Bias and Variance in Statistics and Machine Learning

Part 1 - Bias and Variance in Statistics

Python code in this project :   plot_fig1.txt + embedded snippets.
In this article, I explore the concept of bias and variance, in both statistics and machine learning. These terms are derived using similar methods in both fields, but actually refer to slightly different concepts. For part one, I will first start with how these terms are used in statistics.

Then, I will talk about the machine learning case in part two. I will also touch on the observation of a tradeoff between bias and variance in machine learning algorithms, and how such a tradeoff can be absent from modern algorithms such as neural networks and random forests.

The Statistical Model


Suppose we have a set of data \( D_n = \{ x_1,...,x_n \} \) of size \( n \), where each point \( x_i \) is drawn independently from a distribution \( P \). I am suppressing the \( n \) in \( D_n \), and writing it as just \( D \) to reduce clutter. Let \( y \) be some parameter of \( P \) that we wish to estimate. Let \( h_D \) be an estimator calculated from the data.

Note that I am using the non-standard notation \( h_D \), which stands for "Hypothesis calculated from Data". This is so that our notation is consistent with what we will be using in part 2 of this article. An example of a \( h_D \) that is often calculated from samples would be the sample mean,

\[ h_D = \frac{1}{n} \sum_{i=1}^{n} X_i. \]

To illustrate, I am treating this set of data on public housing resale prices in Singapore as the population. I then took a random sample of \( n = 30 \) data points.


Because each data point in the sample \( D \) is randomly drawn from the population, the sample mean will vary each time we sample. Fir this particular draw, our sample mean was lower than the population mean.




We can see that, even with just \( n = 30 \), the sample mean is (visually) close to the population mean. The code for generating these values and the figure can be found here: plot_fig1.txt

Bias of an Estimator


The bias of estimator \( h_D \) is the difference between its expected value \( \mathbb{E}[h_{D}] \), and the true parameter value \( y \). Note that this expected value taken over all possible \( D \), which are randomly generated sets of data with size \( n \).

\[ \text{Bias}(h_D) = \mathbb{E}[h_D - y]. \]

It is known that the sample mean is an unbiased estimator of the population mean. Which means this quantity is actually zero when \( h_{D} \) is the sample mean. However, the bias might not be zero for other estimators \( h_{D} \).

Consistency

Note that the bias is defined here for a data set \( D \) with fixed finite size \( n \). There is a similar concept for the estimator's behavior as the number of data points \( n \) goes to infinity. Of course, we would like the estimator to get closer and closer to the true parameter value as \( n \) increases. And ideally we should have \( h_D = y \) if we somehow managed to collect "infinite" number of samples. This is known as "consistency" consistency, and estimators with this property are called consistent.

Variance of an Estimator


The variance of an estimator \( h_D \) is its expected squared difference from its own expected value, \[ \text{Var}(h_D) = \mathbb{E}[(h_D - \mathbb{E}[h_D])^2]. \] The variance formalizes the notion of how \( h_D \) can vary from its expected value every time we calculate it from a set of randomly sampled data \( D \).

The variance is defined in terms of the squared difference, rather than the absolute difference. This is a paradigm that we will see everywhere in statistics and machine learning. Two possible reasons for the prevalence of this paradigm could be:

One thing to note is that the square function \( f(x) = x^2 \) increases exponentially in \( x \). So, large deviations from \( y \) contributes disproportionally more to the MSE than smaller deviations.

Standard Deviation

The standard deviation is the square root of the variance. This is often used over the variance as an indicator of how much values can vary. A famous example is the 68-95-99.7 rule which estimates that 68% / 95% / 99.7% of an approximately normally distributed set of data has values within 1 / 2 / 3 standard deviations from the mean.

The Mean Square Error


The mean square error (MSE) is the expected value of the squared difference between the estimator and the parameter. \[ \mathbb{E}[(h_D - y)^2]. \]

We can decompose the MSE into two terms, by using the "adding zero" trick to add and also subtract a \( \mathbb{E}[h_D] \) term into the squared term before expanding.

\begin{align} \mathbb{E}[(h_D - y)^2] &= \mathbb{E}[(h_D - \mathbb{E}[h_D] + \mathbb{E}[h_D] - y)^2] \\[0.5em] &= \mathbb{E}[(h_D - \mathbb{E}[h_D])^2] + (\mathbb{E}[h_D] - y)^2 \\[0.5em] &= \text{Var}(h_D) + \text{Bias}^2(h_D) \end{align}

The derivation is not difficult, but can get messy. Please refer to this wikipedia section for the full derivation: https://en.wikipedia.org/wiki/Mean_squared_error#Proof_of_variance_and_bias_relationship.

Stein's Paradox


After reading my example on the sample and population mean, you might be wondering if the sample mean is the best estimator for the population mean in terms of mean square error?" It surprisingly turns out that the answer is "no"! An even more shocking fact is that no one knows what estimator of the population mean minimizes the mean square error!

This is closely related to Stein's Paradox. A great discussion about the history and the mystery surrounding the search for the best estimator of the population mean can be found in a classic statistics paper by Bradley Efron and Carl Morris.


So, we conclude our brief summary of bias and variance in statistics. We will look at bias and variance from a machine learning perspective, in part 2 of this article,