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Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
concerned with the theory of making decisions based on assigning
probabilities Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
to various factors and assigning numerical consequences to the outcome. There are three branches of decision theory: # Normative decision theory: Concerned with the identification of
optimal decisions Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
, where optimality is often determined by considering an ideal decision-maker who is able to calculate with perfect accuracy and is in some sense fully
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
. # Prescriptive decision theory: Concerned with describing observed behaviors through the use of
conceptual models Conceptual may refer to: Philosophy and Humanities *Concept *Conceptualism *Philosophical analysis (Conceptual analysis) *Theoretical definition (Conceptual definition) * Thinking about Consciousness (Conceptual dualism) *Pragmatism (Conceptual p ...
, under the assumption that those making the decisions are behaving under some consistent rules. # Descriptive decision theory: Analyzes how individuals actually make the decisions that they do. Decision theory is closely related to the field of
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
and is an interdisciplinary topic, studied by economists, mathematicians, data scientists, psychologists, biologists, political and other social scientists, philosophers and computer scientists. Empirical applications of this theory are usually done with the help of
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
and
econometric Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...
methods.


Normative and descriptive

Normative decision theory is concerned with identification of optimal decisions where optimality is often determined by considering an ideal decision maker who is able to calculate with perfect accuracy and is in some sense fully
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
. The practical application of this prescriptive approach (how people ''ought to'' make decisions) is called
decision analysis Decision analysis (DA) is the discipline comprising the philosophy, methodology, and professional practice necessary to address important decisions in a formal manner. Decision analysis includes many procedures, methods, and tools for identifyi ...
and is aimed at finding tools, methodologies, and software (
decision support system A decision support system (DSS) is an information system that supports business or organizational decision-making activities. DSSs serve the management, operations and planning levels of an organization (usually mid and higher management) and ...
s) to help people make better decisions. In contrast, descriptive decision theory is concerned with describing observed behaviors often under the assumption that those making decisions are behaving under some consistent rules. These rules may, for instance, have a procedural framework (e.g.
Amos Tversky Amos Nathan Tversky ( he, עמוס טברסקי; March 16, 1937 – June 2, 1996) was an Israeli cognitive and mathematical psychologist and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his ...
's elimination by aspects model) or an
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
atic framework (e.g. stochastic transitivity axioms), reconciling the Von Neumann-Morgenstern axioms with behavioral violations of the
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
hypothesis, or they may explicitly give a functional form for time-inconsistent
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
s (e.g. Laibson's quasi-hyperbolic discounting). Prescriptive decision theory is concerned with predictions about behavior that positive decision theory produces to allow for further tests of the kind of decision-making that occurs in practice. In recent decades, there has also been increasing interest in "behavioral decision theory", contributing to a re-evaluation of what useful decision-making requires.


