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'''Foundations of statistics''' is the usual name for the [[epistemology|epistemological]] debate in [[statistics]] over how one should conduct [[inductive inference]] from data.  Among the issues considered in [[statistical inference]] are the question of [[Bayesian inference]] versus [[frequentist inference]], the distinction between [[Ronald Fisher|Fisher]]'s "significance testing" and [[Jerzy Neyman|Neyman]]-[[Egon Pearson|Pearson]] "hypothesis testing", and whether the [[likelihood principle]] should be followed.  Some of these issues have been debated for up to 200 years without resolution.{{sfn|Efron|1978}}

Bandyopadhyay & Forster{{sfn|Bandyopadhyah & Forster|2011}} describe four statistical paradigms: "(1) classical statistics or error statistics, (ii) Bayesian statistics, (iii) likelihood-based statistics, and (iv) the [[Akaike information criterion|Akaikean-Information Criterion]]-based statistics".

Savage's text ''Foundations of Statistics'' has been cited over 10000 times on [[Google Scholar]].<ref>[http://scholar.google.co.uk/scholar?cites=9531312933296806388 Citations of Savage (1972)]</ref>  It tells the following.

{{quote|It is unanimously agreed that statistics depends somehow on probability.  But, as to what probability is and how it is connected with statistics, there has seldom been such complete disagreement and breakdown of communication since the Tower of Babel. Doubtless, much of the disagreement is merely terminological and would disappear under sufficiently sharp analysis.}}

==Fisher's "significance testing" vs Neyman-Pearson "hypothesis testing"==
In the development of classical statistics in the second quarter of the 20th 
century two competing models of inductive statistical testing were 
developed.{{sfn|Lehmann|2011}}{{sfn|Gigerenzer|1989}}  Their relative merits were 
hotly debated{{sfn|Louçã|1993}} (for over 25 years) until Fisher's death.  While 
a hybrid of the two methods is widely taught and used, the philosophical 
questions raised in the debate have not been resolved.

===Significance testing===
[[Ronald Fisher|Fisher]] popularized significant testing, primarily in two 
popular and highly influential books.{{sfn|Fisher|1925}}{{sfn|Fisher|1935}}  
Fisher's writing style in these books was strong on examples and relatively weak 
on explanations.  The books lacked proofs or derivations of significance test 
statistics (which placed statistical practice in advance of statistical theory).  
Fisher's more explanatory and philosophical writing was written much 
later.{{sfn|Fisher|1956}}  There appear to be some differences between his
earlier practices and his later opinions.

Fisher was motivated to obtain scientific experimental results without the 
explicit influence of prior opinion.  The significance test is a 
probabilistic version of [[Modus tollens]], a classic form of deductive 
inference.  The significance test might be simplistically stated, "If the 
evidence is sufficiently discordant with the hypothesis, reject the hypothesis".  In 
application, a statistic is calculated from the experimental data, a probability
of exceeding that statistic is determined and the probability is compared to a 
threshold.  The threshold (the numeric version of "sufficiently discordant") is
arbitrary (usually decided by convention).  A common application of the method 
is deciding whether a treatment has a reportable effect based on a comparative 
experiment.  Statistical significance is a measure of probability not practical 
importance.  It can be regarded as a requirement placed on statistical 
(signal/noise).  The method is based on the assumed existence of an imaginary 
infinite population corresponding to the null hypothesis.

The significance test requires only one hypothesis.  The result of the test is
to reject the hypothesis (or not), a simple dichotomy.  The test does not
distinguish between truth of the hypothesis and insufficiency of evidence to 
disprove it (so it is like a criminal trial in which the defendant is assumed innocent until proven guilty).

===Hypothesis testing===
[[Jerzy Neyman|Neyman]] & [[Egon Pearson|Pearson]] collaborated on a different, 
but related, problem – selecting among competing hypotheses based on the 
experimental evidence alone.  Of their joint papers the most cited was from 
1933.{{sfn|Neyman & Pearson|1933}}  The famous result of that paper is the 
[[Neyman-Pearson lemma]].  The lemma says that a ratio of probabilities is an 
excellent criterion for selecting a hypothesis (with the threshold for 
comparison being arbitrary).  The paper proved an optimality of Student's t-test 
(one of the significance tests).  Neyman expressed the opinion that hypothesis 
testing was a generalization of and an improvement on significance testing.  The
rationale for their methods is found in their joint papers.{{sfn|
Neyman & Pearson|1967}}

Hypothesis testing requires multiple hypotheses.  A hypothesis is always 
selected, a multiple choice.  A lack of evidence is not an immediate 
consideration.  The method is based on the assumption of a repeated sampling of 
the same population (the classical frequentist assumption).

===Grounds of disagreement===
The length of the dispute allowed the debate of a wide range of issues regarded 
as foundational to statistics.

