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"Throughout the history of statistics, advances in data analysis, p...
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"The idea of robustness is central to modern statistics, and it’s a...
What are the most important statistical ideas of the past 50 years?
∗
Andrew Gelman
†
and Aki Vehtari
‡
17 Jan 2021
Abstract
We argue that the most important statistical ideas of the past half century are: counterfactual
causal inference, bootstrapping and simulation-based inference, overparameterized models and
regularization, multilevel models, generic computation algorithms, adaptive decision analysis,
robust inference, and exploratory data analysis. We discuss common features of these ideas, how
they relate to modern computing and big data, and how they might be developed and extended
in future decades. The goal of this article is to provoke thought and discussion regarding the
larger themes of research in statistics and data science.
1. The most important statistical ideas of the past 50 years
A lot has happened in the past half century! The eight ideas below represent a categorization based
on our experiences and reading of the literature and are not listed in chronological order or in order
of importance. They are separate concepts capturing different useful and general developments in
statistics.
Each of these ideas has pre-1970 antecedents, both in the theoretical statistics literature and in
the practice of various applied fields. But each has developed enough in the past fifty years to have
become something new.
1.1. Counterfactual causal inference
We begin with a cluster of different ideas that have appeared in statistics, econometrics, psychometrics,
epidemiology, and computer science, all revolving around the challenges of causal inference, and all
in some way bridging the gap between, on one hand, naive causal interpretation of observational
inferences and, on the other, the recognition that correlation does not imply causation. The key idea
is that causal identification is possible, under assumptions, and that one can state these assumptions
rigorously and address them, in various ways, through design and analysis. Debate continues on the
specifics of how to apply causal models to real data, but the work in this area over the past fifty
years has allowed much more precision on the assumptions required for causal inference, and this in
turn has stimulated work in statistical methods for these problems.
Different methods for causal inference have developed in different fields. In econometrics the
focus has been on the interpretation of causal estimates from linear models (Imbens and Angrist,
1994), in epidemiology the focus has been on inference with observational data (Greenland and
Robins, 1986), psychologists have been aware of the importance of interactions and varying treatment
effects (Cronbach, 1975), in statistics there has been work on matching and other approaches to
adjust for and measure differences between treatment and control groups (Rosenbaum and Rubin,
1983), and in computer science there has been research on models for causal attribution in multiple
dimensions (Pearl, 2009). In all this work there has been a common thread of modeling causal
questions in terms of counterfactuals or potential outcomes, which is a big step beyond the earlier
∗
We thank Bin Yu, Brad Efron, Tom Belin, Trivellore Raghunathan, Chuanhai Liu, Sander Greenland, Howard
Wainer, and Anna Menacher for helpful comments.
†
Department of Statistics and Department of Political Science, Columbia University, New York.
‡
Department of Computer Science, Aalto University, Espoo, Finland.
1
standard approach which did not clearly distinguish between descriptive and causal inferences.
Key developments include Neyman (1923), Welch (1937), Rubin (1974), and Haavelmo (1973); see
Heckman and Pinto (2015) for some background.
Ideas and methods of counterfactual causal inference have been influential within statistics and
also in applied research and policy analysis.
1.2. Bootstrapping and simulation-based inference
A trend of statistics in the past fifty years has been the substitution of computing for mathematical
analysis, a move that began even before the onset of “big data” analysis. Perhaps the purest
example of a computationally defined statistical method is the bootstrap, in which some estimator
is defined and applied to a set of randomly resampled datasets (Efron, 1979, Efron and Tibshirani,
1993). The idea is to consider the estimate as an approximate sufficient statistic of the data and to
consider the bootstrap distribution as an approximation to the sampling distribution of the data.
At a conceptual level, there is an appeal to thinking of prediction and resampling as fundamental
principles from which one can derive statistical operations such as bias correction and shrinkage
(Geisser, 1975).
Antecedents include the jackknife and cross validation (Quenouille, 1949, Tukey, 1958, Stone,
1974), but there was something particularly influential about the bootstrap idea in that its generality
and simple computational implementation allowed it to be immediately applied to a wide variety of
applications where conventional analytic approximations failed; see for example Felsenstein (1985).
Availability of sufficient computational resources also helped as it became trivial to repeat inferences
for many resampled datasets.
The increase in computational resources has made other related resampling and simulation based
approaches popular as well. In permutation testing, resampled datasets are generated by breaking
the (possible) dependency between the predictors and target by randomly shuffling the target
values. Parametric bootstrapping, prior and posterior predictive checking (Box, 1980, Rubin, 1984),
and simulation-based calibration (Talts et al., 2020) all create replicated datasets from a model
instead of directly resampling from the data. Sampling from a known data generating mechanism is
commonly used to create simulation experiments to complement or replace mathematical theory
when analyzing complex models or algorithms.