Types of decisions


Choice under uncertainty

The area of choice under uncertainty represents the heart of decision theory. Known from the 17th century (
Blaise Pascal Blaise Pascal ( , , ; ; 19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, and Catholic writer. He was a child prodigy who was educated by his father, a tax collector in Rouen. Pascal's earliest ...
invoked it in his famous wager, which is contained in his ''
Pensées The ''Pensées'' ("Thoughts") is a collection of fragments written by the French 17th-century philosopher and mathematician Blaise Pascal. Pascal's religious conversion led him into a life of asceticism, and the ''Pensées'' was in many ways hi ...
'', published in 1670), the idea of
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
is that, when faced with a number of actions, each of which could give rise to more than one possible outcome with different probabilities, the rational procedure is to identify all possible outcomes, determine their values (positive or negative) and the probabilities that will result from each course of action, and multiply the two to give an "expected value", or the average expectation for an outcome; the action to be chosen should be the one that gives rise to the highest total expected value. In 1738,
Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mecha ...
published an influential paper entitled ''Exposition of a New Theory on the Measurement of Risk'', in which he uses the St. Petersburg paradox to show that expected value theory must be normatively wrong. He gives an example in which a Dutch merchant is trying to decide whether to insure a cargo being sent from Amsterdam to St Petersburg in winter. In his solution, he defines a
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
and computes
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
rather than expected financial value. In the 20th century, interest was reignited by Abraham Wald's 1939 paper pointing out that the two central procedures of sampling-distribution-based statistical-theory, namely
hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis. Hypothesis testing allows us to make probabilistic statements about population parameters. ...
and parameter estimation, are special cases of the general decision problem. Wald's paper renewed and synthesized many concepts of statistical theory, including
loss function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
s,
risk function In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "co ...
s,
admissible decision rule In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it (or at least sometimes better and never worse), in the precise sense of "better" defined ...
s, antecedent distributions, Bayesian procedures, and
minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When ...
procedures. The phrase "decision theory" itself was used in 1950 by E. L. Lehmann. The revival of
subjective probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification o ...
theory, from the work of Frank Ramsey,
Bruno de Finetti Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prévision: ...
,
Leonard Savage Leonard Jimmie Savage (born Leonard Ogashevitz; 20 November 1917 – 1 November 1971) was an American mathematician and statistician. Economist Milton Friedman said Savage was "one of the few people I have met whom I would unhesitatingly call a g ...
and others, extended the scope of expected utility theory to situations where subjective probabilities can be used. At the time, von Neumann and Morgenstern's theory of
expected utility The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on the ...
proved that expected utility maximization followed from basic postulates about rational behavior. The work of
Maurice Allais Maurice Félix Charles Allais (31 May 19119 October 2010) was a French physicist and economist, the 1988 winner of the Nobel Memorial Prize in Economic Sciences "for his pioneering contributions to the theory of markets and efficient utilization o ...
and
Daniel Ellsberg Daniel Ellsberg (born April 7, 1931) is an American political activist, and former United States military analyst. While employed by the RAND Corporation, Ellsberg precipitated a national political controversy in 1971 when he released the '' Pen ...
showed that human behavior has systematic and sometimes important departures from expected-utility maximization ( Allais paradox and
Ellsberg paradox In decision theory, the Ellsberg paradox (or Ellsberg's paradox) is a paradox in which people's decisions are inconsistent with subjective expected utility theory. Daniel Ellsberg popularized the paradox in his 1961 paper, “Risk, Ambiguity, and ...
). The
prospect theory Prospect theory is a theory of behavioral economics and behavioral finance that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics. Based ...
of
Daniel Kahneman Daniel Kahneman (; he, דניאל כהנמן; born March 5, 1934) is an Israeli-American psychologist and economist notable for his work on the psychology of judgment and decision-making, as well as behavioral economics, for which he was award ...
and
Amos Tversky Amos Nathan Tversky ( he, עמוס טברסקי; March 16, 1937 – June 2, 1996) was an Israeli cognitive and mathematical psychologist and a key figure in the discovery of systematic human cognitive bias and handling of risk. Much of his ...
renewed the empirical study of economic behavior with less emphasis on rationality presuppositions. It describes a way by which people make decisions when all of the outcomes carry a risk. Kahneman and Tversky found three regularities – in actual human decision-making, "losses loom larger than gains"; persons focus more on ''changes'' in their utility-states than they focus on absolute utilities; and the estimation of subjective probabilities is severely biased by
anchoring An anchor is a device, normally made of metal , used to secure a vessel to the bed of a body of water to prevent the craft from drifting due to wind or current. The word derives from Latin ''ancora'', which itself comes from the Greek � ...
.


Intertemporal choice

Intertemporal choice is concerned with the kind of choice where different actions lead to outcomes that are realised at different stages over time. It is also described as cost-benefit decision making since it involves the choices between rewards that vary according to magnitude and time of arrival. If someone received a windfall of several thousand dollars, they could spend it on an expensive holiday, giving them immediate pleasure, or they could invest it in a pension scheme, giving them an income at some time in the future. What is the optimal thing to do? The answer depends partly on factors such as the expected rates of interest and
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduct ...
, the person's
life expectancy Life expectancy is a statistical measure of the average time an organism is expected to live, based on the year of its birth, current age, and other demographic factors like sex. The most commonly used measure is life expectancy at birth ...
, and their confidence in the pensions industry. However even with all those factors taken into account, human behavior again deviates greatly from the predictions of prescriptive decision theory, leading to alternative models in which, for example, objective interest rates are replaced by subjective discount rates.


Interaction of decision makers

Some decisions are difficult because of the need to take into account how other people in the situation will respond to the decision that is taken. The analysis of such social decisions is more often treated under the label of
game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
, rather than decision theory, though it involves the same mathematical methods. From the standpoint of game theory, most of the problems treated in decision theory are one-player games (or the one player is viewed as playing against an impersonal background situation). In the emerging field of
socio-cognitive Socio-cognitive or sociocognitive has been used in academic literature with three different meanings: 1) it can indicate a branch of science, engineering or technology, such as ''socio-cognitive research'', or ''socio-cognitive interactions'', 2) ...
engineering, the research is especially focused on the different types of distributed decision-making in human organizations, in normal and abnormal/emergency/crisis situations.