{| style="border: 1px solid darkgray;"
|+ An example exchange from 1955-1956
! scope="col" | Fisher's Attack{{sfn|Fisher|1955}}
! scope="col" | Neyman's Rebuttal{{sfn|Neyman|1956}}
! scope="col" | Discussion
|-
| Repeated sampling of the same population
* Such sampling is the basis of frequentist probability
* Fisher preferred [[fiducial inference]]
| Fisher's theory of fiducial inference is flawed
* Paradoxes are common
| Fisher's attack on the basis of frequentist probability failed, but was not without result.  He identified a specific case (2x2 table) where the two schools of testing reach different results.  This case is one of several that are still troubling.  Commentators believe that the "right" answer is context dependent.{{sfn|Lehmann|1993}}  Fiducial probability has not fared well, being virtually without advocates, while frequentist probability remains a mainstream interpretation.
|-
| Type II errors
* Which result from an alternative hypothesis
| A purely probabilistic theory of tests requires an alternative hypothesis
| Fisher's attack on type II errors has faded with time.  In the intervening years statistics has separated the exploratory from the confirmatory.  In the current environment, the concept of type II errors is used in power calculations for confirmatory hypothesis test [[sample size determination]].
|-
| Inductive behavior
* (Vs [[inductive reasoning]], Fisher's preference)
|
| Fisher's attack on inductive behavior has been largely successful because of his selection of the field of battle.  While ''operational decisions'' are routinely made on a variety of criteria (such as cost), ''scientific conclusions'' from experimentation are typically made on the basis of probability alone.
|}

In this exchange Fisher also discussed the requirements for inductive inference, 
with specific criticism of cost functions penalizing faulty judgments.  Neyman
countered that Gauss and Laplace used them.  This exchange of arguments occurred 
15 years ''after'' textbooks began teaching a hybrid theory of statistical 
testing.

Fisher and Neyman were in disagreement about the foundations of statistics (although united in opposition to the Bayesian view):
* The interpretation of probability
** The disagreement over Fisher's inductive reasoning vs Neyman's inductive behavior contained elements of the Bayesian/Frequentist divide.  Fisher was willing to alter his opinion (reaching a provisional conclusion) on the basis of a calculated probability while Neyman was more willing to change his observable behavior (making a decision) on the basis of a computed cost. 
* The proper formulation of scientific questions with special concern for modeling{{sfn|Louçã|1993}}{{sfn|Lenhard|2006}}
* Whether it is reasonable to reject a hypothesis based on a low probability without knowing the probability of an alternative
* Whether a hypothesis could every be accepted on the basis of data
** In mathematics, deduction proves, counter-examples disprove
** In the Popperian philosophy of science, advancements are made when theories are disproven
* Subjectivity: While Fisher and Neyman struggled to minimize subjectivity, both acknowledged the importance of "good judgment".  Each accused the other of subjectivity.
** Fisher ''subjectively'' chose the null hypothesis.
** Neyman-Pearson ''subjectively'' chose the criterion for selection (which was not limited to a probability).
** Both ''subjectively'' determined numeric thresholds.

Fisher and Neyman were separated by attitudes and perhaps language.  Fisher
was a scientist and an intuitive mathematician.  Inductive reasoning was 
natural.  Neyman was a rigorous mathematician.  He was convinced by deductive
reasoning rather by a probability calculation based on an experiment.{{sfn
|Lehmann|2011}}  Thus there was an underlying clash between applied and 
theoretical, between science and mathematics.

===Related history===
Neyman, who had occupied the same building in England as Fisher, accepted a 
position on the west coast of the United States of America in 1938.  His move
effectively ended his collaboration with Pearson and their development of 
hypothesis testing.{{sfn|Lehmann|2011}}  Further development was continued by 
others.

Textbooks provided a hybrid version of significance and hypothesis testing by 
1940.{{sfn|Halpin|2006}}  None of the principals had any known personal 
involvement in the further development of the hybrid taught in introductory 
statistics today.{{sfn|Gigerenzer|1989}}

Statistics later developed in different directions including decision theory 
(and possibly game theory), Bayesian statistics, exploratory data analysis, 
robust statistics and nonparametric statistics.  Neyman-Pearson hypothesis 
testing contributed strongly to decision theory which is very heavily used (in 
statistical quality control for example).  Hypothesis testing readily 
generalized to accept prior probabilities which gave it a Bayesian flavor.  
Neyman-Pearson hypothesis testing has become an abstract mathematical subject 
taught in post-graduate statistics,{{sfn|Lehmann & Romano|2005}} while most of 
what is taught to under-graduates and used under the banner of hypothesis 
testing is from Fisher.

===Contemporary opinion===
No major battles between the two classical schools of testing have erupted for 
decades, but sniping continues (perhaps encouraged by partisans of other 
controversies).  After generations of dispute, there is virtually no chance that
either statistical testing theory will replace the other in the foreseeable 
future.

The hybrid of the two competing schools of testing can be viewed very 
differently – as the imperfect union of two mathematically complementary ideas
{{sfn|Lehmann|1993}} or as the fundamentally flawed union of philosophically 
incompatible ideas.{{sfn|Hubbard & Bayarri|2003?}}  Fisher enjoyed some 
philosophical advantage, while Neyman & Pearson employed the more rigorous 
mathematics.  Hypothesis testing is 
[[Statistical hypothesis testing#Criticism|controversial]] among some users, but 
the most popular alternative (confidence intervals) is based on the same 
mathematics.

The history of the development left testing without a single citable 
authoritative source for the hybrid theory that reflects common statistical 
practice.  The merged terminology is also somewhat inconsistent.  There is 
strong empirical evidence that the graduates (and instructors) of an 
introductory statistics class have a weak understanding of the meaning of 
hypothesis testing.{{sfn|Sotos|2007}}

===Summary===
* The interpretation of probability has not been resolved (but fiducial probability is an orphan).
* Neither test method has been rejected.  Both are heavily used for different purposes.
* Texts have merged the two test methods under the term hypothesis testing.
** Mathematicians claim that (with some exceptions) that significance tests are a special case of hypothesis tests.
** Others treat the problems and methods as distinct (or incompatible).
* The dispute has adversely affected statistical education.

