This interactive tutorial visualizes Bayesian inference using the Beta-Binomial model: how a Prior belief is updated by Evidence (Likelihood) to produce a Posterior distribution. You literally "see" the probability shift as you add successes or failures (e.g. coin flips).

The chart shows three curves: Prior (blue) — your initial belief about the probability of success; Likelihood (orange) — the data you observed (scaled for visibility); Posterior (purple, thicker) — the updated belief after combining prior and data. Sliders set the prior strength (Prior Successes α and Prior Failures β). Buttons add one success or one failure. The posterior updates via the conjugate rule: α_post = α₀ + successes, β_post = β₀ + failures. A strong prior (large α₀, β₀) resists new data; a weak prior moves quickly toward the data.

Mathematical foundation

1. Bayes' rule

Posterior ∝ Prior × Likelihood. For a probability p of success (e.g. coin fairness), we start with a prior distribution over p, observe data (successes and failures), and compute the posterior distribution that combines both.

2. Beta distribution (Prior and Posterior)

The Beta(α, β) distribution describes a belief over p in [0, 1]. Its PDF is f(p; α, β) ∝ p^α−1(1−p)^β−1, normalized by the Beta function B(α, β). Mean = α/(α+β); Mode = (α−1)/(α+β−2) when α, β > 1. We use a log-Gamma implementation so the PDF remains numerically stable for large α, β.

3. Conjugate update (Beta-Binomial)

For Bernoulli trials (e.g. coin flips), the Binomial likelihood has the form p^s(1−p)^f (s = successes, f = failures). The Beta distribution is the conjugate prior: if Prior is Beta(α₀, β₀) and you observe s successes and f failures, the Posterior is Beta(α₀ + s, β₀ + f). No numerical integration is needed — the update is exact and fast.

4. Likelihood curve

The Likelihood (orange) is the Binomial likelihood of the observed data as a function of p. It is scaled so its peak is visible alongside the Prior and Posterior. Its maximum is at the sample proportion s/(s+f). As you add data, you see the Prior "pulled" toward the Likelihood to form the Posterior.

The Bayesian Cycle

Bayesian inference is a continuous "learning loop". The Likelihood isn’t what gets updated — the Prior is what gets updated to become the Posterior. Then, in the next round of learning, that Posterior becomes your new Prior.

The Hand-off: For the next experiment, your old Posterior becomes your new Prior.   (Posterior_n → Prior_n+1)

5. The formula as a "correction"

You can think of Bayes’ rule as a way to "correct" your initial guess using new data:

• P(H) — the “Old You” (a priori).
• P(D | H) / P(D) — the “Learning Factor” (how much the new data should change your mind).
• P(H | D) — the “New You” (a posteriori).

6. Why the Likelihood isn’t "updated"

In Bayesian terms, the Likelihood is a fixed mathematical function that describes the data-gathering process (e.g. “if the defect rate is 12%, how likely is it that I saw 3 defects?”). It doesn’t change — it is the lens through which you view the data to update your belief. Only the Prior → Posterior transition represents "learning." The Likelihood is always determined by the experiment design and the data you actually observed.

Tab 1 — Theory

Use the controls to explore how the prior is updated by evidence:

1. Set the Prior: Use the sliders "Prior Successes (α)" and "Prior Failures (β)" to choose your initial belief. Equal values (e.g. 2, 2) give a symmetric prior centered at 0.5. Larger values make the prior "stronger" (narrower curve that resists new data).
2. Add Data: Click "Add Success" or "Add Failure" to simulate new trials (e.g. coin flips). The Likelihood (orange) appears and the Posterior (purple) updates: it is pulled toward the data. With many successes the posterior shifts right; with many failures it shifts left.
3. Strong vs Weak Prior: Try a strong prior (e.g. α=20, β=20) and add a few successes — the curve moves only slightly. Then reset evidence, set a weak prior (α=2, β=2), add the same number of successes — the posterior moves much more. This illustrates how prior strength affects the update.
4. Summary Panel: Posterior mean, mode, and 95% credible interval update in real time. The "Live Insight" text explains the current belief in plain language.
5. Reset evidence: Clears successes and failures; the prior is unchanged. Use this to try different data sequences with the same prior.

Tab 2 — Example 1 (Factory Quality Inspection)

A realistic simulation that demonstrates Bayesian inference in action. You receive products from a new supplier and need to estimate the defect rate — a real-world problem with no strong "50/50" prior intuition.

1. Set True Defect Rate: The actual defect rate of the supplier (hidden from the inspector). Default is 8%. Range 1%–50%. This is the value the posterior should converge to.
2. Set Prior α, β: The starting prior. Default is Beta(2, 10) — a mildly informed prior meaning "based on past experience, defect rates are usually around 10–20%" (prior mean ≈ 17%). The posterior will be pulled down toward the true 8% as evidence accumulates. Try Beta(1, 1) for a completely flat prior, or Beta(5, 5) for a strong belief that the rate is near 50%.
3. Inspect items: Click "Inspect 1", "Inspect 10", or "Inspect 100" to draw random items from the supplier. Or click "Auto" to watch continuous inspection (stops at 500 items).
4. Posterior PDF chart: Shows the Prior (blue) and Posterior (purple, shaded) Beta distributions. A red dashed vertical line marks the true defect rate. Watch the posterior narrow and converge as evidence accumulates.
5. Histogram: Shows the observed relative frequencies of Defective vs Good items — the frequentist estimate of the defect rate.
6. Inspection log: A scrollable record of every inspection: D = Defective, G = Good.
7. Reset: Clears all inspections and resets to the prior.

Tab 3 — Example 2 (Discrete + Continuous)

Demonstrates that Bayes' rule applies to any hypothesis given data — not just the Beta-Binomial case.

Discrete Case — Crime Investigation:

• 5 suspects (A–E), each starting with a uniform 20% prior probability.
• Select a suspect and an evidence type (Witness ID, Forensic match, or Alibi confirmed), then click "Add Evidence".
• Witness ID for suspect X: P(evidence | X) = 0.80, P(evidence | other) = 0.05 — strongly points to X.
• Forensic match for suspect X: P(evidence | X) = 0.90, P(evidence | other) = 0.025 — very strong evidence for X.
• Alibi confirmed for suspect X: P(evidence | X) = 0.02, P(evidence | other) ≈ 0.245 — nearly eliminates X.
• The equation panel shows the Bayes calculation for all 5 suspects at each step.

Continuous Case — Weighing an Object:

• Estimate the true weight of an object using noisy measurements.
• Prior: N(50, 15²) — broad initial guess. Measurement noise: σ = 3 kg.
• Uses the Normal-Normal conjugate update: Posterior is also Normal, getting narrower with each measurement.
• Orange dots at the bottom of the chart show individual measurements.
• The red dashed line marks the true weight.

The connection: All three tabs use the same Bayes' rule P(H|D) = P(D|H)·P(H)/P(D). The only thing that changes is the type of distribution used (discrete probabilities, Normal PDF, or Beta PDF).

Visualizations

• Prior (blue): Initial belief before data (Beta, Normal, or uniform bars).
• Likelihood (orange): (Theory tab) Binomial likelihood of the observed data.
• Posterior (purple): Updated belief after combining prior and data.
• True value (red dashed): Ground-truth for comparison (Example 1 and 2).

Summary panel

• Posterior mean: Point estimate for the unknown parameter.
• Posterior mode: Peak of the posterior distribution.
• 95% credible interval / σ: Uncertainty measure (Bayesian analogue of a confidence interval).