Beta Random Variate Generator
Generates continuous random variates on [0, 1] via the ratio of independent Gamma variables X = G_1 / (G_1 + G_2).
Statistics & Probability / Beta Distribution & Priors
FREE TO EXPLORELABORATORY 20 · STATISTICS & PROBABILITY
Beta Distribution, Shape Parameters, and Bayesian Conjugate Priors
Analyze the continuous Beta distribution family on the unit interval [0, 1], model prior uncertainty, perform exact Bayesian conjugate updating under Binomial likelihood, and verify posterior hyperparameter accumulation. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
TOTAL ACCUMULATED HYPERPARAMETER SUM Α + Β—
SAMPLING TELEMETRY
Ready to sample with Batch k = 1.0.
Adjust Batch k, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Conjugate Beta priors assume that prior beliefs can be adequately captured by pseudo-counts, which may restrict complex multi-modal priors. The empirical regression of accumulated hyperparameters against observation batch multiplier k confirms exact conjugate updating with slope m = 4.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Prior-Posterior Density Modeler
Evaluates Beta probability density functions f(θ; α, β) across U-shaped, uniform, unimodal, and skewed configurations.
Bayesian Conjugate Updating Engine
Performs exact analytical Bayesian updates: Beta(α + s, β + f) given s successes and f failures.
Posterior Credible Interval Resolver
Calculates equal-tailed and highest posterior density (HPD) credible intervals for the binomial success parameter θ.
Maximum A Posteriori (MAP) Estimator
Computes mode θ̂_MAP = (α - 1)/(α + β - 2) and posterior mean E[θ] = α/(α + β) to contrast point estimators.
Hyperparameter Accumulation Regression Unit
Performs linear regression of posterior precision metric (α + β) versus batch factor to verify the slope m = 4.000.
FOLLOW THE METHOD
A reason behind every step.
Analyze the continuous Beta distribution family on the unit interval [0, 1], model prior uncertainty, perform exact Bayesian conjugate updating under Binomial likelihood, and verify posterior hyperparameter accumulation. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 batches.
01
Initialize Prior Distribution
Set prior shape parameters to α_0 = 1.0 and β_0 = 1.0 on the Bayesian console, establishing a standard non-informative uniform prior Beta(1, 1) on θ ∈ [0, 1].
02
Execute Primary Bernoulli Observation Batch
Generate a batch of 4 independent Bernoulli trials with true parameter p = 0.50 (resulting in 2 successes and 2 failures).
03
Apply Bayesian Conjugate Update Rule
Update hyperparameters to α_1 = α_0 + 2 = 3.0 and β_1 = β_0 + 2 = 3.0. Verify that the posterior density Beta(3, 3) concentrates around θ = 0.50.
04
Evaluate Posterior Mean and Variance
Calculate posterior mean E[θ] = α / (α + β) = 3 / 6 = 0.50 and posterior variance Var(θ) = αβ / [(α + β)^2 (α + β + 1)] = 9 / (36 · 7) = 0.0357.
05
Step Observation Batches Across Multiplier k = 1 to 6
Increment observation multiplier across standardized settings k ∈ {1, 2, 3, 4, 5, 6}, adding 4 trials per batch (2 successes, 2 failures) and tracking total hyperparameter sum α + β.
06
Perform Linear Regression on Hyperparameter Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = k, y = (α + β) - 2). Confirm that empirical slope m = 4.0000 validates exact hyperparameter accumulation.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Beta Distribution & Priors mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 4.0000 · x
| # | Observation Batch Multiplier k (batches) | Total Accumulated Hyperparameter Sum α + β (Sum α+β) | Observation Batch Multiplier k | Posterior Precision Metric α + β | Remove |
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