Jonah Patel
Bootstrapping is a statistical method used to make inferences about a population from sample data. It involves resampling the sample data and performing inferences about the sample from the resampled data. This is done by treating the inference of the true probability distribution as analogous to an inference of the empirical distribution given the resampled data. The basic idea is that if the population is the sample, then the quality of inference of the true sample from resampled data is measurable. In this context, the term bootstrap refers to the process of pulling from existing data to understand the influence of extreme values. Bootstrapping can be used to estimate a quantity of a population by repeatedly taking small samples, calculating the statistic, and taking the average of the calculated statistics. The bootstrap distribution of a point estimator of a population parameter can be used to produce a bootstrapped confidence interval for the parameter's true value.
| Characteristics | Values |
|---|---|
| Basic idea | Inference about a population from sample data (sample → population) can be modeled by resampling the sample data and performing inference about a sample from resampled data (resampled → sample) |
| Accuracy | Can be assessed because we know Ĵ, the empirical distribution |
| Sample size | Same as the original dataset; can be smaller if the original dataset is enormous and computational efficiency is an issue |
| Number of samples | Should be as many as is reasonable, given available computing power and time; 1000 is a good lower bound, 10,000 is a good rule of thumb |
| Sample values | Can appear more or less frequently in the resampled datasets than in the original dataset |
| Sample repetition | Some samples will be represented multiple times in the bootstrap sample while others will not be selected at all |
| Sample type | Nonparametric and parametric |
| Sample procedure | Sampling with replacement |
| Sample calculation | Calculate the mean of the calculated sample statistics |
| Sample application | Used to estimate a quantity of a population, the skill of a machine learning model, or to assess the reliability of an inferred tree |
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What You'll Learn
- Bootstrapping is a statistical tool to quantify uncertainty associated with a given estimator or learning method
- The bootstrap works by treating the inference of the true probability distribution as analogous to an empirical distribution
- The bootstrap sample is the same size as the original dataset, with some samples represented multiple times and others not at all
- The bootstrap distribution of a point estimator of a population parameter has been used to produce a bootstrapped confidence interval
- Bootstrap aggregating (bagging) is a meta-algorithm based on averaging model predictions obtained from models trained on multiple bootstrap samples
Bootstrapping is a statistical tool to quantify uncertainty associated with a given estimator or learning method
Bootstrapping is a statistical tool that uses random sampling methods to estimate the distribution of an estimator or a learning method. It is a procedure for estimating the distribution of an estimator by resampling the data or a model estimated from the data. This technique allows for the estimation of the sampling distribution of almost any statistic.
The basic idea of bootstrapping is to model inference about a population from sample data by resampling the sample data and performing inference about a sample from resampled data. In other words, it treats the inference of the true probability distribution, given the original data, as analogous to an inference of the empirical distribution given the resampled data. This allows us to assess the accuracy of inferences regarding the empirical distribution using the resampled data because we know the empirical distribution. If the empirical distribution is a reasonable approximation of the true distribution, then the quality of inference on the true distribution can be inferred.
Bootstrapping is often used as an alternative to traditional hypothesis testing procedures. Traditional statistical methods try to make generalizations about a data set based on a single sample, whereas bootstrapping can generate thousands of simulated samples, each with its own statistic of interest. This allows for the development of a more precise confidence interval since it relies on a larger collection of samples. Bootstrapping is also more equipped for calculating standard error since it generates many simulated samples at random, making it easier to determine the means of different samples and estimate a sampling distribution that is more reflective of the larger data set.
The simplest bootstrap method involves taking an original data set and using a computer to sample from it to form a new sample, or "resample," of the same size. This process is repeated a large number of times (typically 1,000 or 10,000 times), and for each of these bootstrap samples, the mean is computed. A histogram of these bootstrap means provides an estimate of the shape of the distribution of the sample mean, from which we can answer questions about how much the mean varies across samples.
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The bootstrap works by treating the inference of the true probability distribution as analogous to an empirical distribution
Bootstrapping is a statistical method that uses resampling with replacement to estimate the properties of a population from a sample. The basic idea is to treat the inference of the true probability distribution as analogous to an inference of the empirical distribution from resampled data. This allows us to make inferences about the population based on the sample, even when the population is unknown.
The bootstrap works by taking a sample of size N and resampling this data with replacement to create a new set of sequences. This new set of sequences is then used to infer the empirical distribution, which can be used to make inferences about the true probability distribution. The accuracy of these inferences can be assessed because we know the empirical distribution, and if it is a reasonable approximation, we can infer the quality of the inference on the true probability distribution. This process is particularly useful when the theoretical distribution of a statistic is complicated or unknown, as it provides an indirect method to assess the underlying distribution and its parameters.
Bootstrapping can be used to construct confidence intervals for hypothesis testing, such as in regression problems. It is often used as an alternative to statistical inference based on the assumption of a parametric model. The bootstrap distribution of a point estimator of a population parameter can be used to produce a bootstrapped confidence interval for the parameter's true value. This is especially useful when the parameter can be written as a function of the population's distribution, and various families of point estimators are available for this purpose.
The bootstrap has desirable asymptotic properties, such as weak consistency and the validity of confidence intervals derived from the bootstrap. It also works well when the distribution is symmetrical and centred on the observed statistic, and the sample statistic is median-unbiased. However, it may not be ideal for small sample sizes, as the basic and reverse percentile confidence intervals tend to be less accurate and some authors discourage their use.
