a.py

import marimo

__generated_with = "0.23.3"
app = marimo.App()


@app.cell
def _():
    import marimo as mo
    import numpy as np
    import matplotlib.pyplot as plt
    from matplotlib.patches import Rectangle

    return Rectangle, mo, plt


@app.cell
def _(mo):
    mo.md("""
    # Bayes' Theorem: Updating Beliefs with Evidence

    Bayes' Theorem is a fundamental concept in probability theory that helps us update our beliefs based on new evidence.

    The formula is:
    $$P(A|B) =     rac{P(B|A)P(A)}{P(B)}$$

    Where:
    - $P(A|B)$ is the **posterior probability**: the probability of event A given that B occurred
    - $P(B|A)$ is the **likelihood**: the probability of observing B given that A is true
    - $P(A)$ is the **prior probability**: our initial belief about A before seeing B
    - $P(B)$ is the **evidence**: the overall probability of observing B
    """)
    return


@app.cell
def _(mo):
    mo.md("""
    ## Medical Testing Example

    Let's say we're testing for a rare disease:
    - Prevalence (prior probability): 0.1% of the population has the disease
    - Test accuracy:
      - If you have the disease, the test correctly identifies it 99% of the time (true positive rate)
      - If you don't have the disease, the test incorrectly says you do 5% of the time (false positive rate)

    What's the probability that someone who tests positive actually has the disease?
    """)
    return


@app.cell
def _(mo):
    # Define the parameters
    prior_disease = 0.001  # P(Disease)
    sensitivity = 0.99  # P(Test+ | Disease)
    false_positive_rate = 0.05  # P(Test+ | No Disease)

    # Calculate the components
    # P(Test+)
    p_test_positive = sensitivity * prior_disease + false_positive_rate * (1 - prior_disease)

    # Apply Bayes' Theorem
    posterior_disease = (sensitivity * prior_disease) / p_test_positive

    # Display results
    mo.md(f"""
    Given:
    - Prior probability of disease: {prior_disease * 100:.1f}%
    - Sensitivity (true positive rate): {sensitivity * 100}%
    - False positive rate: {false_positive_rate * 100}%

    Using Bayes' theorem:

    P(Disease|Test+) = [P(Test+|Disease) × P(Disease)] / P(Test+)

    P(Test+) = P(Test+|Disease) × P(Disease) + P(Test+|No Disease) × P(No Disease)

    P(Test+) = {sensitivity:.2f} × {prior_disease:.3f} + {false_positive_rate:.2f} × {(1 - prior_disease):.3f}

    P(Test+) = {p_test_positive:.4f}

    P(Disease|Test+) = ({sensitivity:.2f} × {prior_disease:.3f}) / {p_test_positive:.4f} = {posterior_disease:.3f}

    **Only {posterior_disease * 100:.1f}% of people who test positive actually have the disease!**
    """)
    return


@app.cell
def _(Rectangle, mo, plt):
    # Create a visualization showing the four categories
    fig, ax = plt.subplots(figsize=(10, 6))

    # Set up the grid
    ax.set_xlim(0, 10)
    ax.set_ylim(0, 8)

    # Draw rectangles for different categories
    # Population (1000 people)
    ax.add_patch(Rectangle((1, 1), 8, 6, fill=False, edgecolor="black", linewidth=2))
    ax.text(5, 7.2, "Population (1000 people)", ha="center", va="bottom")

    # Disease (0.1%)
    disease_count = 1000 * 0.001
    no_disease_count = 1000 - disease_count

    # Draw disease area
    ax.add_patch(Rectangle((1, 1), 8, disease_count / 1000 * 6, facecolor="red", alpha=0.5))
    ax.text(
        5,
        1.5 + disease_count / 1000 * 3,
        f"Disease ({disease_count:.0f} people)",
        ha="center",
        va="center",
        color="white",
        weight="bold",
    )

    # Draw non-disease area
    ax.add_patch(
        Rectangle((1, 1 + disease_count / 1000 * 6), 8, no_disease_count / 1000 * 6, facecolor="blue", alpha=0.5)
    )
    ax.text(
        5,
        1.5 + disease_count / 1000 * 6 + no_disease_count / 1000 * 3,
        f"No Disease ({no_disease_count:.0f} people)",
        ha="center",
        va="center",
        color="white",
        weight="bold",
    )

    # Add labels
    ax.text(0.5, 4, "True Positive\n(99% of Diseased)", ha="right", va="center")
    ax.text(0.5, 2, "False Positive\n(5% of Non-Diseased)", ha="right", va="center")

    # Add test results
    tp = disease_count * 0.99  # True positives
    fp = no_disease_count * 0.05  # False positives

    ax.add_patch(Rectangle((2, 3.5), 2, 1, facecolor="green", alpha=0.7))
    ax.text(3, 4, f"True Positive\n({tp:.0f})", ha="center", va="center")

    ax.add_patch(Rectangle((2, 1.5), 2, 1, facecolor="orange", alpha=0.7))
    ax.text(3, 2, f"False Positive\n({fp:.0f})", ha="center", va="center")

    # Add total positive tests
    total_positives = tp + fp
    ax.add_patch(Rectangle((5, 1.5), 3, 2, facecolor="yellow", alpha=0.5))
    ax.text(6.5, 2.5, f"Total Positive\nTests ({total_positives:.0f})", ha="center", va="center")

    ax.set_xticks([])
    ax.set_yticks([])
    ax.set_title("Bayes' Theorem Visualization: Medical Testing")

    mo.ui.matplotlib(plt.gca())
    return


@app.cell
def _(mo):
    mo.md("""
    ## Why This Matters

    This example shows why Bayes' Theorem is important:

    1. **High false positive rate** combined with **low prevalence** leads to counterintuitive results
    2. **95% accurate tests** can still give misleading results when the condition is rare
    3. **Bayes' Theorem forces us to think about**:
       - Our initial beliefs (prior probability)
       - How likely we are to observe evidence given our beliefs
       - How to update our beliefs in light of new evidence

    This same logic applies to:
    - Spam detection
    - Financial risk assessment
    - Scientific hypothesis testing
    - Machine learning classification
    """)
    return


@app.cell
def _():
    return


@app.cell
def _():
    return


if __name__ == "__main__":
    app.run()