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Another example of Bayesian inference has been immortalized in the words of the
fictional detective Sherlock Holmes, who often said to his sidekick, Doctor Watson:
“How often have I said to you that when you have eliminated the impossible, whatever
remains, however improbable, must be the truth?” (Doyle, 1890, chap. 6) Although this
reasoning was not described by Holmes or Watson or Doyle as Bayesian inference, it is.
Holmes conceived of a set of possible causes for a crime. Some of the possibilities may
have seemed very improbable, a priori. Holmes systematically gathered evidence that
ruled out a number of the possible causes. If all possible causes but one were eliminated,
then (Bayesian) reasoning forced him to conclude that the remaining possible cause was
fully credible, even if it seemed improbable at the start.

Figure 2.1 illustrates Holmes’ reasoning. For the purposes of illustration, we suppose
that there are just four possible causes of the outcome to be explained. We label the
causes A, B, C, and D. The heights of the bars in the graphs indicate the credibility
of the candidate causes. (“Credibility” is synonymous with “probability”; here I use
the everyday term “credibility” but later in the book, when mathematical formalisms
are introduced, I will also use the term “probability.”) Credibility can range from zero
to one. If the credibility of a candidate cause is zero, then the cause is definitely not
responsible. If the credibility of a candidate cause is one, then the cause definitely is
responsible. Because we assume that the candidate causes are mutually exclusive and
exhaust all possible causes, the total credibility across causes sums to one.

Figure 2.1 illustrates Holmes’ reasoning. For the purposes of illustration, we suppose
that there are just four possible causes of the outcome to be explained. We label the
causes A, B, C, and D. The heights of the bars in the graphs indicate the credibility
of the candidate causes. (“Credibility” is synonymous with “probability”; here I use
the everyday term “credibility” but later in the book, when mathematical formalisms
are introduced, I will also use the term “probability.”) Credibility can range from zero
to one. If the credibility of a candidate cause is zero, then the cause is definitely not
responsible. If the credibility of a candidate cause is one, then the cause definitely is
responsible. Because we assume that the candidate causes are mutually exclusive and
exhaust all possible causes, the total credibility across causes sums to one.

Figure 2.1 The upper-left graph shows the credibilitiesof the four possible causes for an outcome. The
causes, labeled A, B, C, and D, are mutually exclusive and exhaust all possibilities. The causes happen
to be equally credible at the outset; hence all have prior credibility of 0.25. The lower-left graph shows
the credibilitieswhen one cause is learned to be impossible.The resulting posterior distribution is used
as the prior distribution in the middle column, where another cause is learned to be impossible. The
posterior distribution from the middle column is used as the prior distribution for the right column.
The remaining possible cause is fully implicated by Bayesian reallocation of credibility.

prior knowledge suggested that rain may be a more likely cause than a newly erupted
underground spring, the present illustration assumes equal prior credibilities of the
candidate causes. Suppose we make new observations that rule out candidate cause A.
For example, if A is a suspect in a crime, we may learn that A was far from the crime
scene at the time. Therefore, we must re-allocate credibility to the remaining candidate
causes, B through D, as shown in the lower-left panel of Figure 2.1. The re-allocated
distribution of credibility is called the posterior distribution because it is what we believe
after taking into account the new observations. The posterior distribution gives zero
credibility to cause A, and allocates credibilities of 0.33 (i.e., 1/3) to candidate causes B,
C, and D.

The posterior distribution then becomes the prior beliefs for subsequent observations.
Thus, the prior distribution in the upper-middle of Figure 2.1 is the posterior
distribution from the lower left. Suppose now that additional new evidence rules out
candidate cause B.

Data are noisy and inferences are probabilistic

In scientific research, measurements are replete with randomness. Extraneous influences
contaminate the measurements despite tremendous efforts to limit their intrusion.
For example, suppose we are interested in testing whether a new drug reduces blood
pressure in humans.We randomly assign some people to a test group that takes the drug,
and we randomly assign some other people to a control group that takes a placebo. The
procedure is “double blind” so that neither the participants nor the administrators know
which person received the drug or the placebo (because that information is indicated by
a randomly assigned code that is decrypted after the data are collected).We measure the
participants’ blood pressures at set times each day for several days. As you can imagine,
blood pressures for any single person can vary wildly depending on many influences,
such as exercise, stress, recently eaten foods, etc. The measurement of blood pressure is
itself an uncertain process, as it depends on detecting the sound of blood flow under a
pressurized sleeve. Blood pressures are also very different from one person to the next.
The resulting data, therefore, are extremely messy, with tremendous variability within
each group, and tremendous overlap across groups.