==Bayesian inference versus frequentist inference==
Two different interpretations of probability (based on objective evidence and 
subjective degrees of belief) have long existed.  Gauss and 
Laplace could have debated alternatives more than 200 years ago.  Two competing
schools of statistics have developed as a consequence.
Classical inferential statistics was largely developed in the second quarter of 
the 20th Century,{{sfn|Gigerenzer|1989}} much of it in reaction to the 
(Bayesian) probability of the time which utilized the ambiguous 
[[principle of indifference]] to establish prior probabilities.  The 
rehabilitation of Bayesian inference was a reaction to the limitations of 
frequentist probability.  More reactions followed.  While the philosophical interpretations are old, the statistical
terminology is not.  The current statistical terms Bayesian and frequentist 
were stabilized in the second half of the 20th Century.{{sfn|Fienberg|2006}}  
The (philosophical, mathematical, scientific, statistical) terminology is 
confusing: the "classical" interpretation of probability is Bayesian while 
"classical" statistics is frequentist.  "Frequentist" also has varying 
interpretations - different in philosophy than in physics.

The nuances of philosophical [[probability interpretations]] are discussed 
elsewhere.  In statistics the alternative interpretations ''enable'' the 
analysis of different data using different methods based on different models to 
achieve slightly different goals.  Any statistical comparison of the competing
schools considers pragmatic criteria beyond philosophical.

===Major contributors===
{{main|History of statistics}}

Two major contributors to frequentist (classical) methods were 
[[Ronald Fisher|Fisher]] and [[Jerzy Neyman|Neyman]].{{sfn|Lehmann|2011}}  
Fisher's interpretation of probability was idiosyncratic (but strongly 
non-Bayesian).  Neyman's views were rigorously frequentist.  Three major 
contributors to 20th century Bayesian statistical philosophy, mathematics and 
methods were [[Bruno de Finetti|de Finetti]],{{sfn|de Finetti|1964}} 
[[Harold Jeffreys|Jeffreys]]{{sfn|Jeffreys|1939}} and 
[[Leonard Jimmie Savage|Savage]].{{sfn|Savage|1954}}  Savage popularized 
de Finetti's ideas in the English-speaking world and made Bayesian mathematics 
rigorous.   In 1965, Dennis Lindley's 2-volume work "Introduction to Probability 
and Statistics from a Bayesian Viewpoint" brought Bayesian methods to a wide 
audience.  Statistics has advanced over the past 3 generations; The 
"authoritative" views of the early contributors are not all current.

===Contrasting approaches===

====Frequentist inference====
{{main|Frequentist inference}}

Frequentist inference is partially and tersely described above in (Fisher's 
"significance testing" vs Neyman-Pearson "hypothesis testing").  Frequentist 
inference combines several different views.  The result is capable of supporting
scientific conclusions, making operational decisions and estimating parameters
with or without [[confidence intervals]].  Frequentist inference is based solely
on the (one set of) evidence.

====Bayesian inference====
{{main|Bayesian inference}}

A classical frequency distribution describes the probability of the data.  The 
use of [[Bayes' theorem]] allows a more abstract concept – the probability of 
a hypothesis (corresponding to a theory) given the data.  The concept was once 
known as "inverse probability".  Bayesian inference updates the probability 
estimate for a hypothesis as additional evidence is acquired.  Bayesian 
inference is explicitly based on the evidence and prior opinion, which allows it 
to be based on multiple sets of evidence.

====Comparisons of characteristics====
Frequentists and Bayesians use different models of probability.  Frequentists
often consider parameters to be fixed but unknown while Bayesians assign 
probability distributions to similar parameters.  Consequently Bayesians speak
of probabilities that don't exist for frequentists; A Bayesian speaks of the 
probability of a theory while a true frequentist can speak only of the 
consistency of the evidence with the theory.  Example: A frequentist does not 
say that there is a 95% probability that the true value of a parameter lies 
within a confidence interval, saying instead that 95% of confidence intervals 
contain the true value.

{| style="border: 1px solid darkgray;"
|+ Efron's{{sfn|Efron|2013}} comparative adjectives
! scope="col" | 
! scope="col" | Bayes
! scope="col" | Frequentist
|-
|
*Basis
*Resulting Characteristic
*_
*Ideal Application
*Target Audience
*Modeling Characteristic
|
*Belief (prior)
*Principled Philosophy
*One distribution
*Dynamic (repeated sampling)
*Individual (subjective)
*Aggressive
|
*Behavior (method)
*Opportunistic Methods
*Many distributions (bootstrap?)
*Static (one sample)
*Community (objective)
*Defensive
|}

