Resampling and Bayesian inference
Bayesian inference joins a prior belief to new observations through Bayes' theorem, and bootstrapping resamples repeatedly from the data set in hand to generate characteristics such as standard deviations and confidence intervals. Reporting guidelines recommend that a report of a Bayesian analysis state its priors, its model and its posterior.
Bayesian updating
The frequentist view is that a result the data cannot show cannot be believed. The Bayesian view is that information beyond the data in hand, specifically a prior probability, should be used. Both views can be argued convincingly. Bayes' theorem allows new observations to be integrated with previous beliefs.
Bayes' theorem projects the probability of events from conditional probabilities: the likelihood of observing event A given event B, and of observing B given A. The probability of A irrespective of B is the marginal probability of A, and the probability of A given B is the conditional, or posterior, probability.
In a worked example, the United States has about 50,000 new cases of oral cancer a year in a population of about 325,000,000, an annual incidence of 0.0154%. Half of these cases occur in people aged 65 or over. The example assumes that 5,000 cases a year are in people aged exactly 65 and that 5 million people, 1.54% of the population, are that age. On those figures, the probability that a person who develops oral cancer within a year is aged exactly 65 is 10%, and Bayes' theorem turns this into a probability of 1 in 1,000 that a 65-year-old develops oral cancer within the year.
For a 65-year-old who had used smokeless tobacco for about 50 years, the likelihood of developing oral cancer would increase 50-fold, because habitual users are at markedly increased probability compared with age-matched nonusers. Multiplying the 1 in 1,000 by 50 gives that user an estimated absolute risk of 1 in 20, or 5%, of developing oral cancer in the next year.
Markov chain Monte Carlo modelling
Markov chain Monte Carlo modelling is one application of Bayesian principles that simulates uncertain events by successive random sampling from a conditional data distribution, and the successive random selections make up a Markov chain. A Markov chain is a stochastic-systems model in which transition probabilities govern continuous or discrete states, and it can treat time as either a continuous or a discrete variable. A Bayesian model should begin with an accurate base case, especially before any probabilistic Markov chains are set in motion.
Reporting a Bayesian analysis
The Statistical Analyses and Methods in the Published Literature (SAMPL) guidelines recommend that a report of a Bayesian analysis:
- Specify the pre-trial probabilities, the priors.
- Explain how the priors were selected.
- Describe the statistical model used.
- Describe the techniques used in the analysis.
- Identify the statistical software program used.
- Summarise the posterior distribution with a measure of central tendency and a credibility interval.
- Assess the sensitivity of the analysis to different priors.
Bootstrapping
Bootstrapping is an iterative form of analysis that seeks to simulate the true population mean from the data in the sample. Its name comes from the sense of pulling oneself up by one's bootstraps. It resamples repeatedly from a data set, selecting each data point of a new set at random, and independently, until the new set has as many data points as the original. A resampled set can differ from the original, because some observations from the original set may be chosen twice, or not at all, in it. Resampling in this way makes it possible to generate new characteristics for the data, such as standard deviations and confidence intervals.