What is the difference between computational methods and statistical methods?
Artificial Intelligence & Machine Learning
In the past five decades, the field of statistics notably shifted towards computational methods over traditional mathematical analysis.
This transition, which predates the era of big data, is exemplified by the bootstrap method. Introduced by Efron in 1979 and further developed by Efron and Tibshirani in 1993, the bootstrap approach relies on resampling datasets to estimate statistical properties.
Moreover, this method treats estimates as approximate sufficient statistics and the bootstrap distribution as an approximation to the data’s sampling distribution, embodying a predictive and resampling-centric view of statistics.
Earlier methods like the jackknife and cross-validation laid the groundwork. But the bootstrap’s wide applicability and simplicity in computation made it a significant tool, particularly in scenarios where traditional analytic approximations were inadequate.
The exponential growth in computational power also popularized other resampling and simulation-based methods. Techniques like permutation testing and parametric bootstrapping, as well as predictive checks, have gained traction. Additionally, these methods create replicated datasets from models or through direct data resampling, enhancing our understanding of complex models and algorithms.
A significant evolution in statistical methodology since the 1970s has been the trend towards models. With a large number of parameters, sometimes exceeding the data points themselves. Regularization procedures are employed to stabilize these richly parameterized models and ensure accurate predictions without overfitting. This approach achieves the flexibility of nonparametric methods while circumventing the pitfalls of overfitting. Thus, often through penalty functions on parameters or predictions.
What is the difference between computational methods and statistical methods?
The 1970s and onwards saw the development of sophisticated models like Markov random fields, splines, Gaussian processes, and neural networks. Alongside these were methodologies like wavelet shrinkage, lasso, horseshoe, and support vector machines. Concurrently, Bayesian nonparametric priors on infinite-dimensional model families emerged, expanding with sample size and incorporating parameters more as part of a predictive system than as direct interpretative elements.
The practical application of these overparameterized models became more feasible with the advent of substantial computational resources. Their development continued in areas like image recognition and deep neural networks. Recent perspectives on these advancements suggest a broader view of regularization, supporting dense model fitting as data allows.
Moreover, the field has seen growth in methods for tuning and combining inferences from multiple models. Techniques like stacking, Bayesian model averaging, boosting, and gradient boosting emerged. Accompanied by a foundational shift in statistics towards prediction-centric approaches.
In conclusion, the integration of computing in statistical analysis not only redefined methodological approaches. However, also expanding the scope and applicability of statistical models, underlining the dynamic and evolving nature of the field.
Written by Columbia MAFN Student Ruslan Polovinko
