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The Bayesian approach to computer-aided diagnosis

This is a section from Martin Bland’s text book *An Introduction to Medical Statistics,
Fourth Edition.* I hope that the topic will be useful in its own right,
as well as giving a flavour of the book. Section references are to the
book.

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22.3 An example: the Bayesian approach to computer-aided diagnosis

Bayes’ theorem may be stated in terms of the probability
of Diagnosis A having observed Data B, as:

PROB(Diagnosis A | Data B)
is proporional to PROB(Data B | Diagnosis A)
× PROB(Diagnosis A)

If we have a large dataset of known diagnoses and
their associated symptoms and signs, we can estimate
PROB(Diagnosis A) easily. It is simply the proportion of
times A has been diagnosed. For a patient, the data B are
the particular combination of signs and symptoms with
which the patient presents. The problem of finding the
probability of a particular combination of symptoms and
signs for each diagnosis is more difficult. We can say that
the probability of a given symptom for a given diagnosis
is the proportion of times the symptom occurs in patients
with that diagnosis. If the symptoms are all independent,
the probability of any combination of symptoms can
be then found by multiplying their individual probabilities
together (Section 6.2). In practice the assumption
that signs and symptoms are independent is most unlikely
to be met and a more complicated analysis would
be required to deal with this. However, some systems
of computer-aided diagnosis have been found to work
quite well with the simple approach.

We thus have the probability of each diagnosis and
the probability of each combination of symptoms and
signs for each diagnosis. When a new patient presents,
we obtain the data and compute

PROB(Data|Diagnosis) × PROB(Diagnosis)

for each diagnosis and sum these. We then divide the
product by this sum for each diagnosis and this gives
us the probability for each diagnosis given the signs and
symptoms.

PROB(Diagnosis A) is called the **prior probability** of
Diagnosis A, because it is the probability of Diagnosis A
before the data are observed. PROB(Diagnosis A|Data B)
is called the **posterior probability** of Diagnosis A given
Data B, the probability of the diagnosis for someone
with the observed signs and symptoms denoted by B.
PROB(Data B|Diagnosis A) is called the **likelihood** of
Diagnosis A for Data B, the probability of the observed
signs and symptoms for someone with the diagnosis.

Adapted from pages 357–358 of
*An Introduction to Medical Statistics* by Martin Bland, 2015,
reproduced by permission of
Oxford University Press.

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Last updated: 7 August, 2015

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