signal is very hard. Nevertheless, by dropping points randomly inside a rectangle containing the shape, Monte Carlo simulations can provide an approximation of the area quite easily! By generating a lot of random numbers, they can be used to model very complicated processes.

These are simply sequences of events that are probabilistically related to one another. Each event comes from a set of outcomes, and each outcome determines which outcome occurs next, according to a fixed set of probabilities. An important feature of Markov chains is that they are memoryless : everything that you would possibly need to predict the next event is available in the current state, and no new information comes from knowing the history of events. A game like Chutes and Ladders exhibits this memorylessness, or Markov Property, but few things in the real world actually work this way.

Markov Chain Monte Carlo

Nevertheless, Markov chains are powerful ways of understanding the world. In the 19th century, the bell curve was observed as a common pattern in nature.

Galton Boards, which simulate the average values of repeated random events by dropping marbles through a board fitted with pegs, reproduce the normal curve in their distribution of marbles:. He thought that interdependent events in the real world, such as human actions, did not conform to nice mathematical patterns or distributions. Andrey Markov, for whom Markov chains are named, sought to prove that non-independent events may also conform to patterns.

One of his best known examples required counting thousands of two-character pairs from a work of Russian poetry. Using those pairs, he computed the conditional probability of each character. That is, given a certain preceding letter or white space, there was a certain chance that the next letter would be an A, or a T, or a whitespace.

Using those probabilities, Markov was ability to simulate an arbitrarily long sequence of characters. This was a Markov chain. Although the first few characters are largely determined by the choice of starting character, Markov showed that in the long run, the distribution of characters settled into a pattern. Thus, even interdependent events, if they are subject to fixed probabilities, conform to an average.

For a more useful example, imagine you live in a house with five rooms. You have a bedroom, bathroom, living room, dining room, and kitchen. Lets collect some data, assuming that what room you are in at any given point in time is all we need to say what room you are likely to enter next. Using a set of probabilities for each room, we can construct a chain of predictions of which rooms you are likely to occupy next. Making predictions a few states out might be useful, if we want to predict where someone in the house will be a little while after being in the kitchen.

Concrete Examples of Monte Carlo Sampling

So Markov chains, which seem like an unreasonable way to model a random variable over a few periods, can be used to compute the long-run tendency of that variable if we understand the probabilities that govern its behavior. To begin, MCMC methods pick a random parameter value to consider. The simulation will continue to generate random values this is the Monte Carlo part , but subject to some rule for determining what makes a good parameter value.

The trick is that, for a pair of parameter values, it is possible to compute which is a better parameter value, by computing how likely each value is to explain the data, given our prior beliefs.


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If a randomly generated parameter value is better than the last one, it is added to the chain of parameter values with a certain probability determined by how much better it is this is the Markov chain part. To explain this visually, lets recall that the height of a distribution at a certain value represents the probability of observing that value. Therefore, we can think of our parameter values the x-axis exhibiting areas of high and low probability, shown on the y-axis. For a single parameter, MCMC methods begin by randomly sampling along the x-axis:.

After convergence has occurred, MCMC sampling yields a set of points which are samples from the posterior distribution. Draw a histogram around those points, and compute whatever statistics you like:.

Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations
Markov Chains: Analytic and Monte Carlo Computations Markov Chains: Analytic and Monte Carlo Computations

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