Bias-Variance Tradeoff Calculator
ANA›Life Services Authority›National Calculator Authority›Bias-Variance Tradeoff Calculator
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Bias-Variance Tradeoff Calculator
Decompose the expected Mean Squared Error (MSE) of a model into its three fundamental components: Bias², Variance, and Irreducible Error (σ²). Enter either (A) direct component values, or (B) a set of model predictions vs. the true value.
### Mode A — Direct Component Input
Bias (systematic error, same units as target)
Variance of predictions (Var[f̂(x)])
Irreducible noise variance (σ²)
### Mode B — Predictions vs. True Value
Enter multiple model predictions (comma-separated) and the true target value. Bias = mean(predictions) − true; Variance = sample variance of predictions.
Model predictions (comma-separated)
True target value (y)
Irreducible noise variance (σ²) for Mode B
Calculate
function biaCalc() { const resultDiv = document.getElementById('bia-result'); resultDiv.innerHTML = '';
// ── helpers ────────────────────────────────────────────────────────────── const val = id => document.getElementById(id).value.trim(); const num = id => parseFloat(val(id)); const fmt = n => isNaN(n) ? '—' : n.toFixed(6); const pct = (part, total) => total === 0 ? '0.00' : (100 * part / total).toFixed(2);
// ── decide mode ────────────────────────────────────────────────────────── const predsRaw = val('bia-preds'); const trueRaw = val('bia-true'); const useB = predsRaw !== '' || trueRaw !== '';
let bias2, variance, noise, mse, biasVal, predsArr, meanPred;
if (useB) { // ── Mode B ────────────────────────────────────────────────────────── if (predsRaw === '') { resultDiv.innerHTML = 'Mode B: please enter at least one prediction.'; return; } if (trueRaw === '') { resultDiv.innerHTML = 'Mode B: please enter the true target value.'; return; }
predsArr = predsRaw.split(',').map(s => parseFloat(s.trim())); if (predsArr.some(isNaN)) { resultDiv.innerHTML = 'Mode B: predictions must be numeric values separated by commas.'; return; } if (predsArr.length Mode B: enter at least one prediction.'; return; }
const trueVal = num('bia-true'); if (isNaN(trueVal)) { resultDiv.innerHTML = 'Mode B: true target value must be a number.'; return; }
const noiseB = num('bia-noise-b'); if (isNaN(noiseB) || noiseB Mode B: irreducible noise variance must be ≥ 0.'; return; }
const n = predsArr.length; meanPred = predsArr.reduce((a, b) => a + b, 0) / n; biasVal = meanPred - trueVal; bias2 = biasVal * biasVal;
// population variance (expectation over draws from the training distribution) // use population variance (divide by n) — consistent with the theoretical decomposition variance = predsArr.reduce((acc, p) => acc + (p - meanPred) ** 2, 0) / n; noise = noiseB; mse = bias2 + variance + noise;
} else { // ── Mode A ────────────────────────────────────────────────────────── const biasA = num('bia-bias'); const varA = num('bia-variance'); const noiseA = num('bia-noise');
if (isNaN(biasA)) { resultDiv.innerHTML = 'Mode A: Bias must be a number.'; return; } if (isNaN(varA) || varA Mode A: Variance must be a non-negative number.'; return; } if (isNaN(noiseA) || noiseA Mode A: Irreducible noise variance must be ≥ 0.'; return; }
biasVal = biasA; bias2 = biasA * biasA; variance = varA; noise = noiseA; mse = bias2 + variance + noise; }
// ── bar widths (visual) ────────────────────────────────────────────────── const maxComp = Math.max(bias2, variance, noise, 1e-12); const barW = v => Math.max(2, Math.round(200 * v / maxComp));
const barStyle = (v, color) => ``;
// ── interpretation ─────────────────────────────────────────────────────── let interp = ''; if (bias2 > variance && bias2 > noise) { interp = '⚠️ High-Bias (Underfitting) regime: the model is too simple or systematically wrong. Consider a more complex model, adding features, or reducing regularisation.'; } else if (variance > bias2 && variance > noise) { interp = '⚠️ High-Variance (Overfitting) regime: the model is too sensitive to training data. Consider regularisation, more training data, pruning, or ensemble methods.'; } else if (noise >= bias2 && noise >= variance) { interp = '✅ Irreducible-noise dominated: both bias and variance are well-controlled. Further improvement requires better data quality or feature engineering.'; } else { interp = '✅ Balanced: bias and variance contribute roughly equally. Fine-tune the complexity–regularisation tradeoff for marginal gains.'; }
// ── Mode B extra detail ──────────────────────────────────────────────────
let modeDetail = '';
if (useB) {
const predList = predsArr.map(p => p.toFixed(4)).join(', ');
modeDetail = Predictions[${predList}]
Mean prediction (Ê[f̂])${fmt(meanPred)}
True value (y)${fmt(num('bia-true'))}
Bias = Ê[f̂] − y${fmt(biasVal)};
} else {
modeDetail = Bias${fmt(biasVal)};
}
resultDiv.innerHTML = ` ### Results
Component Value % of MSE Visual
${modeDetail}
Bias² = (Ê[f̂] − y)² ${fmt(bias2)} ${pct(bias2, mse)}% ${barStyle(bias2, '#e74c3c')}
Variance = Var[f̂(x)] ${fmt(variance)} ${pct(variance, mse)}% ${barStyle(variance, '#3498db')}
Irreducible Noise (σ²) ${fmt(noise)} ${pct(noise, mse)}% ${barStyle(noise, '#2ecc71')}
Expected MSE ${fmt(mse)} 100.00%
${interp}
`; }
#### Formula
The Bias-Variance Decomposition of the expected Mean Squared Error at a point x:
E[(y − f̂(x))²] = Bias²[f̂(x)] + Var[f̂(x)] + σ²
where: Bias[f̂(x)] = E[f̂(x)] − f(x) (systematic error) Var[f̂(x)] = E[(f̂(x) − E[f̂(x)])²] (sensitivity to training set) σ² = Var[ε] (irreducible noise in y = f(x) + ε)
Mode B estimators (given M predictions {f̂₁, …, f̂_M}): Ê[f̂] = (1/M) Σ f̂ᵢ Bias = Ê[f̂] − y Var = (1/M) Σ (f̂ᵢ − Ê[f̂])² (population variance)
#### Assumptions & References
- The true data-generating process is y = f(x) + ε where ε is zero-mean noise with variance σ².
- Bias and variance are properties of the learning algorithm averaged over all possible training sets of a fixed size, not of a single trained model.
- Mode B uses population variance (divide by n) to stay consistent with the theoretical expectation operator; use sample variance (divide by n−1) if estimating from a finite sample of algorithms.
- References:
Geman, S., Bienenstock, E., & Doursat, R. (1992). Neural networks and the bias/variance dilemma. Neural Computation, 4(1), 1–58. - Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning (2nd ed.), §2.9. Springer. - Bishop, C. M. (2006). Pattern Recognition and Machine Learning, §3.2. Springer.
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