Bayesian Theory
Is the soil safe?
A city plans to redevelop land once used as a power plant dumping site.
Before approving construction, environmental tests are conducted to detect chemical contamination.
The city will only proceed if contamination appears limited and manageable.
However, environmental tests are not perfect.
How should the city interpret the results?
Environmental Testing Assumptions
Before redevelopment begins, soil samples are collected and tested for contamination. Based on historical data, approximately 10% of sample sites are contaminated. The test used by the city correctly identifies contaminated land 90% of the time (this is the test’s sensitivity). However, the test is not perfect; even when land is clean, it incorrectly reports contamination 15% of the time (this is the false positive rate).
These uncertainties must be considered when interpreting a positive test result.
Theoretically,
Imagine 1000 sites tested:
900 clean
100 contaminated
85% true negative (or specificity)
15% false positive rate
90% sensitivity
10% false negative rate
135 positive
765 negative
90 positive
10 negative
False negative = 1 - sensitivity
Run Test
Theoretically,
Imagine 1000 sites tested:
900 clean
100 contaminated
85% true negative (or specificity)
15% false positive rate
10% false negative rate
90% sensitivity
False negative = 1 - sensitivity
Run Test
765 negative
135 positive
90 positive
10 negative
Bayes' Theorem
Why 40%?
Even though the tests are 90% sensitive, a positive result only means a 40% chance of contamination because true contamination is relatively uncommon compared to false positives.
Updating our beliefs after observing new evidence
P(+|C)P(C)
___________
P(C|+) =
P(+|C)P(C) + P(+|C )P(C )
(0.1)(0.9)
___________
=>
(0.9)(0.1) + (0.15)(0.9)
135 false positives
90 truly contaminated sites
0.4
What should the city do next?
- 60% of results are false positives
- Shutting the project down may have significant costs
- Retesting would reduce uncertainty dramatically
Why?
90 true positives
135 false positives
Let's retest!
Run test
10% false negative rate
15% false positive rate
20.25 false positives
9 false negatives
81 true positives
114.75 true negatives
90% sensitivity
85% specificity
90 true positives
135 false positives
Let's retest!
This demonstrates Bayesian updating; the posterior from the first round of testing becomes the prior for the second round
Run test
10% false negative rate
15% false positive rate
20.25 false positives
9 false negatives
81 true positives
81.75 true negatives
90% sensitivity
85% specificity
One test is evidence, not truth
Given that a site tests positive after a single round of testing, there is only a 40% chance the site is actually contaminated. After a second round, that chance jumps to about 80%. Repeated testing allows us to increase the threshold of certainty, and Bayesian reasoning in instances like these protects public health and safety.
Ryan Knott - MATH351 Honors Project
Ryan Knott
Created on February 21, 2026
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Transcript
Bayesian Theory
Is the soil safe?
A city plans to redevelop land once used as a power plant dumping site. Before approving construction, environmental tests are conducted to detect chemical contamination. The city will only proceed if contamination appears limited and manageable. However, environmental tests are not perfect. How should the city interpret the results?
Environmental Testing Assumptions
Before redevelopment begins, soil samples are collected and tested for contamination. Based on historical data, approximately 10% of sample sites are contaminated. The test used by the city correctly identifies contaminated land 90% of the time (this is the test’s sensitivity). However, the test is not perfect; even when land is clean, it incorrectly reports contamination 15% of the time (this is the false positive rate). These uncertainties must be considered when interpreting a positive test result.
Theoretically,
Imagine 1000 sites tested:
900 clean
100 contaminated
85% true negative (or specificity)
15% false positive rate
90% sensitivity
10% false negative rate
135 positive
765 negative
90 positive
10 negative
False negative = 1 - sensitivity
Run Test
Theoretically,
Imagine 1000 sites tested:
900 clean
100 contaminated
85% true negative (or specificity)
15% false positive rate
10% false negative rate
90% sensitivity
False negative = 1 - sensitivity
Run Test
765 negative
135 positive
90 positive
10 negative
Bayes' Theorem
Why 40%?
Even though the tests are 90% sensitive, a positive result only means a 40% chance of contamination because true contamination is relatively uncommon compared to false positives.
Updating our beliefs after observing new evidence
P(+|C)P(C)
___________
P(C|+) =
P(+|C)P(C) + P(+|C )P(C )
(0.1)(0.9)
___________
=>
(0.9)(0.1) + (0.15)(0.9)
135 false positives
90 truly contaminated sites
0.4
What should the city do next?
Why?
90 true positives
135 false positives
Let's retest!
Run test
10% false negative rate
15% false positive rate
20.25 false positives
9 false negatives
81 true positives
114.75 true negatives
90% sensitivity
85% specificity
90 true positives
135 false positives
Let's retest!
This demonstrates Bayesian updating; the posterior from the first round of testing becomes the prior for the second round
Run test
10% false negative rate
15% false positive rate
20.25 false positives
9 false negatives
81 true positives
81.75 true negatives
90% sensitivity
85% specificity
One test is evidence, not truth
Given that a site tests positive after a single round of testing, there is only a 40% chance the site is actually contaminated. After a second round, that chance jumps to about 80%. Repeated testing allows us to increase the threshold of certainty, and Bayesian reasoning in instances like these protects public health and safety.