1.3. Overparameterized models and regularization
A major change in statistics since the 1970s, coming from many different directions, is the idea
of fitting a model with a large number of parameters—sometimes more parameters than data
points—using some regularization procedure to get stable estimates and good predictions. The idea
is to get the flexibility of a nonparametric or highly parameterized approach, while avoiding the
overfitting problem. Regularization can be implemented as a penalty function on the parameters or
on the predicted curve (Good and Gaskins, 1971).
Early examples of richly parameterized models include Markov random fields (Besag, 1974),
splines (Wahba and Wold, 1975, Wahba, 1978), and Gaussian processes (O’Hagan, 1978), followed by
classification and regression trees (Breiman et al., 1984), neural networks (Werbos, 1981, Rumelhart,
Hinton, and Williams, 1987, Buntine and Weigend, 1991, MacKay, 1992, Neal, 1996), wavelet
shrinkage (Donoho and Johnstone, 1994), lasso, horseshoe, and other alternatives to least squares
(Dempster, Schatzoff, and Wermuth, 1977, Tibshirani, 1996, Carvalho, Polson, and Scott, 2010), and
support-vector machines (Cortes and Vapnik, 1995) and related theory (Vapnik, 1998). All these
models have the feature of expanding with sample size, and with parameters that did not always
2
have a direct interpretation but rather were part of a larger predictive system. In the Bayesian
approach the prior could be first considered in a function space, with the corresponding prior for
the model parameters then derived indirectly.
Many of these models had limited usage until enough computational resources became easily
available. Overparameterized models have continued to be developed in image recognition (Wu et
al., 2004) and deep neural nets (Bengio, LeCun, and Hinton, 2015, Schmidhuber, 2015). Hastie,
Tibshirani, and Wainwright (2015) have framed much of this work as the estimation of sparse
structure, but we view regularization as being more general in that it also allows for dense models to
be fit to the extent supported by data. Much of this work has been done outside of statistics, with
methods such as nonnegative matrix factorization (Paatero and Tapper, 1994), nonlinear dimension
reduction (Lee and Verleysen, 2007), generative adversarial networks (Goodfellow et al., 2014),
and autoencoders (Goodfellow, Bengio, and Courville, 2016): these are all unsupervised learning
methods for finding structures and decompositions.
Along with a proliferation of statistical methods and their application to larger datasets,
researchers have developed methods for tuning, adapting, and combining inferences from multiple
fits, including stacking (Wolpert, 1992), Bayesian model averaging (Hoeting et al., 1999), boosting
(Freund and Schapire, 1997), gradient boosting (Friedman, 2001), and random forests (Breiman,
2001).
1.4. Multilevel models
Multilevel or hierarchical models have parameters that vary by group, allowing models to adapt
to cluster sampling, longitudinal studies, time-series cross-sectional data, meta-analysis, and other
structured settings. In a regression context, a multilevel model can be viewed as a particular
parametrized covariance structure or as a probability distribution where the number of parameters
increases in proportion to the data.
Multilevel models can be seen as Bayesian in that they include probability distributions for
unknown latent characteristics or varying parameters. Conversely, Bayesian models have a multilevel
structure with distributions for data given parameters and for parameters given hyperparameters.
The idea of partial pooling of local and general information is inherent in the mathematics of
prediction from noisy data and, as such, dates back to Laplace and Gauss and is implicit in the ideas
of Galton. Partial pooling was used in specific application areas such as animal breeding (Henderson
et al., 1959), and its general relevance to multiplicity in statistical estimation problems was given a
theoretical boost by the work of Stein (1955) and James and Stein (1960), ultimately inspiring work
in areas ranging from psychology (Novick et al., 1972) to pharmacology (Sheiner, Rosenberg, and
Melmon, 1972) to survey sampling (Fay and Herriot, 1979). Lindley and Smith (1972) and Lindley
and Novick (1981) supplied a mathematical structure based on estimating hyperparameters of the
multivariate normal distribution, with Efron and Morris (1971, 1972) providing a corresponding
decision-theoretic justification, and then these ideas were folded into regression modeling and applied
to a wide range of problems with structured data (for example, Liang and Zeger, 1986, and Lax and
Phillips, 2012). From a different direction, shrinkage of multivariate parameters has been given an
information-theoretic justification (Donoho, 1995). Rather than considering multilevel modeling as
a specific statistical model or computational procedure, we prefer to think of it as a framework for
combining different sources of information, and as such it arises whenever we wish to make inferences
from a subset of data (small-area estimation) or to generalize data to new problems (meta-analysis).
