Drunkard's Walk - Leonard Mlodinow [57]
As I mentioned, Bayes developed conditional probability in an attempt to answer the same question that inspired Bernoulli: how can we infer underlying probability from observation? If a drug just cured 45 out of 60 patients in a clinical trial, what does that tell you about the chances the drug will work on the next patient? If it worked for 600,000 out of 1 million patients, the odds are obviously good that its chances of working are close to 60 percent. But what can you conclude from a smaller trial? Bayes also asked another question: if, before the trial, you had reason to believe that the drug was only 50 percent effective, how much weight should the new data carry in your future assessments? Most of our life experiences are like that: we observe a relatively small sample of outcomes, from which we infer information and make judgments about the qualities that produced those outcomes. How should we make those inferences?
Bayes approached the problem via a metaphor.4 Imagine we are supplied with a square table and two balls. We roll the first ball onto the table in a manner that makes it equally probable that the ball will come to rest at any point. Our job is to determine, without looking, where along the left-right axis the ball stopped. Our tool in this is the second ball, which we may repeatedly roll onto the table in the same manner as the first. With each roll a collaborator notes whether that ball comes to rest to the right or the left of the place where the first ball landed. At the end he informs us of the total number of times the second ball landed in each of the two general locations. The first ball represents the unknown that we wish to gain information about, and the second ball represents the evidence we manage to obtain. If the second ball lands consistently to the right of the first, we can be pretty confident that the first ball rests toward the far left side of the table. If it lands less consistently to the right, we might be less confident of that conclusion, or we might guess that the first ball is situated farther to the right. Bayes showed how to determine, based on the data of the second ball, the precise probability that the first ball is at any given point on the left-right axis. And he showed how, given additional data, one should revise one’s initial estimate. In Bayesian terminology the initial estimates are called prior probabilities and the new guesses, posterior probabilities.
Bayes concocted this game because it models many of the decisions we make in life. In the drug-trial example the position of the first ball represents the drug’s true effectiveness, and the reports regarding the second ball represent the patient data. The position of the first ball could also represent a film’s appeal, product quality, driving skill, hard work, stubbornness, talent, ability, or whatever it is that determines the success or failure of a certain endeavor. The reports on the second ball would then represent our observations or the data we collect. Bayes’s theory shows how to make assessments and then adjust them in the face of new data.
Today Bayesian analysis is widely employed throughout science and industry. For instance, models employed to determine car insurance rates include a mathematical function describing, per unit of driving time, your personal probability of having zero, one, or more accidents. Consider, for our purposes, a simplified model that places everyone in one of two categories: high risk, which includes drivers who average at least one accident each year, and low risk, which includes drivers who average less than one. If, when you apply for insurance, you have a driving record that stretches back twenty years without an accident or one that goes back twenty years with thirty-seven accidents, the insurance company can be pretty sure which category to place you in. But if you are a new driver, should you be classified as low risk (a kid who obeys the speed limit and volunteers to be the designated driver) or high risk (a kid who