Educational mathematical tool

Predictive Processing Simulation Tool

Explore how a prior expectation and new evidence combine in an exact one-dimensional Gaussian model.

Full scientific scope

Educational scalar Gaussian model—not a brain, biological, or clinical simulation.

A scalar, static, identity-observation, linear-Gaussian teaching model with arbitrary units. The analytical posterior is exact; iteration is an optional numerical demonstration, not physical or neural time. Generator truth is simulation-only, trials are independent, and this is not a brain or clinical simulation.

Interface mode Simple Advanced

Simple explains the core inference. Advanced preserves every Version 3 control and diagnostic.

Observation route Enter an observationChoose observation directly Generate a simulated observationGenerate observation from a true state

Set the observation used by the unchanged scalar Gaussian inference model.

Set the exampleModel controls

Set the exampleInference inputs

Starting expectationPrior mean μ_pPrior mean μ_pU

Center of the prior distribution; it is separate from prior uncertainty.

New evidenceObservation yObservation yU

Selected evidence for direct inference, not a declared true state.

Prior uncertaintyPrior SD σ_pPrior uncertainty (SD) σ_pU

Larger SD means lower prior precision.

Evidence uncertaintyObservation SD σ_sObservation uncertainty (SD) σ_sU

Conditional Gaussian observation-error SD; its inverse variance is observation precision.

Blank or invalid entries preserve the last valid values. Generated observations are not clamped.Blank and invalid entries preserve the last valid model and generator state. Generated observations are never clamped to slider bounds.

Starting expectation40.00 Uprior mean μ_p

Observation70.00 Unew evidence y

Updated estimate64.00 Uanalytical posterior mean

Updated uncertainty4.47 Uconditional posterior SD

See the resultPanel 1 · Analytical distributions

How the estimate changesPrior, likelihood, and posterior

The same prior, likelihood, and exact posterior shown on one shared scale.True density heights over an expanded real-valued axis. A simulation truth marker is evaluation-only.

Exact

Prior densityLikelihoodPosterior density

Prior SD10.000 U

Observation SD5.000 U

Posterior SD4.472 U

Axis: 0.00 to 110.00 U.

See the result

From expectation to updated estimate

An exact movement from the prior mean to the analytical posterior mean. No optimizer step is shown.

Analytical

Starting expectation 40.00 UObservation 70.00 UUpdated estimate 64.00 U

Understand the weightingPanel 3 · Precision and flow

What influenced the resultRelative precision and update flow

The prior and observation weights sum to 100%. Lower uncertainty produces greater relative influence.Absolute precision uses U⁻². Relative weights are normalized proportions.

Derived

Prior 20.0%

Observation 80.0%

Prior influence 20.0%; observation influence 80.0%. Lower uncertainty corresponds to higher precision and therefore greater relative weight.

Precisions, weights, and weighted errors
Quantity	Definition	Value
Prior precision	τ_p = 1/σ_p²	0.010000 U⁻²
Observation precision	τ_s = 1/σ_s²	0.040000 U⁻²
Prior weight	w_p	20.0%
Observation weight	w_s	80.0%
Weighted prior error	τ_pε_p,k	0 U⁻¹
Weighted sensory error	τ_sε_s,k	+1.2 U⁻¹

Panel 4 · Normalized objective descent

Numerical convergence

Iteration k is a numerical index, distinct from independent trial index i and display cadence. The analytical posterior remains the exact target.

Ready

Dynamics view Estimate by iteration Objective and update

Numerical estimate trajectoryAnalytical posterior mean

Estimate by iteration at k = 0. Current estimate 40.000 U; next estimate 46.000 U; analytical target 64.000 U; not yet converged.

Animation cadence

Initialized at prior mean.k = 0 · rₖ = 0.8000

Current numerical dynamics
Quantity	Symbol	Value and unit
Selected view	—	Estimate by iteration
Current numerical estimate	μ_k	40.000 U
Analytical target	μ_post	64.000 U
Next estimate	μ_k+1	46.000 U
Sensory error	y−μ_k	+30.000 U
Prior error	μ_p−μ_k	0.000 U
Weighted sensory contribution	π_s(y−μ_k)	+1.20000 U⁻¹
Weighted prior contribution	π_p(μ_p−μ_k)	0 U⁻¹
Negative gradient	−F′(μ_k)	+1.20000 U⁻¹
Objective derivative	F′(μ_k)	−1.20000 U⁻¹
Total precision / curvature	λ	0.050000 U⁻²
Iteration step	α	0.25
Derived raw step	η=α/λ	5.00000 U²
Numerical increment	Δμ_k	+6.000 U
Current objective	F(μ_k)	18.0000
Minimum objective	F*	3.60000
Objective gap	F(μ_k)−F*	14.4000
Convergence residual	r_k	0.8000
Convergence state	—	Not yet converged

Independent repetitions

Independent repeated trials

Every trial starts from the same prior and uses one newly generated observation. Trials are not sequential updates.

