Maximum Likelihood Estimators

There's probably no more tortured piece of frequentist procedures than Maximum Likelihood Estimators (MLE's). Note that I didn't say they are wrong, they are widely used because they are generally effective. The problem is the philosophical hoops you have to jump through to accept them.

Of course, most practitioners don't concern themselves with that; they just crank their answers. However, those advancing the field really should at least consider that the axioms upon which these advances are based make some sense.

As with the Method of Moments, the idea is simple enough. Data is more likely to be observed in some configurations than others. Given a set of data, it makes sense to look at what configuration is most likely to produce such data. So, write your likelihood as a function of the parameter you care about and find the value of the parameter maximizes that function.

The rub is that unlikely things do happen and just because something isn't the configuration that makes the data the "most likely" doesn't mean it should be dismissed. Furthermore, as with the Method of Moments, the MLE might be a value that makes no sense.

For example, suppose we want to know how likely a certain even is to occur. We observe a dozen trials and it doesn't happen. The MLE for the probability is zero. As in, the event can't possibly happen. Ever. That may be the Maximum Likelihood value but, assuming this isn't some fantasy event like unicorn sightings, it's clearly wrong. You can argue that the experiment was bad because the sample was too small, but that evades the question. The procedure, properly applied, yielded nonsense.

OK, that can happen with any procedure, but what's important here is the reason the procedure fails. It fails because there's no place to interject common sense. The frequentist world view is that there is underlying truth out there and our experiment is trying to uncover that. That's a perfectly fine world view, but it leaves you completely at the mercy of your data. Some think that's a good thing, I do not.

All that said, MLE's are relatively easy to construct and verify. I've used them myself in past and may have cause to use them again. I just don't put much faith in them. Then again, I don't put much faith in results from any single data set. Unconfirmed research isn't much better than a guess in my mind.