The New York Times published an article of interest to statisticians the other day: "The Odds, Continually Updated". Surprisingly for a general-audience newspaper, this article goes into the the distinctions between Bayesian and frequentist statistics, and does so in a very approachable way. Here's an excerpt:
The essence of the frequentist technique is to apply probability to data. If you suspect your friend has a weighted coin, for example, and you observe that it came up heads nine times out of 10, a frequentist would calculate the probability of getting such a result with an unweighted coin. The answer (about 1 percent) is not a direct measure of the probability that the coin is weighted; it’s a measure of how improbable the nine-in-10 result is — a piece of information that can be useful in investigating your suspicion.
By contrast, Bayesian calculations go straight for the probability of the hypothesis, factoring in not just the data from the coin-toss experiment but any other relevant information — including whether you’ve previously seen your friend use a weighted coin.
The article covers the genesis of both frequentist and Bayesian statistics, and includes several examples of Bayesian statistics applications, including healthcare, cosmology, and even search-and-rescue. (This story of how Bayesian analysis directed the rescue of a fisherman lost at sea is nothing short of amazing.)
New York Times: The Odds, Continually Updated