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.
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 His move effectively ended his collaboration with Pearson and their development of hypothesis testing. Textbooks provided a hybrid version of significance and hypothesis testing by 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,  while most of what is taught to under-graduates and used under the banner of hypothesis testing is from Fisher.
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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  or as the fundamentally flawed union of philosophically incompatible ideas. Hypothesis testing is 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.
Two different interpretations of probability based on objective evidence and subjective degrees of belief have long existed.
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Gauss and Laplace could have debated alternatives more than 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,  much of it in reaction to the Bayesian probability of the time which utilized the controversial 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" stabilized in the second half of the 20th century. 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 the philosophical. Two major contributors to frequentist classical methods were Fisher and Neyman. Neyman's views were rigorously frequentist. Three major contributors to 20th century Bayesian statistical philosophy, mathematics and methods were de Finetti ,  Jeffreys  and Savage. In , 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 three generations; The "authoritative" views of the early contributors are not all current. 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. A classical frequency distribution describes the probability of the data.
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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. 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. 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.
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.
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The likelihood principle has been a battleground. 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.
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There is active discussion about combining Bayesian and frequentist methods,   but reservations are expressed about the meaning of the results and reducing the diversity of approaches. Bayesians 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. The frequentist view is too rigid and limiting while the Bayesian view can be simultaneously objective and subjective, etc. Likelihood is a synonym for probability in common usage.
In statistics that is not true. 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.
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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 Fisher for more than 40 years although prior references to the concept exist and Fisher's support was half-hearted. 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.
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.
go here Inferential statistics is based on statistical 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. Some notations may feel more natural for physicists than mathematicians, as for instance the loose handling of changes of variables, e. Introducing unprincipled estimators like the unbiased estimator of the variance out of the blue is somewhat contradictory with a Bayesian perspective, which produces estimators from a decisional perspective and none of them unbiased.
Using the same symbol for the unknown parameter and its estimate see, e. It is however interesting to notice how the author insists on parameterisation and normalisation issues, but I cannot help pointing the book missing the dominating measure aspect of deriving a MAP estimate.
Pages and pages of printed R output are not necessarily very helpful, considering the codes are available on-line.