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In statistics and machine learning, the bias–variance tradeoff describes the relationship between a model's complexity, the accuracy of its predictions, and how well it can make predictions on previously unseen data that were not used to train the model. In general, as we increase the number of tunable parameters in a model, it becomes more ...
The reason that an uncorrected sample variance, S 2, is biased stems from the fact that the sample mean is an ordinary least squares (OLS) estimator for μ: ¯ is the number that makes the sum = (¯) as small as possible. That is, when any other number is plugged into this sum, the sum can only increase.
This is known as the bias–variance tradeoff. Keeping a function simple to avoid overfitting may introduce a bias in the resulting predictions, while allowing it to be more complex leads to overfitting and a higher variance in the predictions. It is impossible to minimize both simultaneously.
Firstly, while the sample variance (using Bessel's correction) is an unbiased estimator of the population variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality.
The bias–variance tradeoff is often used to overcome overfit models. With a large set of explanatory variables that actually have no relation to the dependent variable being predicted, some variables will in general be falsely found to be statistically significant and the researcher may thus retain them in the model, thereby overfitting the ...
Given an r-sample statistic, one can create an n-sample statistic by something similar to bootstrapping (taking the average of the statistic over all subsamples of size r). This procedure is known to have certain good properties and the result is a U-statistic. The sample mean and sample variance are of this form, for r = 1 and r = 2.
This is the bias-variance tradeoff; if h is too small, the estimate exhibits large variation; while at large h, the estimate exhibits large bias. Careful choice of bandwidth is therefore crucial when applying local regression. Mathematical methods for bandwidth selection require, firstly, formal criteria to assess the performance of an estimate.
An example arises in the estimation of the population variance by sample variance. For a sample size of n , the use of a divisor n −1 in the usual formula ( Bessel's correction ) gives an unbiased estimator, while other divisors have lower MSE, at the expense of bias.