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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
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."
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/
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."