{| style="border: 1px solid darkgray;"
|+ Alternative comparison{{sfn|Little|2005}}{{sfn|Yu|2009}}
! scope="col" | 
! scope="col" | Bayesian
! scope="col" | Frequentist
|-
|
 Strengths
|
*Complete
*Coherent
*Prescriptive
*_
*_
*_
*_
*_
*Strong inference from model
|
*Inferences well calibrated
*No need to specify prior distributions
*Flexible range of procedures
**Unbiasness, sufficiency, ancillarity...
**Widely applicable and dependable
**Asymptotic theory
**Easy to interpret
**Can be calculated by hand
*Strong model formulation & assessment
|-
|
 Weaknesses
|
*Too subjective for scientific inference
*Denies the role of randomization for design
*Requires and relies on full specification of a model (likelihood and prior)
*_
*_
*_
*Weak model formulation & assessment
|
*Incomplete
*Ambiguous
*Incoherent
*Not prescriptive
*No unified theory
*(Over?)emphasis on asymptotic properties
*Weak inference from model
|}

===Mathematical results===
Neither school is immune from mathematical criticism and neither accepts it 
without a struggle.  [[Stein's paradox]] (for example) illustrated that finding 
a "flat" or "uninformative" prior probability distribution in high dimensions is 
subtle.{{sfn|Efron|1978}}  Bayesians regard that as peripheral to the core of 
their philosophy while finding frequentism to be riddled with inconsistencies, 
paradoxes and bad mathematical behavior.  Frequentists can explain most.  Some 
of the "bad" examples are extreme situations - such as estimating the weight of 
a herd of elephants from measuring the weight of one ("Basu's elephants"), which 
allows no statistical estimate of the variability of weights.  The 
[[likelihood principle]] has been a battleground.

===Statistical results===
Both schools have achieved impressive results in solving real-world problems.
Classical statistics effectively has the longer record because numerous results 
were obtained with mechanical calculators and printed tables of special 
statistical functions.  Bayesian methods have been highly successful in the 
analysis of information that is naturally sequentially sampled (radar and 
sonar).  Many Bayesian methods and some recent frequentist methods (such as the 
bootstrap) require the computational power widely available only in the last 
several decades.

There is hint that Bayesian philosophy is "book smart" compared to Frequentist
"street smarts".  Bayesian philosophy has sometimes been silent on shuffling the 
cards.  The "design of experiments" teaches the importance of the source of 
statistical data.  Fisher was a major contributor to the theory.

There is active discussion about combining Bayesian and frequentist 
methods,{{sfn|Berger|2003}}{{sfn|Little|2005}} but reservations are expressed 
about the meaning of the results and reducing the diversity of approaches.

===Philosophical results===
Baysians are united in opposition to the limitations of frequentism, but are 
philosophically divided into numerous camps (empirical, hierarchical, objective, 
personal, subjective), each with a different emphasis. 
One (frequentist) philosopher of statistics has noted a retreat from the 
statistical field to philosophical [[probability interpretations]] over the last 
two generations.{{sfn|Mayo|2013}}  There is a perception that successes in 
Bayesian applications do not justify the supporting philosophy.{{sfn|Senn|2011}}  
Bayesian methods often create useful models that are not used for traditional 
inference and which owe little to philosophy.{{sfn|Gelman & Shalizi|2012}}  
None of the philosophical interpretations of probability (frequentist or 
Bayesian) appears robust.  The frequentist view to too rigid and limiting while 
the Bayesian view can be simultaneously objective and subjective, etc.

===Illustrative quotations===
* "carefully used, the frequentist approach yields broadly applicable if sometimes clumsy answers"{{sfn|Cox|2005}}
* "To insist on unbiased [frequentist] techniques may lead to negative (but unbiased) estimates of a variance; the use of p-values in multiple tests may lead to blatant contradictions; conventional 0.95-confidence regions may actually consist of the whole real line. No wonder that mathematicians find it often difficult to believe that conventional statistical methods are a branch of mathematics."{{sfn|Bernardo|2008}}
* "Bayesianism is a neat and fully principled philosophy, while frequentism is a grab-bag of opportunistic, individually optimal, methods."{{sfn|Efron|2013}}   
* "in multiparameter problems flat priors can yield very bad answers"{{sfn|Cox|2005}}
* "[Bayes' rule] says there is a simple, elegant way to combine current information with prior experience in order to state how much is known. It implies that sufficiently good data will bring previously disparate observers to agreement. It makes full use of available information, and it produces decisions having the least possible error rate."{{sfn|Kass|2012?}}  
* "Bayesian statistics is about making probability statements, frequentist statistics is about evaluating probability statements."{{sfn|Gelman|2008}}
* "[S]tatisticians are often put in a setting reminiscent of Arrow’s paradox, where we are asked to provide estimates that are informative and unbiased and confidence statements that are correct conditional on the data and also on the underlying true parameter."{{sfn|Gelman|2008}} (These are conflicting requirements.)  
* "formal inferential aspects are often a relatively small part of statistical analysis"{{sfn|Cox|2005}}
* "The two philosophies, Bayesian and frequentist, are more orthogonal than antithetical."{{sfn|Efron|2013}}

===Summary===
* Bayesian theory has a mathematical advantage
** Frequentist probability has existence and consistency problems 
** But, finding good priors to apply Bayesian theory remains (very?) difficult
* Both theories have impressive records of successful application
* Neither supporting philosophical interpretation of probability is robust
* There is increasing skepticism of the connection between application and philosophy
* Some statisticians are recommending active collaboration (beyond a cease fire)

==The likelihood principle==
{{main|Likelihood principle}}
Likelihood is a synonym for probability in common usage.  In statistics it is
reserved for probabilities that fail to meet the frequentist definition.  A 
probability refers to variable data for a fixed hypothesis while a likelihood 
refers to variable hypotheses for a fixed set of data.  Repeated measurements of 
a fixed length with a ruler generate a set of observations.  Each fixed set of
observational conditions is associated with a probability distribution and each
set of observations can be interpreted as a sample from that distribution – the 
frequentist view of probability.  Alternatively a set of observations may result 
from sampling any of a number of distributions (each resulting from a set of 
observational conditions).  The probabilistic relationship between a fixed 
sample and a variable distribution (resulting from a variable hypothesis) is 
termed likelihood – a Bayesian view of probability.  A set of length 
measurements may imply readings taken by careful, sober, rested, motivated 
observers in good lighting.

