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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.

Here is a simplified illustration of Bayesian inference when data are noisy. Suppose
there is a manufacturer of inflated bouncy balls, and the balls are produced in four
discrete sizes, namely diameters of 1.0, 2.0, 3.0, and 4.0 (on some scale of distance
such as decimeters). The manufacturing process is quite variable, however, because of
randomness in degrees of inflation even for a single size ball. Thus, balls of manufactured
size 3 might have diameters of 1.8 or 4.2, even though their average diameter is 3.0.
Suppose we submit an order to the factory for three balls of size 2.We receive three balls
and measure their diameters as best we can, and find that the three balls have diameters
of 1.77, 2.23, and 2.70. From those measurements, can we conclude that the factory
correctly sent us three balls of size 2, or did the factory send size 3 or size 1 by mistake,
or even size 4?

好资料！

Identify the data relevant to the research questions.What are the measurement scales
of the data? Which data variables are to be predicted, and which data variables are
supposed to act as predictors?

Define a descriptive model for the relevant data. The mathematical form and its
parameters should be meaningful and appropriate to the theoretical purposes of the
analysis.

The second step is to define a descriptive model of the data that is meaningful
for our research interest. At this point, we are interested merely in identifying a basic
trend between weight and height, and it is not absurd to think that weight might be
proportional to height, at least as an approximation over the range of adult weights and
heights. Therefore, we will describe predicted weight as a multiplier times height plus a
baseline. We will denote the predicted weight as ˆy (spoken “y hat”), and we will denote
the height as x. Then the idea that predicted weight is a multiple of height plus a baseline
can be denoted mathematically as follows:

The coefficient, β1 (Greek letter “beta”), indicates how much the predicted weight
increases when the height goes up by one inch.2 The baseline is denoted β0 in
Equation 2.1, and its value represents the weight of a person who is zero inches tall.
You might suppose that the baseline value should be zero, a priori, but this need not be
the case for describing the relation between weight and height of mature adults, who
have a limited range of height values far above zero. Equation 2.1 is the form of a line,

As outlined above, Bayesian data analysis is based on meaningfully parameterized
descriptive models. Are there ever situations in which such models cannot be used or
are not wanted?
One situation in which itmight appear that parameterizedmodels are not used is with
so-called nonparametric models. But these models are confusingly named because they
actually do have parameters; in fact they have a potentially infinite number of parameters.
As a simple example, suppose we want to describe the weights of dogs. We measure the
weights of many different dogs sampled at random from the entire spectrum of dog
breeds. The weights are probably not distributed unimodally, instead there are probably
subclusters of weights for different breeds of dogs. But some different breeds might
have nearly identical distributions of weights, and there are many dogs that cannot be
identified as a particular breed, and, as we gather data from more and more dogs, we
might encounter members of new subclusters that had not yet been included in the
previously collected data. Thus, it is not clear how many clusters we should include
in the descriptive model. Instead, we infer, from the data, the relative credibilities of
different clusterings. Because each cluster has its own parameters (such as location and
scale parameters), the number of parameters in the model is inferred, and can grow
to infinity with infinite data. There are many other kinds of infinitely parameterized
models. For a tutorial on Bayesian nonparametricmodels, see Gershman and Blei (2012);
for a recent review, see Müller and Mitra (2013); and for textbook applications, see
Gelman et al. (2013). We will not be considering Bayesian nonparametric models in
this book.

There are a variety of situations in which it might seem at first that no parameterized
model would apply, such as figuring out the probability that a person has some rare
disease if a diagnostic test for the disease is positive. But Bayesian analysis does apply even
here, although the parameters refer to discrete states instead of continuous distributions.
In the case of disease diagnosis, the parameter is the underlying health status of the
individual, and the parameter can have one of two values, either “has disease” or “does
Introduction: Credibility, Models, and Parameters 31
not have disease.” Bayesian analysis re-allocates credibility over those two parameter
values based on the observed test result. This is exactly analogous to the discrete
possibilities considered by Sherlock Holmes in Figure 2.1, except that the test results
yield probabilistic information instead of perfectly conclusive information. We will do
exact Bayesian computations for this sort of situation in Chapter 5 (see specifically
Table 5.4).

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She will likely be canonized in September to coincide with the 19th anniversary of her death and Pope Francis' Holy Year of Mercy, according to an Italian Catholic newspaper report.

The pontiff marked his 79th birthday on Thursday by approving a decree that the nun had performed a second miracle, the Vatican said in a statement.

The pontiff marked his 79th birthday on Thursday by approving a decree that the nun had performed a second miracle, the Vatican said in a statement.

哈哈

Francis created six new saints in 2014 — two Indians and four Italians — praising their "creative" commitment to helping the poor.

The miracle needed for her canonization concerned the cure of a Brazilian man with a brain illness, according to a report in Avennire, the newspaper of the Italian bishops' conference.

Despite winning global acclaim for her life in helping the poor, some have criticized standards of health care provided by her mission and raised questions over her conservative approach to contraception and medicine.

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