Statistical Toolkit · 10

Markov Chain Monte Carlo: Reconstructing Distributions

Recovering a complex density through random exploration

When a target can be evaluated only up to a constant, MCMC can still generate samples. Their empirical distribution reconstructs the normalized density.

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The essential idea: a long-run sample pattern can recover a distribution whose normalization has no closed form.

Watch the concept

One Concept · One Example

Markov Chain Monte Carlo: Reconstructing Distributions

Presented by Charlotte Moser

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What to notice

The chain favors high-density regions.
Some lower-density moves preserve exploration.
Reconstruction improves after mixing and with more samples.

The idea in 30 seconds

A density can be known without its constant

Target shape

The unnormalized function is easy to evaluate point by point even though its integral is not available analytically.

f (x) = \exp (x + 4 x^{2} - 2 x^{3} - 4 x^{4})

MCMC reconstruction

After burn-in, a histogram of chain states approximates p(x) = f(x)/Z.

samples \to empirical density \approx p (x)

Explore

Build the chain, then reconstruct the density

First reproduce the six proposal, random-number, accept-or-reject decisions from the slides. Then use many chain states to construct the empirical PDF.

EXAMPLE 1 · BUILD THE CHAIN

The first six Metropolis decisions

Start at x₀ = 0. Each proposal uses the Gaussian random move from the slides. Compare the uniform draw r with min(1, α): accept the proposal, or repeat the previous state.

The open circle is the current proposal. A rejection creates a horizontal segment because xₖ = xₖ₋₁.

1-0.64902.39390.7203-0.6490accept

2-1.40750.01140.0923-0.6490reject

3-1.49460.00120.3456-0.6490reject

4-1.20770.29150.5388-0.6490reject

5-0.84591.35780.6852-0.8459accept

6-1.69780.00000.8781-0.8459reject

EXAMPLE 2 · FROM CHAIN TO PDF

Use enough states to reconstruct the density

Once the chain has mixed, its empirical histogram becomes an approximation of the unknown normalized PDF.

reconstructionnormalized target for comparison

KEY TAKEAWAY

Strategic correlated sampling can recover an unknown, multimodal probability density without calculating its normalization constant.