A likelihood is a probability (or not) by another name which exists because of 
the limited frequentist definition of probability.  Likelihood is a concept 
introduced and advanced by [[Ronald A Fisher|Fisher]] for more than 40 years 
(although prior references to the concept exist and Fisher's support was half-hearted).{{sfn|Edwards|1999}}  The concept was accepted and substantially 
changed by [[Harold Jeffreys|Jeffreys]].{{sfn|Aldrich|2002}}  In 1962 
[[Allan Birnbaum|Birnbaum]] "proved" the likelihood principle from premises 
acceptable to most statisticians.{{sfn|Birnbaum|1962}}  The "proof" has been 
disputed by statisticians and philosophers.  The principle says that all of the 
information in a sample is contained in the [[likelihood function]], which is 
accepted as a valid probability distribution by Bayesians (but not by 
frequentists).

Some (frequentist) significance tests are not consistent with the likelihood 
principle.  Bayesians accept the principle which is consistent with their 
philosophy (perhaps encouraged by the discomfiture of frequentists).  "[T]he 
likelihood approach is compatible with Bayesian statistical inference in the 
sense that the posterior Bayes distribution for a parameter is, by Bayes’s 
Theorem, found by multiplying the prior distribution by the likelihood 
function."{{sfn|Edwards|1999}}  Frequentists interpret the principle adversely 
to Bayesians as implying no concern about the reliability of evidence.  "The 
likelihood principle of Bayesian statistics implies that information about the 
experimental design from which evidence is collected does not enter into the 
statistical analysis of the data."{{sfn|Backe|1999}}  Many Bayesians (Savage for 
example){{sfn|Savage|1960|p=585}} recognize that implication as a vulnerability.

The likelihood principle has become an embarrassment to both major 
philosophical schools of statistics; It has weakened both rather than favoring 
either.  Its strongest supporters claim that it offers a better foundation for 
statistics than either of the two schools.  "[L]ikelihood looks very good indeed 
when it is compared with these [Bayesian and frequentist] alternatives."{{sfn|
Forster & Sober|2001}}  These supporters include statisticians and philosophers 
of science.{{sfn|Royall|1997}}  The concept needs further development before 
it can be regarded as a serious challenge to either existing school, but it
seems to offer a promising compromise position.  While Bayesians acknowledge the 
importance of likelihood for calculation, they believe that the posterior 
probability distribution is the proper basis for inference.{{sfn|Lindley|2000}}

==Modeling==
{{main|Statistical model|Structural equation modeling}}
Inferential statistics is based on models.  Much of classical hypothesis 
testing, for example, was based on the assumed normality of the data.  Robust 
and nonparametric statistics were developed to reduce the dependence on that 
assumption.  Bayesian statistics interprets new observations from the 
perspective of prior knowledge – assuming a modeled continuity between past and 
present.  The design of experiments assumes some knowledge of those factors to 
be controlled, varied, randomized and observed.  Statisticians are well aware of 
the difficulties in proving causation (more of a modeling limitation than a 
mathematical one), saying "[[correlation does not imply causation]]".

More complex statistics utilizes more complex models, often with the intent of 
finding a latent structure underlying a set of variables.  As models and data 
sets have grown in complexity,<ref>Some large models attempt to predict the 
behavior of voters in the United States of America.  The population is around 
300 million.   Each voter may be influenced by many factors.  For some of the 
complications of voter behavior (most easily understood by the natives) see: 
http://www.stat.columbia.edu/~gelman/presentations/redbluetalkubc.pdf</ref><ref>
Efron mentions millions of data points and thousands of parameters from 
scientific studies.</ref> foundational questions have been raised about the 
justification of the models and the validity of inferences drawn from them.  The 
range of conflicting opinion expressed about modeling is large.

*Models can be based on scientific theory or on ad-hoc data analysis.  The approaches use different methods.  There are advocates of each.{{sfn|Tabachnick & Fidell|1996}}
*Model complexity is a compromise.  The Akaikean information criterion and Bayesian information criterion are two less subjective approaches to achieving that compromise.{{sfn|Forster & Sober|1994}}  
*Fundamental reservations have been expressed about even simple regression models used in the social sciences.  A long list of assumptions inherent to the validity of a model is typically neither mentioned nor checked.  A favorable comparison between observations and model is often considered sufficient.{{sfn|Freedman|1995}}
*Bayesian statistics focuses so tightly on the posterior probability that it ignores the fundamental comparison of observations and model.{{sfn|Gelman & Shalizi|2012}}
*Traditional observation-based models are inadequate to solve many important problems.  A much wider range of models, including algorithmic models, must be utilized.  "If the model is a poor emulation of nature, the conclusions may be wrong."{{sfn|Breiman|2001}}  
*Modeling is often poorly done (the wrong methods are used) and poorly reported.{{sfn|Chin|?}}

In the absence of a strong philosophical consensus review of statistical 
modeling, many statisticians accept the cautionary words of statistician 
[[George E. P. Box|George Box]], "[A]ll models are wrong, but some are useful." 
(wikiquote)

==Other reading==
For a short introduction to the foundations of statistics, see ch.&nbsp;8 ("Probability and statistical inference") of ''Kendall's Advanced Theory of Statistics'' (6th edition, 1994).

