As the number of training examples goes to infinity, your model trained on that data will have:

Question

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
As sample size increases, statistical estimators and learned model parameters concentrate around their population values by the Law of Large Numbers and the Central Limit Theorem. The bias–variance decomposition of prediction error separates error into bias (systematic error) and variance (sensitivity to training set fluctuations). For a fixed model/hypothesis class and fixed training algorithm, increasing the number of independent training examples reduces the sampling variability (variance) of the learned parameters and predictions.

2. Step-by-step reasoning
Consider a simple, illustrative case: estimating a population mean μ from n i.i.d. samples X1,...,Xn with Var(Xi)=σ^2. The sample mean X̄ is an estimator of μ and its variance is:
Var(X̄) = Var( (1/n) Σ Xi ) = (1/n^2) Σ Var(Xi) = (1/n^2) · n · σ^2 = σ^2 / n.
As n → ∞, σ^2 / n → 0, so the estimator’s variance vanishes. More generally, parameter estimates from empirical risk minimization concentrate: their sampling distribution has covariance that typically shrinks on the order of 1/n (as implied by asymptotic normality). Thus the variance component of prediction error decreases with more data.

3. Calculations / intermediate steps
- Var(X̄) = (1/n^2) ΣVar(Xi) = (1/n^2)(nσ^2) = σ^2/n.
- Limit: lim_{n→∞} σ^2/n = 0.

4. Why the correct answer is correct
Because with more independent training examples the learned model becomes less sensitive to which particular finite sample was drawn; parameter estimates concentrate and their variance decreases toward zero (for a fixed model/hypothesis space and well-behaved estimator).

5. Why the other options are incorrect
- "Higher variance": contradicts the concentration results; more data reduces, not increases, sampling variability.
- "Same variance": incorrect because variance scales down (e.g., ∝1/n) and does not remain constant as n grows.
- "None of the answers are correct": one of the listed options (Lower variance) is correct, so this is false.

6. Conclusion
Therefore, as the number of training examples goes to infinity, your model trained on that data will have lower variance. This matches the provided answer: Lower variance.

As the number of training examples goes to infinity, your model trained on that data will have:单项选择题

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