Similarly, Bayesian inference has been valuable not just as a way of combining prior information
with data but also as a way of accounting for uncertainty for inference and decision making.
3
1.5. Generic computation algorithms
The advances in modeling we have discussed have only become possible due to modern computing.
But this is not just larger memory, faster CPUs, efficient matrix computations, user-friendly
languages, and other innovations in computing. A key component has been advances in statistical
algorithms for efficient computing.
The innovative statistical algorithms of the past fifty years are statistical in the sense of being
motivated and developed in the context of the structure of a statistical problem. The EM algorithm
(Dempster, Laird, and Rubin, 1977, Meng and van Dyk, 1997), Gibbs sampler (Geman and Geman,
1984, Gelfand and Smith, 1990), particle filters (Kitagawa, 1993, Gordon et al., 1993, Del Moral,
1996), variational inference (Jordan et al., 1999), and expectation propagation (Minka, 2001, Heskes
et al., 2005) in different ways make use of the conditional independence structures of statistical
models. The Metropolis algorithm (Hastings, 1970) and hybrid or Hamiltonian Monte Carlo (Duane
et al., 1987) were less directly motivated by statistical concerns—these were methods that were
originally developed to compute high-dimensional probability distributions in physics—but they
have become adapted to statistical computing in the same way that optimization algorithms were
adopted in an earlier era to compute least squares and maximum likelihood estimates. The method
called approximate Bayesian computation, in which posterior inferences are obtained by simulating
from the generative model instead of evaluating the likelihood function, can be useful if the analytic
form of the likelihood is intractable or very costly to compute (Rubin, 1984, Tavar´e et al., 1997,
Marin et al., 2012).
Throughout the history of statistics, advances in data analysis, probability modeling, and
computing have gone together, with new models motivating innovative computational algorithms
and new computing techniques opening the door to more complex models and new inferential ideas,
as we have already noted in the context of high-dimensional regularization, multilevel modeling,
and the bootstrap. The generic automatic inference algorithms allowed decoupling the development
of the models so that changing the model did not require changes to the algorithm implementation.
1.6. Adaptive decision analysis
From the 1940s through the 1960s, decision theory was often framed as foundational to statistics,
via utility maximization (Wald, 1949, Savage, 1954), error-rate control (Tukey, 1953, Scheff´e, 1959),
and empirical Bayes analysis (Robbins, 1959, 1964), and recent decades have seen developments
following up this work, in Bayesian decision theory (Berger, 1985) and false discovery rate analysis
(Benjamini and Hochberg, 1995). Decision theory has also been influenced from the outside by
psychology research on heuristics and biases in human decision making (Kahneman, Slovic, and
Tversky, 1982, Gigerenzer and Todd, 1999).
One can also view decision making as an area of statistical application. Some important
developments in statistical decision analysis involve Bayesian optimization (Mockus, 1974, 2012,
Shariari et al., 2015) and reinforcement learning (Sutton and Barto, 2018), which are related to
a renaissance in experimental design for A/B testing in industry and online learning in many
engineering applications. Recent advances in computation have made it possible to use richly
parameterized models such as Gaussian process and neural networks as priors for functions in
adaptive decision analysis, and to run massive scale reinforcement learning in simulated environments,
for example to create artificial intelligence to control robots, generate text, and play games such as
go (Silver et al., 2017).
4
1.7. Robust inference
The idea of robustness is central to modern statistics, and it’s all about the idea that we can use
models even when they have assumptions that are not true—indeed, an important part of statistical
theory is to develop models that work well, under realistic violations of these assumptions. Early
work in this area was synthesized by Tukey (1960); see Stigler (2010) for a historical review. Following
the theoretical work of Huber (1972) and others, researchers have developed robust methods that
have been influential in practice, especially in economics, where there is acute awareness of the
imperfections of statistical models. In economic theory there is the idea of the “as if” analysis
and the reduced-form model, so it makes sense that econometricians are interested in statistical
procedures that work well under a range of assumptions. For example, applied researchers in
economics and other social sciences make extensive use of robust standard errors (White, 1980) and
partial identification (Manski, 1990).
In general, though, the main impact of robustness in statistical research is not in the development
of particular methods, so much as in the idea of evaluating statistical procedures under what Bernardo
and Smith (1994) call the
M
-open world in which the data-generating process does not fall within
the class of fitted probability models. Greenland (2005) has argued that researchers should explicitly
account for sources of error that are not traditionally included in statistical models. Concerns of
robustness are relevant for the densely parameterized models that are characteristic of much of
modern statistics, and this has implications for model evaluation more generally (Navarro, 2018).