In his book ''Statistics As Principled Argument'', [[Robert P. Abelson]] articulates the position that statistics serves as a standardized means of settling disputes between scientists who could otherwise each argue the merits of their own positions ''[[ad infinitum]]''. From this point of view, statistics is a form of rhetoric; as with any means of settling disputes, statistical methods can succeed only as long as all parties agree on the approach used.

==See also==
{{Portal|Statistics}}
{{colbegin}}
*[[Philosophy of statistics]]
*[[History of statistics]]
*[[Philosophy of probability]]
*[[Philosophy of mathematics]]
*[[Philosophy of science]]
*[[Evidence]]
*[[Probability interpretations]]
*[[Founders of statistics]]
{{colend}}

==Notes==
{{reflist|colwidth=25em}}

==References==
*{{Cite book |last=Abelson |first=Robert P. |authorlink=Robert P. Abelson |title=Statistics as Principled Argument |year=1995 |publisher=Lawrence Erlbaum Associates |isbn=0-8058-0528-1 |quote=... the purpose of statistics is to organize a useful argument from quantitative evidence, using a form of principled rhetoric. }}
*Stuart A., Ord J.K. (1994). ''Kendall's Advanced Theory of Statistics, volume I: Distribution Theory'' (Edward Arnold).

* {{cite journal|last=Aldrich|first=John|title=How likelihood and identification went Bayesian|journal=International Statistical Review|volume=70|issue=1|pages=79–98|year=2002|ref=harv}}
* {{cite journal|last=Backe|first=Andrew|title=The likelihood principle and the reliability of experiments|journal=Philosophy of Science|pages=S354-S361|year=1999|ref=harv}}
* {{cite journal|last=Berger|first=James O.|title=Could Fisher, Jeffreys and Neyman Have Agreed on Testing?|journal=Statistical Science|year=2003|volume=18|issue=1|pages=1–32|ref=harv}} 
* {{cite journal|last=Bernardo|first=Jose M.|title=Comment on Article by Gelman|journal=Bayesian Analysis|year=2008|volume=3|issue=3|page=453|doi=10.1214/08-BA318REJ|ref=harv}} 
* {{cite journal|last=Birnbaum|first= A.|title=On the foundations of statistical inference|journal=J. Amer. Statist. Ass|volume=57|pages=269–326|year=1962|ref=harv}}
* {{cite book | editor1-last = Bandyopadhyay | editor1-first = Prasanta | others = Malcolm Forster | title = Philosophy of statistics |series= Handbook of the Philosophy of Science|volume= 7 | publisher = North-Holland | location = Oxford | year = 2011 | isbn = 978-0444518620 }}  The text is a collection of essays.
* {{cite journal|last=Breiman|first=Leo |title=Statistical Modeling: The Two Cultures|journal=Statistical Science|volume=16|issue=3|pages=199–231|year=2001|ref=harv}}
* {{cite web | last = Chin | first = Wynne W.| title = Structural Equation Modeling in IS Research - Understanding the LISREL and PLS perspective | url = http://disc-nt.cba.uh.edu/chin/ais/|year=?|ref=harv}}  University of Houston lecture notes?  
* {{cite web | last = Cox | first = D. R. | title = Frequentist and Bayesian Statistics: a Critique | url = http://www.physics.ox.ac.uk/phystat05/proceedings/default.htm |year=2005|ref=harv}} Proceedings of the Statistical Problems in Particle Physics, Astrophysics and Cosmology
* {{cite book |last1= de Finetti |first1= Bruno |authorlink1= Bruno de Finetti |editor1-first= H. E. |editor1-last= Kyburg  |others= H. E. Smokler |title= Studies in Subjective Probability |year= 1964 |publisher= Wiley |location= New York |pages= 93–158 |chapter= Foresight: its Logical laws, its Subjective Sources|ref=harv}} Translation of the 1937 French original with later notes added.
* {{cite web | last = Edwards | first = A.W.F. | title = Likelihood | url = http://www.cimat.mx/reportes/enlinea/D-99-10.html |year=1999|ref=harv}}  Preliminary version of an article for the International Encyclopedia of the Social and Behavioral Sciences.
* {{cite journal|last= Efron|first=Bradley|title=A 250-Year Argument: Belief, Behavior, and the Bootstrap|journal=Bulletin (new series) of the
American Mathematical Society|year=2013|volume=50|issue=1|pages= 129–146|doi=|ref=harv}} 
* {{cite journal|last=Efron|first=Bradley|title=Controversies in the foundations of statistics|journal=The American Mathematical Monthly|year=1978|volume=85|issue=4|pages=231–246 |doi=10.2307/2321163| url = http://mathdl.maa.org/images/upload_library/22/Ford/BradleyEfron.pdf |ref=harv}}
* {{cite journal|last=Fienberg|first=Stephen E.|title=|journal=Bayesian Analysis|year=2006|volume=1|issue=1|pages=1–40|ref=harv}}
* {{cite book | last = Fisher | first =  R. A. | title = Statistical Methods for Research Workers | publisher =  Oliver and Boyd | location = Edinburgh | year =  1925 |ref=harv}}
* {{cite book | last = Fisher | first = Sir Ronald A. | title = Design of Experiments | publisher =  Oliver and Boyd | location = Edinburgh | year =  1935 |ref=harv}}
* {{cite journal|last=Fisher|first=R|title=Statistical Methods and Scientific Induction|journal=Journal of the Royal Statistical Society, Series B|year=1955 |volume=17|issue=1|pages=69–78|url=http://www.phil.vt.edu/dmayo/PhilStatistics/Triad/Fisher%201955.pdf|ref=harv}}
* {{cite book | last = Fisher | first = Sir Ronald A. | title = The logic of scientific inference | publisher =  Oliver and Boyd | location = Edinburgh | year =  1956 |ref=harv}}
* {{cite journal|last1=Forster|first1=Malcolm|last2=Sober|first2=Elliott|title=How to Tell when Simpler, More Unified, or Less Ad Hoc Theories will Provide More Accurate Predictions|journal=British Journal for the Philosophy of Science|issue=45|pages=1–36|year=1994|ref=harv}} 
* {{cite journal|last1=Forster|first1=Malcolm|last2=Sober|first2=Elliott|title=Why likelihood|journal=Likelihood and evidence|pages=89–99|year=2001|ref=harv}}
* {{cite journal|last=Freedman|first=David|title=Some issues in the foundation of statistics|journal=Foundations of Science|pages=19–39|volume=1|year=1995/6|ref=harv}}
