Bayes' rule is named after Thomas Bayes (minister, mathematician) who established a mathematical basis for probability inference. Bayesian logic is a foundation of many decision procedures used in the area of artificial intelligence and intelligent systems.

Why should a business keep tests that may have stigmas attached to them confidential? For example, tests such as drug tests, AIDS tests, cancer tests, etc.

Instead of stating opinions, let us quantify the problem and see what qualitative results in the form of general guidance follow from the quantitative analysis.

Assume that the occurrence of cancer in the general population is 0.4%. You take a test for cancer. The test is 98.0% accurate in predicting cancer. The test results are positive. Given that this is the only information available, what is the probability that you have cancer?

Adapted from Paulos, J. (1988). Innumeracy: Mathematical illiteracy and its consequences. New York: Hill and Wang (pp. 89-90).

That is, you take a test for cancer. The test results are positive. This is a Tp or true-positive.

What assumptions are we making?

We are assuming that the information given is the only information available about the situation.

In real life, a doctor would use other additional information to make a diagnosis.

• For a drug test, a manager might use that as the only information available.
• For a certification exam, this might be the only information available.

Let us quantify the cancer test problem using Bayes' rule and use sensitivity analysis to analyze the solution.

Everyone either has cancer or does not have cancer. (two-valued logical assumption)

• Cn is set of people without cancer (Cancer no)
• Cy is set of people with cancer (Cancer yes).

Events Cy and Cn are mutually exclusive and collectively exhaustive.

This is two-valued logic, as opposed to multi-valued (i.e., fuzzy) logic.

What do we know about cancer probabilities?

• Events Cy and Cn are mutually exclusive since P(Cn ∩ Cy) = 0.0.
• Events Cy and Cn are collectively exhaustive since P(Cn ∪ Cy) = 1.0.

Question: Do we know P(Cy) and do we know P(Cn)?

Yes, we know that P(Cy) is 0.004 and that P(Cn) is 0.996.

• P(Cn) = 0.996
• P(Cn) + P(Cy) = 1.000
• P(Cy) = 1.000 - P(Cn) = 0.004

Common error: converting 0.4% to 0.04, not 0.004.

Note: The size of the sets in a Venn diagram do not represent their relative weight.

A test T is either positive (cancer) or negative (free of cancer)

• Tn is a negative test for cancer (free of cancer).
• Tp is a positive test for cancer (cancer).

Events Tp and Tn are mutually exclusive and collectively exhaustive.

We are making certain assumptions.

• Events Tp and Tn are mutually exclusive since P(Tn ∩ Tp) = 0.0.
• Events Tp and Tn are collectively exhaustive since P(Tn ∪ Tn) = 1.0.

Question: Do we know P(Tp)? No, we only know that the probability that a person tests positive for cancer given that they have cancer is 0.98, or 98.0%. This is the conditional probability P(Tp | Cy) and not the probability P(Tp).

This is what we want to determine/calculate!

Prob(A | B) is the probability that A is true given that B is true.

• P(Tp | Cy) = 0.98 = P(Tp ∩ Cy) / P(Tp)
• P(Tp | Cn) = 0.02 = P(Tp ∩ Cn) / P(Tp)
• P(Tn | Cn) = 0.98 = P(Tn ∩ Cy) / P(Tn)
• P(Tn | Cy) = 0.02 = P(Tn ∩ Cn) / P(Tn)

In real life, not all tests performance results are symmetric.

• Make a prediction (decision)
• See what happens (state of nature).

The prediction is the result of a test T for cancer.

The result can be true or false.

• Predict it true and it happens (true), a true-positive.
• Predict it true and it does not happen (false), a false-positive, or Type II error.
• Predict it false but it does happen (true), a false-negative, or Type I error.
• Predict it false and it does not happen (false), a true-negative.

• Test predicts cancer, person has cancer (true-positive, Tp given Cy).
• Test predicts cancer, person does not have cancer (false-positive, Tp given Cn).
• Test predicts no cancer, person has cancer (false-negative, Tn given Cy).
• Test predicts no cancer, person does not have cancer (true-negative, Tn given Cn).

True-positives and true-negatives are not logical problems, although in this case a true-positive is a tremendous personal problem or tragedy.

False-positives and false-negatives are logical problems and can also be tremendous personal problems.

Question: Why can a false-negative result be a problem? A false-negative result means that the test indicates that you do not have cancer, but you do have cancer.

Question: Why can a false-positive result be a problem?

A false-positive result means that the test indicates that you do have cancer, but you do not have cancer. Here is the Venn diagram showing the four possibilities.

We have P(Tp | Cy). We want P(Cy | Tp).

In general, P( Cy | Tp ) is not the same as P( Tp | Cy ).

The following are not the same.

• The probability that you test positive for cancer given that you have cancer (very high).
• The probability that you have cancer given that you test positive for cancer (to be determined).

The probability that someone speaks Spanish
given that they live in Spain
is very high. The probability that someone lives in Spain
given that they speak Spanish
is very low.

The probability that one has cancer, given that one has tested positive for cancer is

P(Cy | Tp)

You are given P(Tp | Cy). What is P(Cy | Tp)? Question: Can we determine P(Cy | Tp)?

Let us expand our definitions and see what happens.

We have:

P(Tp | Cy) = 0.98 = P(Tp ∩ Cy) / P(Tp)

Question: Can we get it?

Note the common denominator P(Tp ∩ Cy) on the right hand side. Let us move terms around in order to remove this denominator.

