Bayes Rule Odds

Building off Odds

What is Bayes’ Rule

• A mathematical rule for updating the probability of a hypothesis based on new evidence.

  • Prior Probability $(P(H))$: The initial probability of the hypothesis.
  • Evidence Probability $(P(E|H))$: The probability of the evidence given that the hypothesis is true.
  • Updated Probability $(P(H|E))$: The new probability of the hypothesis after considering the evidence.

Bayes’ Rule Formula

• Bayes’ Rule can be expressed mathematically as:

$[P(H|E)=\frac{P(E|H)\times P(H)}{P(E)}]$ - Where $P(E)$ is the total probability of the evidence: $[P(E)=P(E|H)\times P(H)+P(E|\neg H)\times P(\neg H)]$

What is a Bayes’ Box

• A visual tool for understanding Bayesian updates.

  • A Bayes’ Box represents the probabilities involved in Bayes’ Rule as areas within a box.

Drawing a Bayes’ Box

• Steps to create a Bayes’ Box:

  • Step 1: Divide the box according to the prior probabilities of $H$ and $\neg H$.
    • Example: If prior odds are 1:3 (for $H$ and $\neg H$ respectively), the box is split accordingly.
  • Step 2: Divide the areas of $H$ and $\neg H$ based on the probability of evidence $E$ if $H$ is true or false.
    • Fill sections representing $P(E|H)$, $P(\neg E|H)$, $P(E|\neg H)$, and $P(\neg E|\neg H)$.
  • Step 3: Use the areas to visually estimate the updated probability.

Interpreting the Bayes’ Box

• EX.

  • a box is divided to show prior odds of $H$ of 1:3, making $P(H)=\frac{1}{4}$ and $P(\neg H)=\frac{3}{4}$.

  • Evidence provided (e.g., a medical test):

    • $P(E|H)=0.75$
    • $P(E|\neg H)=0.25$
    • Updated calculation for $P(H|E)$ given by: [ P(H|E) = \frac{P(E|H) \times P(H)}{P(E|H) \times P(H) + P(E|\neg H) \times P(\neg H)} = \frac{0.75 \times 0.25}{0.75 \times 0.25 + 0.25 \times 0.75} = \frac{0.1875}{0.375} \approx 0.5 ]

• Visual estimation:

  • Compare shaded areas for $E$ within $H$ and $\neg H$. For $P(H|E)$, look at the ratio of shaded areas in the $H$ section to the total shaded area.

Example Interpretation from Exercises

• With Evidence (E):

  • Updated odds: approximately 2:3 (visually), translating to $P(H|E)\approx 2/5$ (probability).

• Without Evidence ($\neg E$):

  • Updated odds: approximately 1:6 (visually), translating to $P(H|\neg E)\approx 1/7$.

Useful Lessons

• Key Bayesian Lessons:

  • Bayes Lesson #1: Always consider the prior probability.
  • Bayes Lesson #2: Likely evidence if hypothesis is true confirms the hypothesis.
  • Bayes Lesson #3: If evidence ($E$) confirms the hypothesis, then lack of evidence ($\neg E$) disconfirms it.
  • Bayes Lesson #4: Yesterday’s updated probability becomes today’s new prior probability for further evidence.

Sensitivity and Specifisity