NOTATION AND ABUSE OF NOTATION
I will do my best to use the simplest possible notations in this book, but in my efforts to do so, I will have to leave out some details and write things that aren't exact (or "abuse" the notation). For example, you probably already noticed that I gave some formulas for probability distributions as p(x), and now I'm using a new notation P(X = A). In the formula P(X = A), I'm writing the probability that the random variable X takes the value A. Usually, I won't bother with this and I'll just write P(X) or p(X), and you'll have to infer from context whether I'm talking about the random variable or the value of the random variable. You'll see that this is easier than it sounds. For example,
All mean the same thing, with varying degrees of detail. I will usually choose the simplest formula I can, unless, for example, it's important to emphasize that there are n observations.
I'll be especially loose with my use of the notation for conditional probabilities. I'll start writing probability distributions as p(X|parameters) or null distributions as p(X|H0 is true) implying that the parameters or hypotheses are actually observations of random variables. You might start thinking that it's a bit strange to write models with parameters as random variables and that this assumption might cast philosophical doubt on the whole idea of modeling as we discussed earlier. In practice, however, it's easier not to think about it—we're using statistical models because they describe our data. We don't really need to worry about exactly what the notation means, as long as we're happy with the predictions and data analysis algorithms that we get.