If you've read Nate silver's "The Signal and the Noise", you need to read this New Yorker article: "What Nate Silver gets wrong" by two NYU professors. If you have not read Silver's book, you should read the article, too. It explains what Bayes's theorem is good for: making prediction when there is previous information available. The professors argue that Nate Silver advocates the use of Bayersian methods when there is no real previous information. They also explain why it's so hard to make predictions in some field.
My favorite example from the article illustrates how Bayesian statistics work:
A Bayesian approach is particularly useful when predicting outcome probabilities in cases where one has strong prior knowledge of a situation. Suppose, for instance (borrowing an old example that Silver revives), that a woman in her forties goes for a mammogram and receives bad news: a “positive” mammogram. However, since not every positive result is real, what is the probability that she actually has breast cancer? To calculate this, we need to know four numbers. The fraction of women in their forties who have breast cancer is 0.014, which is about one in seventy. The fraction who do not have breast cancer is therefore 1 - 0.014 = 0.986. These fractions are known as the prior probabilities. The probability that a woman who has breast cancer will get a positive result on a mammogram is 0.75. The probability that a woman who does not have breast cancer will get a false positive on a mammogram is 0.1. These are known as the conditional probabilities. Applying Bayes’s theorem, we can conclude that, among women who get a positive result, the fraction who actually have breast cancer is (0.014 x 0.75) / ((0.014 x 0.75) + (0.986 x 0.1)) = 0.1, approximately. That is, once we have seen the test result, the chance is about ninety per cent that it is a false positive. In this instance, Bayes’s theorem is the perfect tool for the job.