Some Thoughts on Probability
By Kevin
Mark Miller is trying to make probability sensible:
What does it mean to say there is a 10% chance of rain tomorrow? Or that there is a 7% chance Iraq will be a peaceful democracy in two years? These numbers appear silly: these are things that either will happen or won't....Well, I wouldn't go that far. Probability is an operational concept: a guess, constrained by several simple rules, about what will happen in a very specific context.To me, the traditional definition simply does not make sense. Instead, I think of probability just as a way of being systematic about our ignorance of the world....
In the end, I think probability is not something that can be concretely defined. It is a useful concept simply because it allows us to make sense of and systematize a complex world.
Those rules are the axioms of probability: namely, there is no such thing as negative probability, either something or nothing (an "event") will happen with probability 1, and if two things can't occur together, the probability of one thing or another thing happening is the sum of both.
Of course, all this is done very formally -- with logic and hard thought and, if you're like me, luck. But note what the rules leave out: they don't tell you which events in reality can happen in any given situation, they don't tell you how events relate to one another in sequence, and they certainly don't tell you how to determine what the probability of any event is.
For your regular Joe, a probability is a way to describe the evidence supporting the conclusion that an event will occur. These numbers do not fall off of trees (though they often are plucked out of thin air with Blink-thought). To be truly useful, they must be created with effort; somebody must 1) decide what's important to measure, 2) pull out of the historical record (and personal bias) all that is relevant, 3) make predictions about the environment up until the event is supposed to occur, and 4) figure out how to put it all together.
Here's one way: somebody can say, hey, event X1 looks like it belongs a class {X} with a known historical distribution of occurence. That is, somebody makes a personal judgement that X1 looks like it comes from {X}. When he does that, he says that there is a ten percent chance of rain tomorrow (X1). Why? Because in reasonably similar cloud-cover/pressure/storm-front situations at similar time periods in the past ({X}), it has rained 150 out of 1500 days. In other words, given what we know today about conditions tomorrow AND looking at how all this worked out in some situation we think similar, we come to probability figure.
That's useful when it works, but often the evidence does not present itself in such simple ways, and cannot be coralled, into a suitable form for this type of model. And to me, that's all probability is, a model converting what we know and what we're uncertain about into what we think will happen.
For instance, it is not possible to say that 7 out of 100 times the U.S. has invaded countries similar to Iraq, they've become stable democracies within five years. So what can it possibly mean to assign a probability to Iraq becoming stable even with no historical evidence to construct a class of which it is a hypothetical member?
Well, it can mean anything -- some people call this "degree of belief", but I don't think model outputs are beliefs. Anyway, as long as you have a way of transforming the convoluted mass of prior belief, uncertainty, and measured evidence into something that meets the laws of probability, then you've got a probability. Whether that metric is useful for decisionmaking is another matter.
Meanwhile, Naveen Mandava is "still looking for a powerful and simple way that the idea of distributions can make sense to a Philosophy undergrad or my mommy. Do you know of any?"
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