Experiment 1
Keep estimates fixed. Increase only visual standard deviation from 1 to 4 cm.
Check the prediction
Vision now gets 20% weight and the estimate moves to 13.2 cm. Reliability concerns spread, not which sense has a privileged name.
Inference and perception
Reliable cues earn more weight.
Derive a two-cue estimate and identify when correlated errors or different causes invalidate it.
Start with: Averages and variance. Standard deviation measures spread; variance is its square. Precision is inverse variance.
01 · Commit to a prediction
What do you expect?
Under an unbiased independent Gaussian model, the more precise cue receives more weight. Vision gets 80%, so the estimate is 10.8 cm.
Responses stay in this page only. Reloading or closing may discard them. Nothing is transmitted, saved or synchronized.
02 · Change an assumption
Predict. Change. Explain.
Before changing a control, say what should move and why. Start with the experiments below. Reset restores the starting model; it preserves your written responses.
Experiment 2
Reset. Make touch standard deviation 1 cm.
Check the prediction
Weights become equal. The combined estimate is 12 cm with standard deviation 1/√2 ≈ 0.7071 cm under independence.
0 to 20 · step 0.5
0 to 20 · step 0.5
0.5 to 5 · step 0.5
0.5 to 5 · step 0.5
Calculated model output. The table gives the same values. Displayed values are rounded; calculations keep full precision.
Read the plotted data as a table
03 · Connect the mechanism
From the picture to the quantities.
Suppose both cues measure the same unknown position with unbiased independent Gaussian errors of known variances. Multiplying their likelihoods gives a narrower Gaussian centered at a precision-weighted average. The plotted curves are normalized likelihood shapes over possible position, not neural recordings. Their vertical axis is a density, which can exceed one per unit length.
wv = (1/σᵥ²) / (1/σᵥ² + 1/σₜ²)
μ = wᵥxᵥ + (1−wᵥ)xₜ
σ² = 1 / (1/σᵥ² + 1/σₜ²)
μ = wᵥxᵥ + (1−wᵥ)xₜ
σ² = 1 / (1/σᵥ² + 1/σₜ²)
xv and xt are visual and touch observations, 0 to 20 cm here. σv and σt are positive error standard deviations, 0.5 to 5 cm. μ is the combined estimate in cm; σ² is combined variance in cm². wv is a dimensionless weight. With a flat prior over position this is also a Bayesian posterior mean; a nonflat prior would contribute additional information.
Worked example
Visual precision is 1/1² = 1 cm⁻²; touch precision is 1/2² = 0.25 cm⁻². Normalize 1:0.25 to get weights 0.8:0.2. Then μ = 0.8 × 10 + 0.2 × 14 = 10.8 cm. Combined variance is 1/1.25 = 0.8 cm² and standard deviation is √0.8 ≈ 0.894427 cm.
Assumptions and limits
This is an ideal-observer model. Ernst and Banks tested a particular visual-haptic task and found behavior consistent with reliability-weighted integration. That does not prove that every brain computes these formulas or that perception is always optimal. Cue bias, correlated errors, uncertain reliability and cues from different objects can break the model. The plot extends four standard deviations beyond the extreme cue means and truncates tiny tails for display.
The annotated sources distinguish established results from this lesson’s original examples.
04 · Follow the structure
Where else does this apply?
Sensory cue integration
Two modalities may jointly constrain the same object property.
Boundary: The brain must also infer whether the cues have a common cause; large discrepancies can lead to segregation.
Sensor fusion
Independent unbiased sensors can reduce estimation variance when combined.
Boundary: A shared calibration bias is not removed by multiplying independent likelihoods.
05 · Retrieve without hints
Close the explanation. Try a new case.
Write an answer before opening its feedback. Later, return directly here without rereading above. Recognition, explanation and transfer are separate outcomes. No page action or answer reveal measures mastery.
Recognition and explanation
Reveal reasoning and rubric
No. Variance quadruples, so precision becomes one quarter.
Self-check: Square the standard deviation before taking its inverse.
Calculation
Reveal reasoning and rubric
The mean is 6 cm. Variance is 1/(1/4+1/4) = 2 cm², so standard deviation is √2 ≈ 1.414214 cm.
Self-check: Distinguish variance from standard deviation and retain units.
Novel transfer
Reveal reasoning and rubric
No. Their errors are perfectly correlated and there is only one measurement. The copy adds no evidence. Use the covariance structure or treat it as one cue.
Self-check: Identify dependence and reject an artificial precision gain.
On a later day, try again and record actual evidence in the curriculum. A later unaided explanation and a fresh transfer problem give stronger evidence than immediate familiarity. No reminder is scheduled.
Sources & scope
What supports the lesson?
- Ernst and Banks (2002), Nature 415, 429 to 433. A specific visual-haptic experiment consistent with reliability-weighted integration, not universal optimal perception.
Original teaching examples. Reference links need a connection; the lesson itself does not. Built 2026-10-11. Learner understanding is not assessed.