kindly explain to this to me (like im 5). what they are, the difference between Monte Carlo and MCMC simulations, how they’re used in Physics and Astronomy. I’ve been wondering since I heard about this from my seniors.
In: 5
A Markov process is a random process where the probabilities of changing to a new state depend only on the current state.
Imagine a frog in a pond sitting on one of a bunch of lily pads. The frog will jump from one lily pad to the next – it prefers nearer ones, and some are too far to reach, but it chooses randomly. The only thing that matters is which pad it is on, and where the pads are (which doesn’t change). Importantly, the frog is also dumb as a rock and doesn’t remember where it has been and doesn’t care where it is going.
So, if the frog is on pad number 3, it might jump as follows: pad 1 10% of the time, pad 2 20%, pad 4 30%, pad 5 30%, pad 6 10%. When the frog is on pad 1, there is a different set of probabilities, and so on.
Markov processes have a variety of analytical techniques, which can be used to solve certain problems. For example, if you want to find out what the probability of where the frog will be after 100 jumps, then there is an equation to do this. There are also analytical techniques for things like determining steady states. This means that if you can model something as a Markov process, then you have mathematical tools to analyse it.
Monte Carlo techniques are a method of analysing random processes, by simulating the process over and over, each time taking a different set of random decisions. This allows you to collect statistics and determine what the likely results of your process is.
For example, a builder wants to build a house. He has an estimated cost, but wants to know how likely he is to make a profit, or a loss. There are lots of things that could happen – the price of wood might fluctuate, there might be a shortage of electricians, the weather might be bad, the cost of pipes may change, interest rates on his loan may change, there may be delays which add extra costs. Some of these processes are more likely than others to happen, and some are more expensive than others.
In a Monte Carlo process, the builder simulates building 100 houses. He takes 100 estimates for the price of wood, 100 estimates for the price of electricians, 100 forecasts of what the weather might be like, and so on. He then looks at the 100 calculated total prices, and he can tell how many of those simulated houses make a profit, and how many make a loss. This informs him whether his project is a good risk or a bad risk.
The great thing about the Monte Carlo process, is that you can use it to simulate all sorts of random processes. You can simulate Markov chains too – maybe you want an analysis which doesn’t have an easy technique, or maybe your Markov chain is so big and complex that it’s not practical to solve the equations directly.
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