# Is bias and variance equivalent to accuracy vs precision?

I looked at some graphs of high bias and high variance, and I don't really understand how the bias vs variance tradeoff is any different from accuracy vs precision?

• In what ways do you think bias v. variance is the same as accuracy v. precision? You will need to walk is through your rationale. Feb 7, 2021 at 20:10
• “Accuracy” and “precision” and used here in the sense from science, not as the terms in machine learning classification problems. And I would say that they are similar, yes.
– Dave
Feb 7, 2021 at 20:51
• Sample mean $\bar X$ estimates population mean $\mu.$ If $\bar X$ is not systematically above $\mu$ or below $\mu,$ then your estimation of $\mu$ by $\bar X$ is said to be 'accurate'. If sample standard deviation $S$ is small then your estimation is said to be 'precise'. Often a 95% confidence interval for $\mu$ is of the form $\bar X \pm 2S/\sqrt{n}$ with a normal sample size $n > 30.$ Good precision leads to short confidence intervals. Good accuracy leads to CIs that are correctly centered near $\mu.$ Feb 7, 2021 at 21:39
• The ISO has recommended a use of "accuracy" that considers issues besides bias. See the Wikipedia page for example.
– EdM
Feb 7, 2021 at 21:51

Putting aside the use of these words in the context of classification, the word "precision" is generally taken to represent "a description of random errors, a measure of statistical variability." (Wikipedia) Even "in the sense from science," though, there are two uses of the word "accuracy" distinguished on the Wikipedia page: a common use of the term and one adopted by the ISO. In the common use, "accuracy" is

a description of systematic errors, a measure of statistical bias... ISO calls this trueness.

Under the ISO usage, accuracy is:

a combination of both types of observational error above (random and systematic), so high accuracy requires both high precision and high trueness.

In that latter ISO usage, measures like mean-squared or mean absolute error could represent "accuracy."

The bias-variance tradeoff in statistical modeling is often presented in terms of the expected error when applying a model to a new test set. Quoting from ISLR, pages 33 to 34, where $$\hat f(x_0)$$ represents the model's prediction for an input value of $$x_0$$:

... the expected test MSE, for a given value $$x_0$$, can always be decomposed into the sum of three fundamental quantities: the variance of $$\hat f(x_0)$$, the squared bias of $$\hat f(x_0)$$ and the variance of the error terms $$\epsilon$$. That is, $$E\left( y_0 − \hat f(x_0)\right)^2 = \text{Var}\left( \hat f(x_0) \right) + \left[ \text{Bias} \left( \hat f(x_0) \right) \right]^2 + \text{Var}(\epsilon)$$ Here the notation $$E\left( y_0 − \hat f(x_0)\right)^2$$ defines the expected test MSE, and refers to the average test MSE that we would obtain if we repeatedly estimated $$f$$ using a large number of training sets, and tested each at $$x_0$$. The overall expected test MSE can be computed by averaging $$E\left( y_0 − \hat f(x_0)\right)$$ over all possible values of $$x_0$$ in the test set.

I see "bias" and "variance" in this modeling context as somewhat different from the situation where one is describing the "accuracy" and "precision" of a scientific measuring instrument. One might roughly consider: common "accuracy" or ISO "trueness" related to "bias" in modeling, "variance" related to the inherent "precision" of the instrument, and $$\text{Var}(\epsilon)$$ related to inherent variability in what you're trying to measure.

Yet the nature of bias and variance in modeling differs from that in evaluating a scientific measuring instrument. In the process of modeling there is a tradeoff that needs to be made between bias and variance to improve performance on a new data set. In setting up a model, one can introduce bias in a way that decreases variance enough to lower the overall mean squared error (a measure of "accuracy" in the ISO sense). The bias-variance tradeoff is key. That's not typically the situation with designing or using a scientific measuring device.