What is the difference between prior and posterior probability?

The prior probability P(A) is the probability before observing evidence, while the posterior probability P(A|B) is the updated probability after evidence B is observed. With repeated observations, the posterior can become the new prior for the next Bayesian update.

What happens when the false positive rate is high?

The higher the false positive rate, the lower the probability that a positive result is truly positive (positive predictive value). This is especially important in screening for rare diseases because the prior probability is low.

How is Bayes' theorem used in CAE?

It is used in model uncertainty quantification (UQ). A typical example is Bayesian estimation, where simulation results and experimental data are compared to update the probability distributions of model parameters.

Bayes' Theorem Visualizer

Q: What is Bayes' theorem?

P(A|B)=P(B|A)P(A)/P(B). It calculates how the probability of hypothesis A changes when new evidence B is observed. It is widely used in medical diagnosis, machine learning, and quality control.

Parameters

P(A) Prior probability (disease prevalence) 0.300

P(B|A) Sensitivity (true positive rate) 0.900

P(B|¬A) False positive rate 0.100

P(A|B) Posterior probability (positive predictive value)

75.0%

Results

P(B) Positive test rate

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Bayes Factor K

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Specificity 1-P(B|¬A)

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Negative predictive value P(¬A|¬B)

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Probability Tree

Prior to posterior

Population Grid

Tree

Probability tree. Line thickness represents probability magnitude.

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That is the base-rate fallacy. In a population of 1000 people with 1% prevalence, a test with 99% sensitivity and 95% specificity finds about 9.9 true positives among 10 patients, but also about 49.5 false positives among 990 healthy people. Of the 59.4 positives, only 9.9 are true positives, about 17%. Even with high sensitivity, positive predictive value stays low when the prior probability is low.

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I heard Bayes' theorem can be used repeatedly. What does that mean?

Theory & Key Formulas

$P(A|B) = \dfrac{P(B|A)\,P(A)}{P(B)}$
$P(B) = P(B|A)P(A) + P(B|\neg A)P(\neg A)$
Bayes factor: $K = \dfrac{P(B|A)}{P(B|\neg A)}$

Frequently Asked Questions

If the prior probability is set to 0% or 100%, the posterior probability will not change regardless of the presence or absence of evidence (test results). Also, if the true positive rate and false positive rate are the same, the evidence has no discriminatory power, so the posterior probability matches the prior probability. Please return the slider to an intermediate value and try again.

Because the prevalence (prior probability) is as low as 1%, the absolute number of false positives (healthy individuals incorrectly identified as positive) exceeds that of true positives. Specifically, out of 1000 people, about 10 of the 10 patients are true positives, and about 10 of the 990 healthy people are false positives, resulting in a positive predictive value of approximately 50%.

You can simulate sequential Bayesian updates by manually setting the posterior probability calculated in one step as the prior probability on the slider for the next step. For example, by setting the positive predictive value obtained from the first test as the prior probability and then inputting the result of a second test, you can experience the effect of accumulating evidence.

In CAE, 'Bayesian calibration' is performed to probabilistically correct the parameters of a simulation model using experimental data. The flow of 'prior probability → evidence → posterior probability' learned with this tool is precisely the process of updating the uncertainty of design parameters based on experimental results.

What is Bayes Theorem Visualizer?

Bayes Theorem Visualizer is a fundamental topic in engineering and applied physics. This interactive simulator lets you explore the key behaviors and relationships by directly manipulating parameters and observing real-time results.

By combining numerical computation with visual feedback, the simulator bridges the gap between abstract theory and physical intuition — making it an effective learning tool for students and a rapid-verification tool for practicing engineers.

Physical Model & Key Equations

The simulator is based on the governing equations behind Bayes' Theorem Visualizer. Understanding these equations is key to interpreting the results correctly.

Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.

Real-World Applications

Engineering Design: The concepts behind Bayes' Theorem Visualizer are applied across mechanical, structural, electrical, and fluid engineering disciplines. This tool provides a quick way to estimate design parameters and sensitivity before committing to full CAE analysis.

Education & Research: Widely used in engineering curricula to connect theory with numerical computation. Also serves as a first-pass validation tool in research settings.

CAE Workflow Integration: Before running finite element (FEM) or computational fluid dynamics (CFD) simulations, engineers use simplified models like this to establish physical scale, identify dominant parameters, and define realistic boundary conditions.

Common Misconceptions and Points of Caution

Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.

Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.

Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.