1.8. Exploratory data analysis
The statistical ideas discussed above all involve some mixture of intense theory and intense compu-
tation. From a completely different direction, there has been an influential back-to-basics movement,
eschewing probability models and focusing on graphical visualization of data. The virtues of statis-
tical graphics were convincingly argued in influential books by Tukey (1977) and Tufte (1983), and
many of these ideas entered statistical practice through their implementation in the data analysis
environment S (Chambers et al., 1983), a precursor to R, which is currently the dominant statistics
software in many areas of statistics and its application.
Following Tukey (1962), the proponents of exploratory data analysis have emphasized the
limitations of asymptotic theory and the corresponding benefits of open-ended exploration and
communication (Cleveland, 1985) along with a general view of data science as going beyond statistical
theory (Chambers, 1993, Donoho, 2017). This fits into a view of statistical modeling that is focused
more on discovery than on the testing of fixed hypotheses, and as such has been influential not just
in the development of specific graphical methods but also in moving the field of statistics away from
theorem-proving and toward a more open and, we would say, healthier perspective on the role of
learning from data in science. An example in medical statistics is the much-cited paper by Bland
and Altman (1986) that recommends graphical methods for data comparison in place of correlations
and regressions.
In addition, attempts have been made to formalize exploratory data analysis: Gelman (2003)
connects data display and visualization to Bayesian predictive checks, and Wilkinson (2005) formalizes
the comparisons and data structures inherent in statistical graphics, in a way that Wickham (2016)
was able to implement into a highly influential set of R packages that has transformed statistical
practice in many fields.
Advances in computation have allowed practitioners to build large complicated models quickly,
leading to a process in which ideas of statistical graphics are useful in understanding the relation
between data, fitted model, and predictions. The term “exploratory model analysis” (Unwin,
5
Volinsky, and Winkler, 2003, Wickham, 2006) has sometimes been used to capture the experimental
nature of the data analysis process, and efforts have been made to include visualization within the
workflow of model building and data analysis (Gabry et al., 2019, Gelman et al., 2020).
2. What these ideas have in common and how they differ
2.1. Ideas lead to methods and workflows
We consider the ideas listed above to be particularly important in that each of them was not so much
a method for solving an existing problem, as an opening to new ways of thinking about statistics
and new ways of data analysis.
To put it another way, each of these ideas was a codification, bringing inside the tent an approach
that had been considered more a matter of taste or philosophy than statistics:
•
The counterfactual framework placed causal inference within a statistical or predictive frame-
work in which causal estimands could be precisely defined and expressed in terms of unobserved
data within a statistical model, connecting to ideas in survey sampling and missing-data
imputation (Little, 1993, Little and Rubin, 2002).
• The bootstrap opened the door to a form of implicit nonparametric modeling.
•
Overparameterized models and regularization formalized and generalized the existing practice
of restricting a model’s size based on the ability to estimate its parameters from the data,
which is related to cross validataion and information criteria (Akaike, 1973, Mallows, 1973,
Watanabe, 2010).
•
Multilevel models formalized “empirical Bayes” techniques of estimating a prior distribution
from data, leading to the use of such methods with more computational and inferential stability
in a much wider class of problems.
•
Generic computation algorithms make it possible for applied practitioners to fit quickly
advanced models for causal inference, multilevel analysis, reinforcement learning, and many
other areas, leading to a broader impact of core ideas in statistics and machine learning.
•
Adaptive decision analysis connects engineering problems of optimal control to the field of
statistical learning, going far beyond classical experimental design.
•
Robust inference formalized intuitions about inferential stability, framing these questions in a
way that allowed formal evaluation and modeling of different procedures to handle otherwise
nebulous concerns about outliers and model misspecification, and ideas of robust inference
have informed ideas of nonparametric estimation (Owen, 1988).
•
Exploratory data analysis moved graphical technique and discovery into the mainstream of
statistical practice, just in time for the use of these tools to better understand and diagnose
problems of new complex classes of probability models that are being fit to data.
2.2. Advances in computing
Meta-algorithms—workflows that make use of existing models and inferential procedures—have
always been with us in statistics: consider least squares, the method of moments, maximum
likelihood, and so forth. One characteristic aspect of many of the machine learning meta-algorithms
6
that have been developed in the past fifty years is that they involve splitting the data or model in
some way. The learning meta-algorithms are associated with divide-and-conquer computational
methods, most notably variational Bayes and expectation propagation.
Meta-algorithms and iterative computations are an important development in statistics for
two reasons. First, the general idea of combining information from multiple sources, or creating
a strong learner by combining weak learners, can be applied broadly, beyond the examples where
such meta-algorithms were originally developed. Second, adaptive algorithms play well with online
learning and ultimately can be viewed as representing a modern view of statistics in which data and
computation are dispersed, a view in which information exchange and computational architecture
are part of the meta-model or inferential procedure (Efron and Hastie, 2016).
