Breast Cancer


"Subjectivity (Ed, i.e., the prior) is sometimes seen as a deficiency of Bayesian inference. Others regard it as a powerful advantage; it permits us to express our personal experience mathematically and combine it with data in a principled and transparent way. Bayesโ€™s rule informs our reasoning in cases where ordinary intuition fails us or where emotion might lead us astray. We will demonstrate this power in a situation familiar to all of us.

Suppose you take a medical test to see if you have a disease, and it comes back positive. How likely is it that you have the disease? For specificity, letโ€™s say the disease is breast cancer, and the test is a mammogram."

Pearl, Judea. The Book of Why: The New Science of Cause and Effect (pp. 104-105). Basic Books. Kindle Edition.

Representing the Problem Domain as a Bayesian Network

  • We implement this example as a causal Bayesian network, which means the arc between Breast Cancer and Mammogram represents a causal relationship.

  • The resulting Bayesian network in XBL format is available here: BoW_BreastCancer.xbl

Should I worry about a positive test result?

"Suppose a forty-year-old woman gets a mammogram to check for breast cancer, and it comes back positive. The hypothesis, D (for โ€œdiseaseโ€), is that she has cancer. The evidence, T (for โ€œtestโ€), is the result of the mammogram. How strongly should she believe the hypothesis? Should she have surgery?" (Pearl, p. 105)

Calculating the Cancer Risk with BayesiaLab

  • We use the probabilities described by Pearl to set the parameters of the Causal Bayesian Network:

    • For a typical forty-year-old woman, the probability of getting breast cancer in the next year is about one in seven hundred, 0.14%. We use that as our prior;

    • The sensitivity (true-positive) of a mammogram is 73%;

    • The specificity (true-negative) of a mammogram is 88%.

  • Notice the Input component Breast Cancerโ€”Your Prior Estimate in the WebSimulator. This allows you to set your own initial belief that a patient has breast cancer.

  • Upon setting Mammogram=Positive as Hard Evidence, the probability of Breast Cancer=True increases from 0.14% to 0.86%.

Counterintuitive Results

"The conclusion is startling. I think that most forty-year-old women who have a positive mammogram would be astounded to learn that they still have less than a 1 percent chance of having breast cancer. Figure 3.3 might make the reason easier to understand: the tiny number of true positives (i.e., women with breast cancer) is overwhelmed by the number of false positives."(Pearl, p. 106)

Should I worry now?

"However, the story would be very different if our patient had a gene that put her at high risk for breast cancerโ€”say, a one-in-twenty chance within the next year. [...]

For a woman in this situation, the chances that the test provides lifesaving information are much higher. That is why the task force continued recommending annual mammograms for high-risk women.

This example shows that P(disease | test) is not the same for everyone; it is context-dependent (Ed: it depends on the prior). If you know that you are at high risk for a disease to begin with, Bayesโ€™s rule allows you to factor that information in. Or if you know that you are immune, you need not even bother with the test!" (Pearl, pp. 107โ€“108)

Recalculating the Risk

  • To answer this question with BayesiaLab, you can either modify the model by setting the prior of Breast Cancer to 5% via the Node Editor, or you can set a Probabilistic Evidence via the Monitor.

  • In the WebSimulator, you would set the Input Breast Cancerโ€”Your Prior Estimate (initial belief) to 5%.

  • Upon setting Mammogram=Positive, the probability of Breast Cancer=True increases to 24.25%.

Visualizing the Impact of the Prior

  • To illustrate the impact of the prior (or prevalence), we added a parent node to Breast Cancer for defining such prior. This is what we call a "hyperparameter."

  • The updated network, including the hyperparameter, is available here: BoW_BreastCancer_Prevalence.xbl

  • You can now set Mammogram=Positive as Hard Evidence.

  • With this evidence set, you can use Target Mean Analysis to explore a range of values for the prior, from 0% to 100%: Main Menu > Analysis > Visual > Target > Target's Posterior > Curves > Total Effects.

  • You will obtain a plot in which the x-axis represents the prior of Breast Cancer=True, i.e., the hyper-parameter.

  • The y-axis represents the updated probability of Breast Cancer=True given a positive mammogram result.

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