What we have discussed so far this semester is known as the frequentist approach to statistics, but another approach is called Bayesian statistics. Two lines of argument show the rationale for taking a Bayesian approach.
First, recall what we calculate in frequentist statistics. We calculate p-values, the probability of observing a given statistic (or a more extreme value) if the null hypothesis is true. We also calculate likelihoods, the probability of observing of given statistic if some hypothesis is true, where we might often calculate likelihoods for two or more hypotheses and select the best-supported one using model selection. Both of these might strike you as backwards, in that we are using what is unknown (whether the hypothesis is true or not) to make inferences about what we already know (the observed statistic). What we'd really like, though, is the opposite: we'd like to know the support for a hypothesis provided by the data.
Second, the way we go about hypothesis testing in the frequentist framework is that every test stands on its own, but that's not how we work as a a scientist. For example, suppose you perform a statistical test on natural selection. If you reject it, saying that the data are not consistent with natural selection, it's not as if the entire edifice of evolutionary theory crumbles. It's just one piece of evidence, one that happens to conflict with enormous body of previous evidence in support of natural selection. What would be better is if our approach could explicitly take into account this previous work and reevaluate the hypothesis in light of our new data.
Bayesian statistics provides both of these, a way to measure the support for a hypothesis given some data, and a way to account for the support for a hypothesis before we performed our experiment and after.
Although Bayesian statistics has been around as long as the frequentist approaches we’ve discussed this semester, they have become much more common in the past decade. Statisticians are often sharply divided between frequentists and Bayesians, and the arguments are commonly heated.
The core of Bayesian statistics lies in applying a relationship known as Bayes’ Rule:
Bayes’ Rule links four probabilities. P(H) is the probability that the hypothesis is true. P(D) is the probability that our data would exist. The remaining two probabilities are conditional probabilities, that is, they are probabilities contingent on something else being true. The first, P(D|H), is the probability of observing the data if the hypothesis is true, what we have called likelihood, and which formed the basis of calculating a p-value. The second, P(H|D) is the probability that the hypothesis is true given the data that we have collected, and this is what we are after.
We will use Bayes’ Rule to convert from what we have been calculating, P(D|H), to what we would really like to know, how well a given hypothesis is supported, P(H|D).
Bayes’ Rule can produce some insightful, yet initially counterintuitive results, which is one of its attractions. Interpreting medical test results is one common example.
Suppose a woman had a mammogram to screen for breast cancer, and she received a positive test result. A positive test is certainly alarming, and should certainly be taken seriously, but Bayes’ Rule can put those results in perspective.
Using Bayes’ Rule from above, we can think of H, the hypothesis, being that the woman has breast cancer, and D, the data, being the positive mammogram test results. What she would like to know is the probability that she actually has cancer given that she tested positive for it.
After some searching in the medical literature, we find some recent data that we need (Nelson et al. 2016). First, the disease is relatively rare, with 2,963 cases of breast cancer out of 405,191 women screened. Thus, P(H), the probability of having breast cancer is about 0.0073. Second, mammograms produce false positives (indicating cancer when there is none) at a rate of about 121 per 1000 women. Thus, P(D|!H), the probability of getting a positive mammogram result when cancer-free is 0.121. Third, mammograms produce false negatives (where the mammogram fails to detect cancer in a patient who has cancer) at a rate of about 1 in 1000. Thus, P(D|H) is 1 - (1/1000), or 0.999. These give us directly the two terms in the numerator of Bayes’ Rule, P(D|H) and P(H), and the ability to solve for the denominator, P(D).
To find the probability of getting a positive result on a mammogram, P(D), we have to consider the two possible states: having cancer (H) and not having cancer (!H). Importantly, these are mutually exclusive: you can't simultaneously have breast cancer and not have it. Therefore, the probability of the data (a positive mammogram result) is the sum of the probabilities of those two states, each multiplied by the probability of a positive mammogram result data given that state:
Furthermore, because the two states are mutually exclusive, we know that the sum of their probabilities is one, so we can make a substitution:
Substituting the probabilities, we get:
Simplifying gives P(D)=0.127. Knowing that, and from above, knowing that P(D|H)=0.999 and P(H)=0.0073, solving for P(H|D) gives 0.057; in other words, there is only a 5.7% probability that a woman who tests positive on a mammogram actually has cancer. This is still of course serious, but it is far better than concluding that a positive mammogram is sure sign of cancer.
Less seriously, Bayes’ Rule is helpful in solving the Monty Hall problem.
Using Bayes’ rule for statistics in general immediately presents us with two problems. First, we may not know the probability of observing our data, P(D), especially when it is not contingent on any particular hypothesis. Second, we don't know the probability that the hypothesis is true, P(H); moreover, that is what we would ultimately like to know.
Because the probability of the data, P(D), does not depend any particular hypothesis, one approach is treat it as a constant, which means we can rewrite Bayes’ Rule as follows:
Furthermore, we can set the probability of the hypothesis, P(H), to reflect our current knowledge about the hypothesis, and it is therefore called the prior probability, or simply, the prior. Similarly, the term P(H|D) is called the posterior probability, and it is the result of the product of the prior and the likelihood, P(D|H). In other words, this simplified version of Bayes’ rule gives us the support for a hypothesis after we have collected the data (that is, the posterior probability), based on our support for the hypothesis before we collected the data (the prior probability) updated by the likelihood of the data given the hypothesis. Bayesians like this formulation because it is an analogy of how we do science: we update our existing view of the world as new data comes in. New data may result in a different hypothesis becoming our favored model of the world.
To a Bayesian, probability does not have a frequentist interpretation (positive outcomes divided by possible outcomes), but is instead a measure of the degree of belief in a hypothesis. In a frequentist’s view, one cannot talk about the probability that a hypothesis is true; it either is true or it is not, we just do not know which. Indeed, in a frequentist view of probability, discussing the probability of a hypothesis makes no sense. Frequentist instead talk about the probability of an outcome if a hypothesis is true. Bayesians, however, always talk about the probabilities of various hypotheses, and by that, they mean the support for those hypotheses. Bayesian probabilities indicate the various degrees to which the hypotheses are supported not only by the data, but also by our prior knowledge. For scientists, this is an appealing way to think about our task.
It should be obvious that the posterior probability depends highly on the prior probability. It is possible to set the prior probabilities such that no new data could change your view of the favored hypothesis, but this is not a legitimate thing to do. You could set the prior probabilities to be the same across all models, and these are called uninformed priors, weak priors, or flat priors. This is exactly what frequentists do: they start from scratch every time. Bayesians, on the other hand, see it as essential that you include prior knowledge when you evaluate a hypothesis. Bayesians are split into two camps on how these prior probabilities should be established. Objectivists view prior probabilities as coming from an objective statement about knowledge. Subjectivists view prior probabilities as coming from personal belief.
Bolker, B.M., 2008. Ecological models and data in R. Princeton University Press, 396 p.
Hilborn, R., and M. Mangel, 1997. The ecological detective: Confronting models with data. Monographs in Population Biology 28, Princeton University Press, 315 p.
Nelson, H.D., E.S. O'Meara, K. Kerlikowski, S. Balch, and D. Miglioretti, 2016. Factors associated with rates of false-positive and false-negative results from digital mammography screening: an analysis of registry data. Annals of Internal Medicine 164(4):226–235.