How does the computer find the Posterior? · MCMC samples rather than fully computing — the walker in the fog

How does the computer find the Posterior?

Where are we? In the previous tools we defined the two building blocks of our model: in the Likelihood Compass the likelihood (how the data come about) and in the Prior Compass and Prior Lab the prior (what we consider plausible beforehand). The next logical step follows: based on these two ingredients, we want to explore the probability space — that is, find out which parameter values are plausible after seeing the data. Likelihood × prior gives the posterior (at first only up to a constant) — and that is exactly what MCMC samples.

The metaphor: Imagine standing in dense fog on a hilly landscape, tasked with figuring out what the landscape looks like. You can't fly and can't see — but you can feel the ground beneath your feet. If you walk cleverly and record where you've been, after many steps a map of the landscape emerges. That is exactly MCMC. The landscape is the posterior — and your map is the collection of samples.

Why can't you simply calculate the Posterior?

In frequentist thinking there is a formula: data in, estimator out. In the Bayesian framework the goal is an entire distribution over all plausible parameter values — the posterior P(θ | data).

Bayes' theorem states: P(θ|D) ∝ P(D|θ) · P(θ). The proportionality sign hides an integral in the denominator that cannot be solved in realistic models. With 5 parameters and 100 grid points per parameter that would be 100⁵ = 10¹⁰ computations. Impossible.

MCMC solves this by sampling from the posterior rather than computing it fully.

How does the walker move?

At each position the walker makes a random proposal: "What if I stepped over there?"

Is the new location more probable? → always go there.
Is it less probable? → sometimes go anyway — proportional to the difference. A location that is half as probable is visited in 50% of cases.

This prevents the walker from getting stuck in a corner and ensures the whole landscape is explored.

Stages – Please pick a stage: ↓

Start

Choose a stage above and click ▶ Start or → One Step.

Last Step

Current position—

Proposal—

P(Proposal)—

P(Current)—

Ratio α—

Random number u—

No step yet.

Acceptance rate

0 % Good: 20–50 %

          Adjust step size to optimise exploration.
        

Legend

          ● Warmup (discarded)

          ● Sampling (recorded)

          ○ Proposal accepted

          ○ Proposal rejected

          - - Proposal distribution N(θ, s)

HistogramApproximation of Posterior

Trace — θ over stepsGood: «caterpillar»

Step size s 0.80

Warmup 150

Speed 30/s

Steps: 0

Phase: —

Position: —

Samples: 0

Acceptance rate: —

Step size s — what really happens

The step size s determines how far the walker can jump at each step.

s very small: Almost every proposal lands close to the current position — almost always accepted. But: the walker only shuffles, barely explores. Even at 90% acceptance the exploration can be poor.

s very large: Proposals land far away, often in very flat regions — frequently rejected. The walker often stands still.

The optimal step size is the one that yields good exploration with a reasonable acceptance rate. The rule of thumb is 20–50 % — but what matters is the trace plot, not the rate alone. brms adjusts s automatically during warmup (adaptive sampling).

Warmup, Trace Plot & Convergence

Warmup (burn-in): The chain starts somewhere and needs time to find the high-probability region. These first steps are discarded — they do not yet represent the posterior.

Reading the trace plot: A converged chain looks like a «fuzzy caterpillar» — randomly jumping around a stable level. A visible upward trend or long plateaus → warmup not long enough or s not optimal.

R̂ (Gelman-Rubin): In brms multiple chains run in parallel. When all explore the same distribution they converge. R̂ < 1.01 = good. Divergences → often a sign of poorly chosen priors or difficult posterior geometry — increasing adapt_delta usually helps.

▸ Metropolis-Hastings — the mathematics

At each step three rules apply:

1. Proposal: New value θ* = θ + ε, ε ~ Normal(0, s).

2. Ratio: α = P(θ*|data) / P(θ|data). How much more plausible is the proposal?

3. Decision: Here we distinguish two cases.

Case α ≥ 1 (proposal at least as plausible as the current position): the proposal is always accepted. The walker thus always moves uphill (or stays at the same height). No randomness is needed here.

Case α < 1 (proposal less plausible): now chance decides. The computer draws a random number u between 0 and 1 (all values equally likely) and compares it with α:
• u < α → accept (θ* becomes the new position)
• u ≥ α → reject (the walker stays put; the old value is counted once more)
Since u is uniformly distributed, u falls below α with exactly probability α — that is what "accept with probability α" means.
Example: If the proposal is half as plausible, α = 0.5. If the computer draws u = 0.31, then 0.31 < 0.5 → accepted. With u = 0.82, 0.82 ≥ 0.5 → rejected. In the long run, every second such proposal is accepted.

You can see exactly this comparison in the inspector: ratio α, random number u and the outcome (✓ accepted / ✗ rejected).

The trick: Only ratios are needed — the intractable normalising constant cancels out.

Why does this converge? The transition rule satisfies detailed balance: after infinitely many steps the visit frequency exactly matches the posterior distribution. Mathematically provable.

ℹ MCMC Visualizer — Help

What will I learn here?

Bayesian inference requires the posterior P(θ|data) — a distribution over all plausible parameter values. Because the denominator of Bayes' theorem involves an intractable integral, the posterior is sampled via MCMC rather than computed directly. This tool shows step by step how the Metropolis-Hastings algorithm works.

Recommended exploration

Start with Stage 1, click through individual steps using → One Step — watch the Inspector: ratio α and random number u decide acceptance or rejection
Vary step size s — too large: many rejections; too small: the walker barely moves. Target: acceptance rate 20–50 %
Stage 2 (bimodal posterior) — observe whether the walker finds both peaks or gets trapped in one
Stage 3 (2D: μ and σ) — see how brms estimates two parameters simultaneously; histograms show the marginal posteriors
Compare warmup vs. sampling — orange points (warmup) are discarded, blue points (sampling) form the posterior estimate

Key displays

Main plot: Posterior landscape + walker's path

Inspector: Last step in detail — ratio α = P(proposal)/P(current); if α ≥ u → accept

Acceptance rate: Green zone (20–50 %) = optimal step size

Histogram + Trace: Growing posterior approximation and trajectory over all steps — a «caterpillar» trace signals good convergence

Narrative banner: Explains each step in plain language — ideal for step-by-step learning