Bayes In the Real World: The Metropolis-Hastings AlgorithmIf we know that a probability distribution is properly normalized, as in A, we can compute the probability of getting a value in a specific region, here in blue, simply by integrating and finding the area under the curve. But, if the total area under the curve is unknown, we have no way of distinguishing between A and the other panels, which show varying degrees of unexpected behavior. This is the core problem that Metropolis-Hastings has to solve: how can we figure out which one of these pictures is correct while having to evaluate as few times as possible? How do we know that we're looking at the most important region of the probability distribution, and not some peripheral area?
I highly recommend taking a second to think about this problem yourself. There are many correct answers, all of which have pros and cons, so your odds are pretty good.
If we know that a probability distribution is properly normalized, as in A, we can compute the probability of getting a value in a specific region, here in blue, simply by integrating and finding the area under the curve. But, if the total area under the curve is unknown, we have no way of distinguishing between A and the other panels, which show varying degrees of unexpected behavior. This is the core problem that Metropolis-Hastings has to solve: how can we figure out which one of these pictures is correct while having to evaluate as few times as possible? How do we know that we're looking at the most important region of the probability distribution, and not some peripheral area?
I highly recommend taking a second to think about this problem yourself. There are many correct answers, all of which have pros and cons, so your odds are pretty good.