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Richard Redding
Richard Redding
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MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal* to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

I say equal although in general I should talk about measurable sets. A layperson, however, probably wouldn't be interested in this

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.52 (or any other probability that is greater than 0), I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal* to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

I say equal although in general I should talk about measurable sets. A layperson, however, probably wouldn't be interested in this

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.5, I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal* to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

I say equal although in general I should talk about measurable sets. A layperson, however, probably wouldn't be interested in this

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.2 (or any other probability that is greater than 0), I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

In response to Glen_b's request for more "meat" in the answer rather than just linking to an article :)
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Richard Redding
Richard Redding
  • 679
  • 5
  • 11

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equalequal* to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

I say equal although in general I should talk about measurable sets. A layperson, however, probably wouldn't be interested in this

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.5, I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.5, I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal* to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

I say equal although in general I should talk about measurable sets. A layperson, however, probably wouldn't be interested in this

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.5, I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

In response to Glen_b's request for more "meat" in the answer rather than just linking to an article :)
Source Link
Richard Redding
Richard Redding
  • 679
  • 5
  • 11

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.5, I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

MCMC is typically used as an alternative to crude Monte Carlo techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally. You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

MCMC is typically used as an alternative to crude Monte Carlo simulation techniques. Both MCMC and other Monte Carlo techniques are used to evaluate difficult integrals but MCMC can be used more generally.

For example, a common problem in statistics is to calculate the mean outcome relating to some probabilistic/stochastic model. Both MCMC and Monte Carlo techniques would solve this problem by generating a sequence of simulated outcomes that we could use to estimate the true mean.

Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal to the modelled probability of that outcome. Therefore, by generating enough simulations, the results produced by both methods will be accurate.

However, while crude Monte Carlo involves producing many independent simulations, each of which is distributed according to the modelled distribution, MCMC involves generating a random walk that in the long-run "visits" each outcome with the desired frequency.

The trick to MCMC, therefore, is picking a random walk that will "visit" each outcome with the desired long-run frequencies.

A simple example might be to simulate from a model that says the probability of outcome "A" is 0.5 and of outcome "B" is 0.5. In this case, if I started the random walk at position "A" and prescribed that in each step it switched to the other position with probability 0.5, I could be sure that after a large number of steps the random walk would have visited each of "A" and "B" in roughly 50% of steps--consistent with the probabilities prescribed by our model.

This is obviously a very boring example. However, it turns out that MCMC is often applicable in situations in which it is difficult to apply standard Monte Carlo or other techniques.

You can find an article that covers the basics of what it is and why it works here:

http://wellredd.uk/basics-markov-chain-monte-carlo/

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Richard Redding
Richard Redding
  • 679
  • 5
  • 11
Post Made Community Wiki by Richard Redding