Math Visualization List

Bayes' Theorem Interactive Experiment

Through a medical testing case, intuitively understand how new evidence updates our probability judgments

🎯 Key Question: Does a positive test result mean you definitely have the disease?

Core Formula

$$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$$

$P(A|B)$ - Posterior Probability: Updated belief about event $A$ after seeing evidence $B$.

$P(A)$ - Prior Probability: Background probability before seeing evidence (initial belief).

$P(B|A)$ - Likelihood: Probability of observing evidence $B$ given $A$ occurs.

$P(B)$ - Normalizing Constant: Total probability of evidence occurring (total probability formula).

Bayes' Theorem Formula

$$P(患病|阳性) = \frac{P(阳性|患病) \times P(患病)}{P(阳性)}$$

Posterior P(Disease|Positive): Probability of actually having disease given positive test

Prior P(Disease): Prevalence before testing (base rate)

Likelihood P(Positive|Disease): Probability of positive test given disease (sensitivity)

Evidence P(Positive): Total probability of testing positive

Medical Testing Case

Suppose a disease has a prevalence of 5% in the population. Now there's a test available:

✓ Sensitivity 90%: 90% of people with the disease test positive
✗ False Positive Rate 10%: 10% of people without the disease test positive incorrectly

💡 Key Insight:

Even if the test is positive, you're not necessarily 100% likely to have the disease. Multiple factors like prevalence and test accuracy need to be considered.

Interactive Parameters

Adjust the sliders to observe how probabilities change. You'll notice:
• Lower prevalence means positive results are more likely false positives
• Higher false positive rate reduces test reliability

Test Result Interpretation

Real-time Calculation

Probability of disease given positive test

Calculated from current parameters

Population Distribution Visualization

True Positive

False Negative

False Positive

True Negative

Prior Probability (P(Disease))

Base Prevalence

Sensitivity (P(Positive|Disease))

True Positive Rate

Specificity (P(Negative|Healthy))

True Negative Rate

Positive Predictive Value

Probability after positive test

💡 Bayesian Thinking: How to Update Beliefs?

Prior Belief

Initial judgment based on existing knowledge and experience before acquiring new information. For example: the base prevalence of a disease in the population.

Gather Evidence

Acquire new data through observation, experimentation, or testing. For example: a positive medical test result.

Update Belief

Update the original belief using Bayes' theorem based on new evidence to obtain a more accurate posterior probability.