​​The Bayesian Radiologist

If I were to anoint the Patron Saint of Radiology he would not be Wilhelm Rontgen or Henry Pancoast. He would be Reverend Thomas Bayes, an eighteenth-century English minister whose insights about inverse probability is known as Bayes’ Theorem.  

The derivation of Bayes’ theorem is cumbersome. However, Bayesian logic, or Bayesianism, is simple to understand. It is crucial to diagnostic imaging. Many radiologists are Bayesians without realizing, although some are more Bayesian than others.

The components of Bayesian logic

When simplified, stripped to its bare essentials, Bayesian logic comprises three elements.

a)    Probability of A conditional upon B. P (A/B).
b)    Probability of B conditional upon A.P (B/A).
c)    Probability of A. P (A).

P (A/B)

The operative word is “conditional.” The probability of an event A conditional on B is the probability of A given the presence of B. This sounds clumsy but it is important to understand and be so good at articulating “conditional” that it is part of our vocabulary.

Here is an example of conditional. There was a certain probability that Andy Murray would win the 2012 Wimbledon tennis tournament. Let’s call the probability of Murray winning, P (A). The event, B, is Roger Federer beating Novak Djokovic in the semi-finals. P (A/B) is the probability that Murray would win the cup given that Federer beats Djokovic in the semi-finals. It is fair to say that whatever was P (A), and the home crowd cheering Murray surely elevated P (A), P (A/B) was lower, or not the same, as P (A).

When the probability of an event, such as Murray winning the tournament, given another event or information about that event, is not the same as the probability of the event in the absence of the information, the information is valuable.

This means that if P (A/ B) ≠ P (A) then B is valuable to know. Note, it is not necessary that P (A/B) be higher than P (A) for B to be valuable information. Information is valuable when it alters, that is it increases or decreases, the probability of an event conditional upon it.

If P (A/B) = P (A) then A and B are independent. B is redundant information. This information may not be worth pursuing. If the probability of rain in the Napa Valley conditional upon a heat wave in Mongolia is the same as the chances of rain in Napa Valley without a heat wave in Mongolia, the two events are independent.

It is possible that the chances of Murray winning the 2012 Wimbledon was independent of whether Federer or Djokovic won the semi-final, although Federer might have disputed that independence, as perhaps might have Djokovic.

P (B/A)

P (B/A), or the chances of B given A, is also a conditional probability. It is easy to conflate P (A/B) and P (B/A), but they are not the same. The chances of A conditional on B are not the same as the chances of B conditional on A. An extreme example will clarify. The chances that we open our umbrellas if it rains are not the same as the chances that it will rain just because we open our umbrellas.

If P (A/B) is the same as P (B/A) then A and B are perfectly correlated and go hand in hand. This means that knowing B is knowing A, the absence of B excludes A with certainty, and B imputes A without doubt. In imaging, and clinical medicine, we rarely encounter such a relationship.

More commonly in imaging we must distinguish between P (A/B) and P (B/A). Let us take thickened colonic wall on a CT scan. Let’s call the thickened colonic wall B and mesenteric ischemia, a known cause of wall thickening, A.

The chances that the colonic wall is thickened in mesenteric ischemia, P (B/A), are not the same as the chances that there is mesenteric ischemia given wall thickening, P (A/B).

The probability of mesenteric ischemia given wall thickening is an inverse probability problem. Inverse probability lies in the soul of valuable information. Clinicians aren’t as interested in knowing what the chances of bowel wall thickening given the patient is known to have mesenteric ischemia, as they are in knowing the chances of mesenteric ischemia given the bowel wall is thickened. The effect is only interesting because the cause is worth knowing. The cause of wall thickening is worth knowing because it is dangerous if untreated. It is through inverse probability that information, effect, becomes valuable by revealing its cause.

Bayesian logic states that P (A/B) is proportional to P (B/A). This is intuitive. The more likely there is bowel wall thickening with mesenteric ischemia, the more likely there is mesenteric ischemia with bowel wall thickening.

P (A)

The third component of Bayesian logic is P (A), which is the probability of A, or its prior probability. P (A/B) depends on P (A). This, again, is intuitive but has implications. It is worth thinking slowly about this to understand the implications.

The probability of mesenteric ischemia given the presence of bowel wall thickening depends on the probability of mesenteric ischemia before we knew about the thickened wall - the prior probability of mesenteric ischemia. Prior probability is also known as pre-test probability or “prior.” Prior depends on demographics, clinical history, physical examination and laboratory tests.

No matter how high is P (B/A), if P (A) is low, extremely low, P (A/B) is also low. That is P (A) can bring P (B/A) to its knees. A positive, or negative, diagnostic test is only as correct as the reason for getting the test is right. This is rather humbling for radiologists.

Bayesian logic explains the aphorism “rubbish in equals rubbish out.” The “rubbish in’” is a low P (A). The “rubbish out” is a misguided attempt by P (B/A) to compensate for the low P (A). But note a second implication. The reason that P (A) is important is because P (B/A) is not P (A/B). Prior probability is important because diagnostic imaging is imperfect. If diagnostic tests were perfect we could care less about priors.

Revision of probabilities

P (A) is the prior, B is new information and P (A/B) is the posterior probability or the revised probability. P (A/B) then becomes the new prior.

Here is an example of revising probabilities. There is a certain prior probability that a 65 year old man with abdominal pain has mesenteric ischemia. This is revised by knowing that he suffers from atrial fibrillation, because he can have thrombus in the left atrial appendage which has embolized to the superior mesenteric artery. The prior is altered by the fact that he is on Coumadin and has a therapeutic INR because the chances that thombus has formed are lower. The prior is again altered because he has a normal serum lactate.

He has a CT angiogram to look for mesenteric ischemia. The prior probability of mesenteric ischemia just before the CT is the probability revised from the clinical information up till that point. A negative CT scan alters that prior, yielding a posterior probability, a new prior.

Radiologists are Bayesian Generators

Diagnostic imaging provides information so that probabilities can be revised. In the Bayesian relationship, the radiologist looks for B, which in the current example is bowel wall thickening. A competent radiologist is accurate at P (B/A).

However, meaningful interpretation lists not just B, but P (A/B). A meaningful interpretation reports the posterior probability, qualitatively or quantitatively, of the condition suspected. This obliges the interpreter to incorporate P (A), the prior probability, such as that of mesenteric ischemia.

Bayesian logic explains why imaging should not be interpreted in a clinical vacuum. If imaging is read without knowledge of the patient, P (A) will not be known. If P (A) is not known, P (B/A) is a shot in the dark - a shot which may reach its target, or may not.

Which type of radiologist do you wish to be – the generator of P (B/A) or P (A/B)?

Further Reading

  1. The best explanation of Bayes’ Theorem I have read is in The Signal and the Noise, by Nate Silver. Pages 240-261. This is an enjoyable book, even beyond an explanation of Bayes.
  2. Bayesian analysis of electrocardiographic stress testing. NEJM.

Lecture 2

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