* {{cite journal|last1=Gelman|first1=Andrew|title=Rejoiner|journal=Bayesian Analysis|year=2008|volume=3|issue=3|pages=467–478|doi=10.1214/08-BA318REJ|ref=harv}} A joke escalated into a serious discussion of Bayesian problems by 5 authors (Gelman, Bernardo, Kadane, Senn, Wasserman) on pages 445-478.
* {{cite journal|last1=Gelman|first1=Andrew|last2=Shalizi|first2=Cosma Rohilla|title=Philosophy and the practice of Bayesian statistics|journal=British Journal of Mathematical and Statistical Psychology|year=2012|doi=10.1111/j.2044-8317.2011.02037.x|ref=harv}}
* {{cite book|title=The Empire of Chance: How Probability Changed Science and Everyday Life|last=Gigerenzer|first=Gerd|coauthors=Zeno Swijtink, Theodore Porter, Lorraine Daston, John Beatty, Lorenz Kruger|year=1989|publisher=Cambridge University Press|chapter=Part 3: The Inference Experts|isbn=978-0-521-39838-1|pages=70–122|ref=harv}}
* {{cite journal|last=Halpin|first=P F|title=Inductive Inference or Inductive Behavior: Fisher and Neyman: Pearson Approaches to Statistical Testing in Psychological Research (1940–1960)|journal=The American Journal of Psychology|date=Winter 2006 |volume=119|issue=4|pages=625–653|jstor=20445367|doi=10.2307/20445367|pmid=17286092|last2=Stam|first2=HJ|ref=harv}}
* {{cite web | last1 = Hubbard | first1 = Raymond | last2 = Bayarri | first2 = M. J. | title = P Values are not Error Probabilities
 | url = http://ftp.isds.duke.edu/WorkingPapers/03-26.pdf |year=2003?|ref=harv}}  A working paper that explains the difference between Fisher's evidential p-value and the Neyman–Pearson Type I error rate <math>\alpha</math>.
* {{cite book | last = Jeffreys | first = H. | title =  The theory of probability | publisher = Oxford University Press | year = 1939 |ref=harv}}
* {{cite web | last = Kass | first = | title = Why is it that Bayes’ rule has not only captured the attention of so many people but inspired a religious devotion and contentiousness, repeatedly across many years? | url = http://www.stat.cmu.edu/~kass/papers/about-bayes-rule.pdf |year=2012?|ref=harv}}
* {{cite journal|last=Lehmann|first=E. L.|title=The Fisher, Neyman-Pearson Theories of Testing Hypotheses: One Theory or Two?|journal=Journal of the American Statistical Association|volume=88|issue=424|pages=1242–1249|date=December 1993|ref=harv}}
* {{cite book | last = Lehmann | first = E. L. | title = Fisher, Neyman, and the creation of classical statistics | publisher = Springer | location = New York | year = 2011 | isbn = 978-1441994998 |ref=harv}}
* {{cite book|title=Testing Statistical Hypotheses|edition=3E|isbn=0-387-98864-5|last1=Lehmann|first1=E.L.|first2=Joseph P.|last2=Romano|year=2005|publisher=Springer|location=New York|ref=harv}}
* {{cite journal|last=Lenhard|first=Johannes|title=Models and Statistical Inference: The Controversy between Fisher and Neyman–Pearson|journal=Brit. J. Phil. Sci.|volume=57|pages=69–91|year=2006|ref=harv}}
* {{cite journal | last = Little | first = Roderick J. | title = Calibrated Bayes: A Bayes/Frequentist Roadmap |volume=60 |issue=3|year= 2006|ref=harv}}
*{{Cite book |last=Lindley |first=D.V.| author-link=Dennis Lindley |year=2000| title=The philosophy of statistics| journal=[[Journal of the Royal Statistical Society, Series D]]| volume=49| pages=293–337|doi=10.1111/1467-9884.00238|issue=3}}
* {{cite web | last = Louçã | first = Francisco | title = Should The Widest Cleft in Statistics-How and Why Fisher opposed Neyman and Pearson
 | url = http://www.repository.utl.pt/bitstream/10400.5/2327/1/wp022008.pdf |year=2008|ref=harv}}  Working paper contains numerous quotations from the original sources of the dispute.
* {{cite journal|last=Mayo|first=Deborah G.|title=Discussion: Bayesian Methods: Applied? Yes. Philosophical Defense? In Flux|journal=The American Statistician|date=February 2013|volume=67|issue=1|pages=11–15|doi=10.1080/00031305.2012.752410|ref=harv}}
* {{cite journal|last=Neyman|first=J|title=On the Problem of the most Efficient Tests of Statistical Hypotheses|journal=Phil. Trans. R. Soc. Lond. A|date=January 1, 1933|volume=231|issue=694–706|pages=289–337|doi=10.1098/rsta.1933.0009|last2=Pearson|first2=E. S.|ref=harv}}
* {{cite book |last=Neyman|first=J|title=Joint statistical papers of J.Neyman and E.S.Pearson|year=1967|publisher=Cambridge University Press|ref=harv}}
* {{cite journal|last=Neyman|first=Jerzy|title=Note on an Article by Sir Ronald Fisher|journal=[[Journal of the Royal Statistical Society, Series B]]|year=1956|volume=18|issue=2|pages=288–294|ref=harv}}
* {{cite book | last = Royall | first = Richard | title = Statistical evidence : a likelihood paradigm | publisher = Chapman & Hall | location = London New York | year = 1997 | isbn = 978-0412044113 }}
* {{Cite book |last = Savage  |first = L.J.  |authorlink = Leonard Jimmie Savage |year = 1972  |title = Foundations of Statistics (second edition) |ref=harv}}
* {{cite journal|last=Senn|first=Stephen|title=You May Believe You Are a Bayesian But You Are Probably Wrong|journal=RMM|year=2011|volume=2|pages=48–66|ref=harv}}
* {{cite journal|last1=Sotos|first1=Ana Elisa Castro|last2=Vanhoof|first2=Stijn|last3=Noortgate|first3=Wim Van den|last4=Onghena|first4=Patrick|title=Students' Misconceptions of Statistical Inference: A Review of the Empirical Evidence from Research on Statistics Education|journal=Educational Research Review|volume=2|pages=98–113|year=2007|ref=harv}}
* {{cite book |last1=Tabachnick|first1=Barbara G. |last2=Fidell|first2=Linda S. |title=Using Multivariate Statistics |edition=3rd |year=1996 |isbn=0-673-99414-7}} "Principal components is an empirical approach while factor analysis and structural equation modeling tend to be theoretical approaches."  p 27
* {{cite web |last=Yu |first=Yue |title=Bayesian vs. Frequentist |url=http://imyy.net/research/BSTT566__Slides.pdf |format=pdf |year=2009 |ref=harv}} Lecture notes? University of Illinois at Chicago

