Here are two techniques that I've found particularly useful in teaching about maximum likelihood estimation.
(1) What is the difference between "probability theory" and "statistical inference"?
Most students have been exposed to the binomial distribution, and those who have not quickly get the general idea when asked to consider flipping a possibly biased coin some fixed number of times, say 5. I display a table such as that below, in which the binomial distribution is used to calculate the probability of zero through five heads in 5 tosses of a coin for 5 possible values of a bias parameter:
|Bias of coin towards Heads|
I tell the class that if we are dealing with probability theory, then we look at columns of the table. For example, if we have a fair coin, we look at the column labelled ".5". But as statisticians, we deal with situations in which we have already performed an experiment, and want to make an inference about what state of nature probably produced the outcome. Therefore, statisticians look at the rows of the table.
We identify the row corresponding to the outcome that was observed. If we observe four heads, then the numbers in that row of the table show the likelihood function, calculated for these five possible values of the true underlying proportion. Since any value of the proportion between zero and one is possible, we must imagine a similar table with more and more values; or in the limit, a graph showing the shape of the function.
Since the numbers in a row do not add up to one (or, considering the true case of infinite values, do not integrate to one), these are not probabilities; that is why the different name "likelihood" was used by Ronald Fisher to describe these numbers. We can see that higher values of the likelihood correspond to underlying proportions that are more likely to produce the observed outcome. We naturally define the maximum likelihood estimate to be the number for which this likelihood is the largest. Since the biggest likelihood with 4 heads in this table is associated a bias of .7, then the maximum likelihood estimate is that the coin bias is .7.
(2) Maximizing a function is like climbing a hill with a bucket on your head.
There are frequently many parameters to be estimated at once in a general function. We must discover the estimates for all the parameters that simultaneously maximize the likelihood. A frequently-used analogy for finding maxima of general functions is climbing a hill.
Students who are told this don't see what is so difficult about this; they think that all you have to do is look at the hill, and see where the highest point is. What they don't understand is that in maximizing a general function, you don't usually get to see the whole hill at once, but only a small part of it. So I describe the process as one of being blindfolded and led to some point, and then having the blindfold removed while a bucket is placed over your head. The bucket allows you to see a small area around you, but you can't see the whole landscape. Based on what you see "nearby" by looking down, you have to decide what direction to move in, and how far. You are then transported to that spot (blindfolded again), and are then allowed to repeat the process. You must decide when you have reached the top of the hill.
If students can visualize the process this way, they can see more of the complexities involved, including possible multiple hill tops, some of which are higher than others.
City University of New York Graduate School
Explaining maximum likelihood estimation.Rindskopf D. Rasch Measurement Transactions, 1998, 12:3 p. 645.
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