README.md

metadata

title: MCMC Dashboard
emoji: 📊
colorFrom: indigo
colorTo: blue
sdk: static
pinned: false
short_description: Interactive Metropolis-Hastings sampling visualization

📊 MCMC Dashboard

An interactive visualization of Metropolis-Hastings MCMC sampling from a 2D correlated Gaussian distribution. Watch the Markov chain explore probability space in real-time.

🎮 How to Use

Visualizations

Panel	Description
2D Landscape	Main view showing the target distribution (cyan cloud) and walker trajectory
Trace Plot	X-coordinate over time with running mean (yellow line)
Histogram	Marginal distribution of X samples vs. ideal Normal (dashed)

Controls

Parameter	Effect
Target Correlation	Covariance between X and Y (-0.95 to 0.95). Higher = more elongated ellipse
Proposal Width	Standard deviation of the proposal distribution. Too small = slow mixing, too large = high rejection
Speed	Steps per animation frame (1-50)

Diagnostics

• Steps: Total MCMC iterations
• Acceptance: Percentage of proposals accepted (ideal: 20-50%)
• Running Mean: Cumulative average of X samples (should → 0)
• ESS: Effective Sample Size - accounts for autocorrelation
• AutoCorr: Lag-1 autocorrelation (lower = better mixing)

🔬 The Algorithm

The Metropolis-Hastings algorithm samples from a target distribution π(x):

1. Propose: x* ~ N(x_current, σ²)
2. Accept ratio: α = π(x*) / π(x_current)
3. Accept/Reject:
   • If α ≥ 1: Always accept
   • If α < 1: Accept with probability α

Target Distribution

The target is a bivariate Gaussian with covariance:

Σ = [[1, ρ], [ρ, 1]]

where ρ is the correlation parameter.

📈 Key Insights

Proposal Width Tuning

• Too narrow: Walker takes tiny steps → high acceptance but slow exploration
• Too wide: Walker proposes jumps too far → high rejection rate
• Optimal: ~23% acceptance rate for Gaussian targets (Roberts et al., 1997)

Correlation Effects

• High |ρ|: Creates elongated "banana-like" region → harder to sample efficiently
• Low |ρ|: Near-circular distribution → easier for isotropic proposals

Effective Sample Size (ESS)

ESS estimates how many independent samples the chain represents:

ESS ≈ N × (1 - ρ₁) / (1 + ρ₁)

where ρ₁ is lag-1 autocorrelation. Green = good (>200), Red = needs more samples.

🧪 Experiments to Try

1. Slow mixing: Set correlation to 0.95, proposal width to 0.3. Watch the walker struggle.
2. Over-rejection: Set proposal width to 4.0. See acceptance rate plummet.
3. Sweet spot: Set correlation to 0.5, width to 1.0. Observe healthy mixing.

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Interactive demonstration of Bayesian computation fundamentals