It is no surprise that new methods take advantage of new technical tools: as computing improves
in speed and scope, statisticians are no longer limited to simple models with analytic solutions and
simple closed-form algorithms such as least squares. We can outline how the above-listed ideas make
use of modern computation:
•
Several of the ideas—bootstrapping, overparameterized models, and machine learning meta-
analysis—directly take advantage of computing speed and could not easily be imagined in a
pre-computer world. For example, the popularity of neural networks increased substantially
only after the introduction of efficient GPU cards and cloud computing.
•
Also important, beyond computing power, is the dispersion of computing resources: desktop
computers allowed statisticians and computer scientists to experiment with new methods and
then allowed practitioners to use them.
•
Exploratory data analysis began with pencil-and-paper graphs but has completely changed
with developments in computer graphics.
•
In the past, Bayesian inference was constrained to simple models that could be solved
analytically. With the increase in computing power, variational and Markov chain simulation
methods have allowed separation of model building and development of inference algorithms,
leading to probabilistic programming that has freed domain experts in different fields to
focus on model building and get inference done automatically. This resulted in an increase in
popularity of Bayesian methods in many applied fields starting in the 1990s.
•
Adaptive decision analysis, Bayesian optimization, and online learning are used in compu-
tationally and data-intensive problems such as optimzing big machine learning and neural
network models, real-time image processing, and natural language processing.
•
Robust statistics are not necessarily computationally intensive, but their use was associated
with a computation-fueled move away from closed-form estimates such as least squares. The
development and understanding of robust methods was facilitated by a simulation study that
used extensive computation for its time (Andrews et al., 1972).
•
Shrinkage for multivariate inference can be justified not just by statistical efficiency but also on
computational grounds, motivating a new kind of asymptotic theory (Donoho, 2006, Cand`es,
Romberg, and Tao, 2008).
•
The key ideas of counterfactual causal inference are theoretical, not computational, but in recent
years causal inference has advanced by the use of computationally intensive nonparametric
methods, leading to a unification of causal and predictive modeling in statistics, economics,
and machine learning (Hill, 2011, Wager and Athey, 2018, Chernozhukov et al., 2018).
7
2.3. Big data
In addition to the opportunities opened up for statistical analysis, modern computing has also
yielded big data in ways that have inspired the application and development of new statistical
methods: examples include gene arrays, streaming image and text data, and online control problems
such as self-driving cars. Indeed, one reason for the popularity of the term “data science” is because,
in such problems, data processing and efficient computing can be as important as the statistical
methods used to fit the data.
This is related to the saying of Hal Stern that the most important aspect of a statistical analysis
is not what you do with the data but what data you use. A common feature of all the ideas discussed
in this paper and they facilitate the use of more data, compared to previously existing approaches:
•
The counterfactual framework allows causal inference from observational data using the same
structure used to model controlled experiments.
•
Bootstrapping can be used for bias correction and variance estimation for complex surveys,
experimental designs, and other data structures where analytical calculations are not possible.
•
Regularization allows users to include more predictors in a model without such concern about
overfitting.
•
Multilevel models use partial pooling to incorporate of information from different sources,
applying the principle of meta-analysis more generally.
•
Generic computation algorithms allow users to fit larger models, which can be necessary to
connect available data to underlying questions of interest.
•
Adaptive decision analysis makes use of stochastic optimization methods developed in numerical
analysis.
•
Robust inference allows more routine use of data with outliers, correlations, and other aspects
that could get in the way of conventional statistical modeling.
•
Exploratory data analysis opens the door to visualization of complex datasets and has
motivated the development of tidy data analysis and the integration of statistical analysis,
computation, and communication.
The past fifty years have also seen the development of statistical programming environments,
most notably S (Becker, Chambers, and Wilks, 1988) and then R (Ihaka and Gentleman, 1996),
and general-purpose inference engines beginning with BUGS (Spiegelhalter et al., 1994) and its
successors (Lunn et al., 2009). More recently, ideas of numerical analysis, automated inference, and
statistical computing have started to mix, in the form of reproducible research environments such
as Jupyter notebooks and probabilistic programming environments such as Stan, Tensorflow, and
Pyro (Stan Development Team, 2020, Tensorflow, 2020, Pyro, 2020). So we can expect at least
some partial unification of inferential and computing methods, as demonstrated for example by the
use of automatic differentiation for optimization, sampling, and sensitivity analysis.
