Differences between the Monte Carlo Method and Bootstrapping
Overview
Monte Carlo methods involve repeating simulations with arbitrary data to test new techniques, while bootstrapping involves resampling from actual data to solve problems more cost-effectively.
Definitions
Monte Carlo Method is a method of finding point estimates for a target by drawing random samples.
Bootstrap is a method that involves resampling from a sample to understand the distribution of a target.
Explanation
The primary confusion may come from the fact that both methods involve drawing many samples and performing many trials, but they serve entirely different purposes.
Monte Carlo methods are used when we want to see how accurately a statistic estimates a true value, assuming that we know what the true value is. The question “How could we possibly know the true value?” may arise, and the answer is surprisingly simple: if the experimenter has specified the parameters and distribution and generated the data, then it is normal to know what the theoretical true value is. By repeating simulations with contrived data, it may be possible to check how closely new methods or statistics approximate reality.
On the other hand, bootstrapping starts by drawing samples from a given sample to increase the size of the sample. In statistics, the larger the sample, the better, so increasing the sample size is generally a welcome process. When conducting an experiment is too expensive and difficult, or it’s impossible to acquire more data, this method can be a significant aid to a project. The question then becomes, “How can we trust this?” It’s normal for someone with a proper statistical education to feel skeptical upon hearing about it.
To use bootstrapping, it’s assumed that the given sample adequately reflects the characteristics of the population. Even if it does reflect the population well, one must consider that biased data might appear occasionally. Moreover, the results obtained through bootstrapping can indicate the distribution but can’t explicitly reveal what the exact value being sought is.
Let’s say we’re looking for a true value $\alpha$ using an estimator $\hat{ \alpha}$. Bootstrapping involves repeatedly calculating the estimate of $\hat{ \alpha}$ through resampling, which becomes $\hat{ \alpha} ^{ \ast }$. If the difference between the value we want and the estimate is denoted by $$ \begin{cases} d = \hat{\alpha} - \alpha \\ d^{ \ast } = \hat{\alpha}^{ \ast } - \hat{ \alpha } \end{cases} $$, then the variances of each become $$ \begin{cases} \text{Var} (d) = \text{Var} ( \hat{\alpha} - \alpha ) = \text{Var} ( \hat{\alpha}) \\ \text{Var} ( d^{ \ast } ) = \text{Var} ( \hat{\alpha}^{ \ast } - \hat{ \alpha } ) = \text{Var} ( \hat{\alpha}^{ \ast } ) \end{cases} $$, considering $\alpha$ and $\hat{ \alpha}$ as constants. So, even after determining $\text{Var} (\hat{\alpha}^{ \ast })$, we still won’t know what $\alpha$ is; we can only guess its distribution.
For example, if we use bootstrapping for regression coefficients, we might get a reasonably accurate standard error for those coefficients, but we can’t be sure whether the coefficients were correctly obtained from a well-behaved sample. If the obtained coefficients are inaccurate, conducting a t-test is meaningless, and therefore, the significance of the regression coefficients remains unknown, rendering it useless.
In conclusion, Monte Carlo methods and Bootstrap differ not just in their advantages and disadvantages but in all aspects, aside from the process.