Prior Belief Density Engine
Models subjective or empirical prior distributions p(θ) ~ N(μ_0, σ_0^2 = 1/τ_0) across parameter domains.
Statistics & Probability / Bayesian Conjugate Updating
FREE TO EXPLORELABORATORY 39 · STATISTICS & PROBABILITY
Bayes' Theorem, Prior Precision, and Normal-Normal Conjugate Updating
Implement Thomas Bayes' rule of inverse probability, formulate conjugate Normal-Normal Bayesian updating, analyze the precision-weighted combination of prior beliefs and empirical likelihood, and verify posterior precision accumulation. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
POSTERIOR PRECISION METRIC Τ_POST—
SAMPLING TELEMETRY
Ready to sample with Precision k = 1.0.
Adjust Precision 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
In small-sample regimes, strongly informative subjective priors can bias posterior estimates away from true empirical reality. The empirical regression of posterior precision against sample multiplier k confirms exact linear precision additivity with slope m = 1.0000 and R² = 1.000000.
Statistical tools & sampling modules.
Gaussian Likelihood Profiler
Computes sample likelihood L(θ; x) = ∏ N(x_i; θ, σ^2) and evaluates sample precision n/σ^2.
Precision-Weighted Posterior Updater
Evaluates exact posterior parameters: τ_post = τ_0 + n τ_data and μ_post = (τ_0 μ_0 + n τ_data x̄) / τ_post.
Bayesian Credible Interval Resolver
Calculates exact equal-tailed 95% credible intervals μ_post ± 1.96 / √τ_post.
Bayes Factor Model Comparator
Computes marginal likelihoods to evaluate Bayes Factors BF_{10} = p(y | M_1) / p(y | M_0).
Precision Accumulation Regression Unit
Performs linear regression of posterior precision versus sample factor k to confirm the unit slope m = 1.000.
FOLLOW THE METHOD
A reason behind every step.
Implement Thomas Bayes' rule of inverse probability, formulate conjugate Normal-Normal Bayesian updating, analyze the precision-weighted combination of prior beliefs and empirical likelihood, and verify posterior precision accumulation. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 factor.
01
Initialize Prior Distribution and Precision
Set prior mean μ_0 = 0.0 and prior precision τ_0 = 1.0 (prior variance σ_0^2 = 1.0) on the Bayesian updating console.
02
Synthesize Gaussian Data Observations
Draw an initial sample of n = 1.0 observations from N(μ_true = 2.0, σ^2 = 1.0). Record the empirical sample mean x̄.
03
Execute Exact Normal-Normal Conjugate Update
Compute posterior precision τ_1 = τ_0 + n = 1 + 1 = 2.0 and posterior mean μ_1 = (1 · 0.0 + 1 · x̄) / 2.0. Verify Gaussian shape.
04
Evaluate Bayesian Compromise Behavior
Observe that the posterior mean lies strictly between the prior mean and sample mean, weighted proportionally by their respective precisions.
05
Step Observation Count Across Multiplier k = 1 to 6
Increment observation multiplier across standardized settings k ∈ {1.0, 2.0, 3.0, 4.0, 5.0, 6.0}, recording the linear growth of posterior precision.
06
Conduct Linear Regression on Precision Accumulation
Fit a linear regression line y = mx to coordinate pairs (x = k, y = τ_post - 1.0). Confirm that empirical slope m = 1.0000 validates linear precision additivity.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Bayesian Conjugate Updating mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.0000 · x
| # | Sample Size Precision Multiplier k (factor) | Posterior Precision Metric τ_post (Precision τ_post) | Sample Multiplier k | Posterior Precision Metric τ_post | Remove |
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