Simulation only

No batch generated yet.

Observation sampling distribution

Empirical histogram (density)Theoretical density

Empirical and theoretical repeated-trial comparison
Summary	Mean	SD
Theoretical
Empirical

Conditional posterior uncertainty: . Across-trial posterior-mean variability: . These are different quantities.

Simple interface initialized with the direct observation route.

Model scope, limitations, and sourcesEquations, distinctions, limitations, and sources

Mathematics and interpretation

One exact target, one numerical update path

Quadratic objective and analytical target

For precision, π is an exact alias of the inherited τ: π_p ≡ τ_p = 1/σ_p² and π_s ≡ τ_s = 1/σ_s².

F(μ) = ½π_s(y−μ)² + ½π_p(μ−μ_p)²F′(μ) = π_s(μ−y) + π_p(μ−μ_p)F″(μ) = λ = π_p + π_s > 0μ_post = (π_pμ_p + π_sy) / (π_p+π_s)

F is a quadratic negative-log-posterior objective up to an additive constant. Its unique minimizer is the unchanged analytical posterior mean. The numerical estimate approximates that known target; it does not replace the analytical solution.

Normalized precision-weighted gradient descent

μ_k+1 = μ_k + α/(π_p+π_s)[π_s(y−μ_k) + π_p(μ_p−μ_k)]μ_k+1 = μ_k + α(μ_post−μ_k)

The two forms are algebraically equivalent and are cross-checked numerically. The derived raw step η = α/(π_p+π_s) is displayed but never independently controlled. The public iteration step is restricted to 0.05 ≤ α ≤ 1.

Closed form, stability, and objective contraction

μ_k = μ_post + (1−α)^k(μ₀−μ_post)0 < α < 2 ⇔ 0 < η < 2/λF(μ_k)−F* = ½λ(μ_k−μ_post)²F(μ_k+1)−F* = (1−α)²[F(μ_k)−F*]

The full mathematical interval includes sign-alternating overshoot for 1 < α < 2. The interface intentionally excludes that region. Numerical convergence means the inherited dimensionless residual is below 10⁻⁶, subject to the 10,000-step safety limit.

Generation, independent trials, and indices

z_i ~ N(0,1) ε_i = σ_sz_i y_i = x* + ε_iμ_post,i|x* ~ N(w_pμ_p + w_sx*, w_s²σ_s²)

Trial/draw index i labels independent observations. Numerical iteration k labels optimizer steps for one active observation. Animation cadence is wall-clock display timing only. Batches remain independent analytical one-observation posteriors with no trajectory or shared update state.

Reproducibility boundary

Version 3 preserves the Version 2 PCG32 XSH-RR and Box–Muller sequence exactly. The finite sequence is pseudorandom and deterministic, not exact continuous randomness or a cryptographic facility.

Scope controls

Limitations

One scalar latent variable and one scalar observation.
The model is static, not a temporal world model.
The prior and observation error are Gaussian.
The observation mapping is fixed and identical.
Precisions remain fixed during a trajectory.
Each trajectory begins at the prior mean by convention.
Numerical iteration is not physical or neural time.
The analytical posterior is known, so iteration is demonstrative rather than necessary.
The raw gradient step is normalized by total precision.
The stable oscillatory region 1 < α < 2 is excluded from the interface.
Independent trials do not learn from or update one another.
The model excludes hierarchy, action, active inference, and parameter learning.
It does not establish a neural implementation or biological realism.
It does not represent psychological, psychedelic, or clinical states.
It is educational, not diagnostic, therapeutic, or validated scientific software.
It uses finite-precision deterministic computation and pseudorandom sampling.

Primary and authoritative sources

Murphy, Conjugate Bayesian analysis of the Gaussian distribution — Gaussian conjugacy and precision weighting.
Boyd & Vandenberghe, Convex Optimization — quadratic objectives and gradient descent.
Rao & Ballard (1999) — historical predictive-coding context, not biological validation of this tool.
Bogacz (2017) — mathematical context for error-based optimization.
Friston (2005) — broader theoretical context, not demonstrated implementation.
NIST TN 1900 — observation and measurement-uncertainty models.
Box & Muller (1958) and O’Neill (2014) with the official PCG reference — preserved reproducible generator.
NIST SP 800-22 Rev. 1a — limits of statistical generator tests.
NIST TN 1297 — unbounded Gaussian support.
Ernst & Banks (2002) — task-specific reciprocal-variance integration evidence.
WCAG 2.2 — accessibility conformance target.