2.4. Connections and interactions among these ideas
Stigler (2016) has argued for the relevance of certain common themes underlying apparently disparate
areas of statistics. This idea of interconnection can be seen to apply to recent developments as
well. For example, what is the connection between robust statistics (which focuses on departures
8
from particular model assumptions) and exploratory data analysis (which is traditionally presented
as being not interested in models at all)? Exploratory methods such as residual plots and hang-
ing rootograms can be derived from specific model classes (additive regression and the Poisson
distribution, respectively) but their value comes in large part from their interpretability without
reference to the models that inspired them. One can similarly consider a method such as least
squares on its own terms, as an operation on data, then study the class of data generating processes
for which it will perform well, and then use the results of such a theoretical analysis to propose
more robust procedures that extend the range of useful applicability, whether defined based on
breakdown point, minimax risk, or otherwise. Conversely, purely computational methods such as
Monte Carlo evaluation of integrals can fruitfully be interpreted as solutions to statistical inference
problems (Kong et al., 2003).
For another connection, the potential outcome framework for causal inference, which allows a
different treatment effect for each unit in the population, lends itself naturally to a meta-analytic
approach in which effects can vary, and this can be modeled using multilevel regression in the
analyses of experiments or observational studies. Work on the bootstrap can, in retrospect, give
us a new perspective on empirical Bayes (multilevel) inference as a nonparametric approach in
which a normal distribution or other parametric model is used for partial pooling but final estimates
are not restricted to any parametric form. And research on regularizing wavelets and other richly
parameterized models has an unexpected connection to the stable inferential procedures developed
in the context of robustness.
Other methodological connections are more obvious. Regularized overparameterized models
are optimized using machine-learning meta-algorithms, which in turn can yield inferences that
are robust to contamination. To draw these connections another way, robust regression models
correspond to mixture distributions which can be viewed as multilevel models, and these can be
fitted using Bayesian inference. Deep learning models are related to a form of multilevel logistic
regression and relates to reproducing kernel Hilbert spaces, which are used in splines and support
vector machines (Kimeldorf and Wahba, 1971, Wahba, 2002).
Highly parameterized machine learning methods can be framed as Bayesian hierarchical models,
with regularizing penalty functions corresponding to hyperpriors, and unsupervised learning models
can be framed as mixture models with unknown group memberships. In many cases the choice of
whether to use a Bayesian generative framework depends on computation, and this can go in both
ways: Bayesian computational methods can help capture uncertainty in inference and prediction,
and efficient optimization algorithms can be used to approximate model-based inference.
Many of the ideas we have been discussing involve rich parameterizations followed by some
statistical or computational tools for regularization. As such, they can be considered as more
general implementations of the idea of sieves—models that get larger as more data become available
(Grenander, 1981, Geman and Hwang, 1982, Shen and Wong, 1994).
2.5. Theory motivating application and vice versa
It would be tempting to say that a common feature of all these methods is catchy names and good
marketing. But we suspect that the names of these methods are catchy only in retrospect. Terms
such as “counterfactual,” “bootstrap,” “stacking,” and “boosting” could well sound jargony rather
than impressive, and we suspect it is the value of the methods that has made the names sound
appealing, rather than the reverse.
Innovative ideas often meet resistance, and this was the fate of some of the influential ideas
discussed in the present article. If a new idea originates in an applied field, it can be a challenge
to convince theoreticians of its value; conversely, new methods can be criticized as being useful in
9
theory but not in practice.
We should clarify that by “resistance,” we do not necessarily mean active opposition. In
comparison with some other academic fields, statistics is not very political: there has been a live-
and-let-live attitude regarding statistical developments within academia, government, and industry,
and even fringe ideas are allowed the space to develop. Many of the methods discussed here, such as
bootstrap, lasso, and multilevel models, were immediately popular in statistics and various applied
fields, but even these ideas faced resistance in the sense that outsiders needed to be convinced of
the necessity of expanding the bounds of statistics in some particular way.
Theoretical statistics is the theory of applied statistics, and this has been clear thanks in part to
influential texts such as Cox (1958), Box and Tiao (1973), Cox and Hinkley (1974), Box, Hunter,
and Hunter (1978) that straddled that divide. There is no pure statistics in the same way that there
is pure mathematics. Yes, some statistical ideas are accessible, deep, and beautiful (that elusive
trifecta which is characteristic of the best problems in mathematics) and, as with mathematics,
these ideas have fundamental links—consider, for example, the connections between regression to
the mean, least squares, and partial pooling (Stigler, 1983)—but they are still tied to particular
topics. Like a plucked apple, research in theoretical statistics tends to dry up after it has been
removed from its source of nourishment. This is said of mathematics too, but it seems that ideas in
pure mathematical stay fresh longer and can benefit from isolated research in a way that statistical
ideas cannot.
The benefit of application to statistical theory is clear. What about the benefits the other way?