In summary, the bootstrap works by treating the inference of the true probability distribution as analogous to an inference of the empirical distribution from resampled data. This allows us to make inferences about the population based on the sample and assess the accuracy of these inferences by comparing the empirical distribution to the true probability distribution. Bootstrapping provides a flexible and powerful tool for statistical inference, especially in cases where the theoretical distribution is unknown or complex.
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The bootstrap sample is the same size as the original dataset, with some samples represented multiple times and others not at all
The bootstrap method is a statistical technique used to estimate the precision of sample statistics by sampling from the original dataset with replacement. In other words, it involves randomly selecting samples from the original dataset, allowing some samples to be included in the new sample multiple times, while others may not be included at all. This process creates a new sample, known as a bootstrap sample, of the same size as the original dataset.
The key idea behind the bootstrap is to treat the dataset as a population and draw random samples from it to simulate the process of data collection. By drawing samples with replacement, the bootstrap method captures the variability in the sample estimates, providing a distribution of possible values for the statistic of interest. This distribution can then be used to calculate confidence intervals or perform hypothesis tests.
When creating a bootstrap sample, the number of times each sample is selected follows a binomial distribution. This means that some samples may be selected multiple times, resulting in duplicate entries in the bootstrap sample, while others may not be selected at all. The probability of a sample being selected is equal to the proportion of that sample in the original dataset.
It is important to note that the bootstrap sample is not simply a random subset of the original dataset. The presence of duplicate entries and the omission of some samples introduce variability into the sample estimates, allowing for the assessment of precision and uncertainty. By repeating the bootstrap process multiple times, a distribution of values for the statistic of interest is obtained, providing valuable insights into the variability and reliability of the estimates.
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The bootstrap distribution of a point estimator of a population parameter has been used to produce a bootstrapped confidence interval
Bootstrapping is a statistical technique used to make inferences about a population from sample data. It involves resampling the sample data and performing inference about the sample from the resampled data. The basic idea is that by treating the sample as the "population", we can measure the quality of inference of the "true" sample from the resampled data. This is because the true error in a sample statistic against its population value is unknown when the population is unknown.
The bootstrap distribution of a point estimator of a population parameter is a powerful tool in statistics. It is used to produce a bootstrapped confidence interval for the parameter's true value. This is done by treating the inference of the true probability distribution as analogous to an inference of the empirical distribution given the resampled data. If the empirical distribution is a reasonable approximation of the true distribution, then the quality of the inference on the true distribution can be inferred.
There are several popular families of point estimators that can be used to estimate population parameters, including mean-unbiased minimum-variance estimators, median-unbiased estimators, Bayesian estimators, and maximum-likelihood estimators. The choice of estimator depends on the specific problem and the characteristics of the data. For example, a Bayesian point estimator or a maximum-likelihood estimator is theoretically ideal when the sample size is infinite, but for practical problems with finite samples, other estimators may be more suitable.
The bootstrap method can also be used to estimate the skill of a machine learning model. This is done by training the model on the sample and evaluating its performance on the out-of-bag samples, or the samples not included in the training set. By repeating this process multiple times, we can assess the variation and accuracy of the model's performance. This procedure is known as bootstrap aggregating or bagging, and it has certain desirable properties, such as the validity of confidence intervals derived from the bootstrap.
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Bootstrap aggregating (bagging) is a meta-algorithm based on averaging model predictions obtained from models trained on multiple bootstrap samples
Bootstrap aggregating, also known as bagging, is a machine learning (ML) ensemble meta-algorithm that improves the stability and accuracy of ML classification and regression algorithms. It is derived from the concept of bootstrapping, which was developed by Bradley Efron, and proposed by Leo Breiman, who also coined the term "bagging".
Bagging is a special case of the ensemble averaging approach, which combines multiple models to improve overall prediction accuracy and model stability. It involves training multiple models independently on random subsets of the data, and then aggregating their predictions through voting or averaging. Each model is trained on a random subset of the data sampled with replacement, meaning that the individual data points can be chosen more than once. This random subset is known as a bootstrap sample.
The process of bagging reduces the variance of the individual models and avoids overfitting by exposing the constituent models to different parts of the dataset. The predictions from all the sampled models are then combined through a simple averaging to make the overall prediction. This way, the aggregated model incorporates the strengths of the individual ones and cancels out their errors.
Bagging is particularly effective in reducing variance and overfitting, making the model more robust and accurate, especially in cases where the individual models are prone to overfitting on the training set, which can lead to wrong predictions on new data. It is also useful for handling high variability, which is common in algorithms like decision trees.
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Frequently asked questions
Bootstrapping is a statistical method used to make inferences about a population from sample data. It involves resampling the sample data and performing inference about a sample from resampled data. The basic idea is to treat the inference of the true probability distribution as analogous to an inference of the empirical distribution given the resampled data.
Bootstrapping is performed by constructing samples through drawing observations from a large data sample one at a time and returning them to the data sample after they have been chosen. This process, known as sampling with replacement, allows a given observation to be included in a given small sample more than once. The size of the bootstrap sample is typically the same as the original dataset, and only contains values that exist in the original set.
Bootstrapping is a powerful statistical tool that offers several advantages. It enables the quantification of uncertainty associated with a given estimator or statistical learning method. It is also useful for assessing the skill of a machine learning model and can be applied to a wide range of problems. Additionally, bootstrapping can handle outliers effectively and does not require the assumption of independent and identically distributed random variables.