P(Tp | Cy) = 0.98 = P(Tp ∩ Cy) / P(Tp)

= ( P(Tp) * P(Tp | Cy) ) / P(Cy)

= P(Cy) * P(Tp | Cy) / P(Tp)

= 0.004 * 0.98 / P(Tp)

Question: But what is P(Tp)?

We do not know, but we can use a truth table tautology to indirectly determine P(Tp). Note: Cn = ¬ Cy

P(Tp)

= P(Cy & Tn) + P((¬ Cy) & Tp)
= P(Cy & Tn) + P(Cn & Tp)

Question: How do we find the relevant intersections?

By using the conditional probability rule.

P( Tp | Cy ) = P( Cy ∩ Tp ) / P( Cy )

Multiplying both sides by P( Cy ):

P( Tp | Cy ) * P( Cy ) = P( Cy ∩ Tp )

Reverse the equality:

P( Cy ∩ Tp ) = P( Tp | Cy ) * P( Cy )

We can now calculate the intersection probabilities:

P(Cn ∩ Tn) = P(Tn | Cn) * P(Cn) = 0.02 * 0.996

P(Cy ∩ Tn) = P(Tn | Cy) * P(Cy) = 0.02 * 0.004

P(Cn ∩ Tp) = P(Tp | Cn) * P(Cn) = 0.98 * 0.996

P(Cy ∩ Tp) = P(Tp | Cy) * P(Cy) = 0.98 * 0.004

P(Cn ∩ Tn) = 0.02 * 0.996 ; P(Cy ∩ Tn) = 0.02 * 0.004

P(Cn ∩ Tp) = 0.98 * 0.996 ; P(Cy ∩ Tp) = 0.98 * 0.004

Question: Which Intersection probabilities do we need?

Since the test is positive, we need P(Cy) ∩ P(Tn) and we need P(Cy) ∩ P(Tp). Now expand the terms. Now simplify. Is this simple enough?

Question: Why is the last formula the most commonly used one?

Because it is the most impressive. Also, in the real world, people are more comfortable estimating conditional probabilities than intersection probabilities.

We have a simpler form of Bayes Rule.

P(Cy |Tp)

= (0.98 * 0.004) / ((0.98 * 0.004) + (0.02 * 0.996))
= 0.00392 / (0.00392 + 0.01992)
≈ 0.004 / (0.004 + 0.02)
= 0.004 / 0.024
= 4 / 24
= 1 / 6
≈ 0.167.
= 16.7%

It is not 98.0%!

Well-known result: This happens when

• the false-positive rate is high (in an absolute sense) and
• when a disease is rare (in a relative sense, such as 0.004), and
• even if the test has a high accuracy (in a relative sense, such as 0.98).

We assumed that the information given was the only information available.

In most medical diagnoses, there is often additional (nonindependent) information with which to make diagnostic decisions.

Question: Why should a business keep tests that may have stigmas attached to them confidential. For example, tests such as drug tests, AIDS tests, cancer tests, etc. A business should keep such tests confidential because the results must be interpreted appropriately. In this case, the probability that the person has cancer given that they tested positive for cancer is not 98.0% but 16.7%.

Question: What is the appropriate action to take when someone tests positive on a test that has a stigma attached to it. 1. Do additional testing. 2. Keep the information confidential.

Do such situations occur in real life?

What worries some about drug tests is their reputed inaccuracy. John P. Morgan, professor of pharmacology at the City University of New York Medical School says the false-positive rate on a typical unconfirmed drug test is 10 to 15 percent. Hawkins, D. (September 15, 1997). "Who's watching now?", in U.S. News & World Report, p. 56-58.

What does this mean? Our results were for a test with an accuracy of 0.98.

A stacked bar chart displays the before and after results of a positive test for cancer with a test accuracy of 0.98.

Notice that a positive test for cancer has increased the probability from 0.004 to 0.167, an increase in probability by a factor of 42 times.

A positive test for cancer dramatically increases the probability that the patient does have cancer, but the probability is not 98.0%, which would be an increase of 24,500 times.

A sensitivity analysis answers the question, "What happens if input parameters of the problem are varied?".

We can perform a sensitivity analysis of this problem by asking the question, "What happens if the accuracy of the test varies?".

To do a sensitivity analysis, we need to generalize our result for a test accuracy of 98.0% to handle a range of accuracies. For example, 90.0% to 100.0%.

Specific result: P(Cy | Tp) = (0.98 * 0.004) / ((0.98 * 0.004) + (0.02 * 0.996))

P(Cy | Tp)

= (0.98 * 0.004) / ((0.98 * 0.004) + (0.02 * 0.996))
= P(Tp | Cy) * P(Cy) /
( P(Tp | Cy) * P(Cy) + P(Tp | Cn) * P(Cn) )

The probability increases to 100.0% only as the test accuracy approaches 100.0%.

Question: What happens as the accuracy of the test decreases from 100.0%? The probability that one has cancer given that the test is positive decreases exponentially.

How do these results apply to lie detector tests where the accuracy of the test is much less?

Note: This analysis does not include factors such as the probability that the lab made a mistake. Factors such as these are important in DNA evidence used in legal court cases.

In general, Bayes' Rule allows information of the form

P(A | B)

to be converted to the form

P(B | A)

• Clemen, R. (1991). Making hard decisions: an introduction to decision analysis. Boston, MA: PWS-Kent Publishing.



• Heckerman, D. (1990). Probabilistic similarity networks. Cambridge, MA: The MIT Press.
• Pearl, J. (1988). Probabilistic reasoning systems, 2nd revised printing. San Francisco, CA: Morgan Kaufman.