Most directly, one can view theory as a shortcut to computation. Such shortcuts will always be
needed: demands for modeling inevitably grow with computing power, hence the value of analytic
summaries and approximations. In addition, theory can help us understand how a statistical method
works, and the logic of mathematics can inspire new models and approaches to data analysis.
2.6. Links to other new and useful developments in statistics
Where do particular statistical models fit into our story? Here we are thinking of influential work
such as hazard regression (Cox, 1972), generalized linear models (Nelder, 1977, McCullagh and
Nelder, 1989), spatial autoregression (Besag, 1974, 1986), structural equation models (Baron and
Kenny, 1986), latent classification (Blei, Ng, and Jordan, 2003), Gaussian processes (O’Hagan,
1978, Rasmussen and Williams, 2006), and deep learning (Hinton, Osindero, and Teh, 2006, Bengio,
LeCun, and Hinton, 2015, Schmidhuber, 2015). As discussed above, the past half century has seen
many important developments in statistical inference and computing that have both been inspired
by and have motivated the new models and inferential ideas discussed above. The models, methods,
applications, and computing all go together.
To discuss the connections among different conceptual advances is not to deny that debates
remain regarding appropriate use and interpretation of statistical methods. For example, there
is a duality between false discovery rate and multilevel modeling, but procedures based on these
different principles can give different results. Multilevel models are typically fit using Bayesian
methods, and nothing is pooled all the way to zero in the posterior distribution. In contrast, false
discovery rate methods are typically applied using
p
-value thresholds, with the goal of identifying
some small number of statistically significantly nonzero results. For another example, in causal
inference, there is increasing interest in densely-parameterized machine learning predictions followed
by poststratification to obtain population causal estimates of specified exposures or treatments, but
in more open-ended settings there is the goal of discovering nonzero causal relationships. Again,
different methods are used, depending on whether the aim is dense prediction or sparse discovery.
Finally, we can connect research in statistical methods to trends in the application of statistics
10
within science and engineering. An entire series of articles could be written just on this topic; here
we mention one such area, the replication crisis or reproducibility revolution in biology, psychology,
economics, and other sciences where variation is large enough that conclusions need to be made
from statistical evidence. Landmark papers in the reproducibility revolution include Meehl (1978)
outlining the philosophical flaws in the standard use of null hypothesis significance testing to make
scientific claims, Ioannidis (2005) arguing that most published studies in medicine were making
claims unsupported by their statistical data, and Simmons, Nelson, and Simonsohn (2011) explaining
how “researcher degrees of freedom” can enable researchers to routinely obtain statistical significance
even from data that are pure noise. Some of the proposed remedies are procedural (for example,
Amrhein, Greenland, and McShane, 2019), but there have also been suggestions that some of the
problems with nonreplicable research can be resolved using multilevel models, partially pooling
estimates toward zero to better reflect the population of effect sizes under study (van Zwet, Schwab,
and Senn, 2020). Questions of reproducibility and stability also relate directly to bootstrapping and
robust statistics (Yu, 2013).
3. What will be the important statistical ideas of the next few decades?
3.1. Looking backward
In considering the most important developments since 1970, it could also make sense to reflect
upon the most important statistical ideas of 1920–1970 (these could include quality control, latent-
variable modeling, sampling theory, experimental design, classical and Bayesian decision analysis,
confidence intervals and hypothesis testing, maximum likelihood, the analysis of variance, and
objective Bayesian inference—quite a list!), 1870–1920 (classification of probability distributions,
regression to the mean, phenomenological modeling of data), and previous centuries, as studied by
Stigler (1986) and others.
In this article we have attempted to offer a broad perspective, reflecting the different perspectives
of the authors. But others will have their own takes on what are the most important statistical
ideas of the past fifty years. Indeed, the point of asking this question is not so much to answer it, as
to stimulate discussion of what it means for a statistical idea to be important. In the present article,
we have avoided ranking papers by citation counts or other numerical measure, but implicitly we
are measuring intellectual influence in a page-rank-like way, in that we are trying to focus on the
ideas that have influenced the development of methods that have influenced statistical practice.
We are interested in others’ views on what are the most influential statistical ideas of the last
half century and how these ideas have combined to affect the practice of statistics and scientific
learning.
3.2. Looking forward
What will come next? We agree with Karl Popper that one can’t anticipate all future scientific
developments, but we might have some ideas about how current trends will continue.
The safest bet is that there will be continuing progress on existing combinations of methods:
causal inference with rich models for potential outcomes, estimated using regularization; complex
models for structured data such as networks evolving over time, robust inference for multilevel
models; exploratory data analysis for overparameterized models (Mimno, Blei, and Engelhardt,
2015); subsetting and machine-learning meta-algorithms for different computational problems; and
so forth. In addition we expect progress on experimental design and sampling for structured data.