==Further reading==
*{{Cite book |last=Barnett | first=Vic | year=1999 | title=Comparative Statistical Inference | edition= 3rd| publisher=Wiley | isbn=978-0-471-97643-1}}

*{{Cite book |last=Cox | first = David R. | author-link=David Cox (statistician)| title=Principles of Statistical Inference | publisher=[[Cambridge University Press]] | year=2006|isbn=978-0-521-68567-2 }}
*{{Cite book |last= Efron | first=Bradley | author-link=Bradley Efron | title=Why Isn't Everyone a Bayesian? (with discussion)| journal= [[The American Statistician]] | volume= 40 | issue= 1 | year=1986 | pages= 1–11 | doi= 10.2307/2683105 |  jstor=2683105}}
*{{Cite book |last=Good | first=I. J. | author-link=I. J. Good | title=The Interface Between Statistics and Philosophy of Science | journal=Statistical Science | volume= 3| issue=4 | year=1988 |pages=386–397 | doi=10.1214/ss/1177012754 | jstor=2245388}}
*[[Joseph Born Kadane|Kadane J.B.]], Schervish M.J., Seidenfeld T. (1999), ''Rethinking the Foundations of Statistics'' ([[Cambridge University Press]]).  [Bayesian.]
*{{Cite book |last=Mayo |first=Deborah G.| title=Did Pearson reject the Neyman-Pearson philosophy of statistics? | journal=Synthese |volume=90 |pages=233–262 |year= 1992 | doi=10.1007/BF00485352| issue=2}}

==External links==
*[http://scholar.google.co.uk/scholar?cites=9531312933296806388 Citations of Savage (1972)] at [[Google Scholar]]. [Over 10000 citations.]
*Stanford Encyclopedia of Philosophy [http://plato.stanford.edu/entries/probability-interpret/ entry] on probability interpretations.

[[Category:Philosophy of science]]
[[Category:Statistical inference]]
[[Category:Statistics]]