Another general area that is ripe for development is model understanding, sometimes called
11
interpretable machine learning (Murdoch et al., 2019, Molnar, 2020). The paradox here is that the
best way to understand a complicated model is often to approximate it with a simpler model, but
then the question is, what is really being communicated here? One potentially useful approach is
to compute sensitivities of inferences to perturbations of data and model parameters (Giordano,
Broderick, and Jordan, 2018), combining ideas of robustness and regularization with gradient-based
computational methods that are used in many different statistical algorithms.
Finally, given that just about all new statistical and data science ideas are computationally
expensive, we envision future research on validation of inferential methods, taking ideas such as
unit testing from software engineering and applying them to problems of learning from noisy data.
As our statistical methods become more advanced, there will a continuing need to understand the
links between data, models, and substantive theory.
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This is a key point: "Different methods for causal inference have developed in different fields." Note the years in which different fields are cited here (1994, 1986, 1975, 2009), as causal inference as a topic became popular, "was discovered", independently in different fields at very different dates. Today, the field is still very active!
"Multilevel models can be seen as Bayesian in that they include probability distributions for unknown latent characteristics or varying parameters. Conversely, Bayesian models have a multilevel structure with distributions for data given parameters and for parameters given hyperparameters."
Indeed, multilevel models are central to bayesian modelling and data analysis.
This is an important point - "A trend of statistics in the past fifty years has been the substitution of computing for mathematical analysis, a move that began even before the onset of “big data” analysis. Perhaps the purest example of a computationally defined statistical method is the bootstrap, in which some estimator is defined and applied to a set of randomly resampled datasets."
Note that even before "Big Data" and fast computers, these methods were developed and identified as being very useful.
Andrew Gelman is an American statistician, professor of statistics and political science at Columbia University. Aki Vehtarii is a Finnish professor in computational probabilistic modeling at Aalto University.
Together, Andrew and Aki helped coauthor the bible for bayesian data analysis: http://www.stat.columbia.edu/~gelman/book/ and are among the creators/maintainers of the Stan probabilistic programming language.
Andrew Gelman also runs the very fun and successful blog: https://statmodeling.stat.columbia.edu/
And Aki Vehtarii also runs this very popular class on bayesian data analysis:
https://avehtari.github.io/BDA_course_Aalto/index.html
“Correlation does not imply causation” has to be one of the most famous/widely shared lines from statistics,
For examples of “Correlation does not imply causation”: https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation
More annotated papers on decision theory, prospect theory, and judgement under uncertainty:
Judgment under uncertainty heuristics and biases:
https://fermatslibrary.com/s/judgment-under-uncertainty-heuristics-and-biases
Prospect Theory: an analysis of decision under risk
https://fermatslibrary.com/s/prospect-theory-an-analysis-of-decision-under-risk
Judea Pearl has a great book on the science of cause and effect that is very accessible to a general audience, called the "Book of Why": http://bayes.cs.ucla.edu/WHY/
I think "correlation does not imply causation" has been quite a damaging phrase for the field of statistics though. It is used as a thoughtless phrase to wave away correlations as worthless, or undermine the field entirely.
Correlation is an invitation for further analysis. I feel the phrase would be better suited with an aside, "... but it strongly suggests causation or a confounding variable".
Demand for hamburgers are correlated to deaths, not because hamburgers kill, but because they both have a relationship to population growth. If you stop at "correlation does not imply causation" you miss out on going down a path of discovery and nuance, and maybe figuring out how many hamburgers you'll eat before you pass.
"The idea of robustness is central to modern statistics, and it’s all about the idea that we can use models even when they have assumptions that are not true—indeed, an important part of statistical theory is to develop models that work well, under realistic violations of these assumptions."
These algorithms (Metropolis Hastings, and Hamiltonian MC) are fundamental to MCMC and Bayesian computation, are the computational workhorse of many popular probabilistic programming languages, like Stan.
Metropolis Hastings:
https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm
Hamiltonian MC: https://en.wikipedia.org/wiki/Hamiltonian_Monte_Carlo
Interesting to think about deep neural networks in the same class of methods as other over parameterized methods like regularized regression.
"Throughout the history of statistics, advances in data analysis, probability modeling, and computing have gone together, with new models motivating innovative computational algorithms and new computing techniques opening the door to more complex models and new inferential ideas, as we have already noted in the context of high-dimensional regularization, multilevel modeling, and the bootstrap. The generic automatic inference algorithms allowed decoupling the development of the models so that changing the model did not require changes to the algorithm implementation."