Complex decisions

Other areas of decision theory are concerned with decisions that are difficult simply because of their complexity, or the complexity of the organization that has to make them. Individuals making decisions are limited in resources (i.e. time and intelligence) and are therefore boundedly rational; the issue is thus, more than the deviation between real and optimal behaviour, the difficulty of determining the optimal behaviour in the first place. One example is the model of economic growth and resource usage developed by the
Club of Rome The Club of Rome is a nonprofit, informal organization of intellectuals and business leaders whose goal is a critical discussion of pressing global issues. The Club of Rome was founded in 1968 at Accademia dei Lincei in Rome, Italy. It consists ...
to help politicians make real-life decisions in complex situations. Decisions are also affected by whether options are framed together or separately; this is known as the distinction bias.


Heuristics

Heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate ...
s in decision-making is the ability of making decisions based on unjustified or routine thinking. While quicker than step-by-step processing, heuristic thinking is also more likely to involve fallacies or inaccuracies. The main use for heuristics in our daily routines is to decrease the amount of evaluative thinking we perform when making simple decisions, making them instead based on unconscious rules and focusing on some aspects of the decision, while ignoring others. One example of a common and erroneous thought process that arises through heuristic thinking is the Gambler's Fallacy — believing that an isolated random event is affected by previous isolated random events. For example, if a fair coin is flipped to tails for a couple of turns, it still has the same probability (i.e., 0.5) of doing so in future turns, though intuitively it seems more likely for it to roll heads soon. This happens because, due to routine thinking, one disregards the probability and concentrates on the ratio of the outcomes, meaning that one expects that in the long run the ratio of flips should be half for each outcome. Another example is that decision-makers may be biased towards preferring moderate alternatives to extreme ones. The ''Compromise Effect'' operates under a mindset that the most moderate option carries the most benefit. In an incomplete information scenario, as in most daily decisions, the moderate option will look more appealing than either extreme, independent of the context, based only on the fact that it has characteristics that can be found at either extreme.


Alternatives

A highly controversial issue is whether one can replace the use of probability in decision theory with something else.


Probability theory

Advocates for the use of probability theory point to: * the work of
Richard Threlkeld Cox Richard Threlkeld Cox (August 5, 1898 – May 2, 1991) was a professor of physics at Johns Hopkins University, known for Cox's theorem relating to the foundations of probability.. Biography He was born in Portland, Oregon, the son of attorney Lew ...
for justification of the probability axioms, * the
Dutch book In gambling, a Dutch book or lock is a set of odds and bets, established by the bookmaker, that ensures that the bookmaker will profit—at the expense of the gamblers—regardless of the outcome of the event (a horse race, for example) on which ...
paradoxes of
Bruno de Finetti Bruno de Finetti (13 June 1906 – 20 July 1985) was an Italian probabilist statistician and actuary, noted for the "operational subjective" conception of probability. The classic exposition of his distinctive theory is the 1937 "La prévision: ...
as illustrative of the theoretical difficulties that can arise from departures from the probability axioms, and * the complete class theorems, which show that all
admissible decision rule In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it (or at least sometimes better and never worse), in the precise sense of "better" defined ...
s are equivalent to the Bayesian decision rule for some utility function and some
prior distribution In Bayesian statistical inference, a prior probability distribution, often simply called the prior, of an uncertain quantity is the probability distribution that would express one's beliefs about this quantity before some evidence is taken into ...
(or for the limit of a sequence of prior distributions). Thus, for every decision rule, either the rule may be reformulated as a
Bayesian Thomas Bayes (/beɪz/; c. 1701 – 1761) was an English statistician, philosopher, and Presbyterian minister. Bayesian () refers either to a range of concepts and approaches that relate to statistical methods based on Bayes' theorem, or a followe ...
procedure (or a limit of a sequence of such), or there is a rule that is sometimes better and never worse.


Alternatives to probability theory

The proponents of
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...
,
possibility theory Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecess ...
, quantum cognition, Dempster–Shafer theory, and
info-gap decision theory Info-gap decision theory seeks to optimize robustness to failure under severe uncertainty,Yakov Ben-Haim, ''Information-Gap Theory: Decisions Under Severe Uncertainty,'' Academic Press, London, 2001.Yakov Ben-Haim, ''Info-Gap Theory: Decisions Unde ...
maintain that probability is only one of many alternatives and point to many examples where non-standard alternatives have been implemented with apparent success; notably, probabilistic decision theory is sensitive to assumptions about the probabilities of various events, whereas non-probabilistic rules, such as
minimax Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When ...
, are
robust Robustness is the property of being strong and healthy in constitution. When it is transposed into a system, it refers to the ability of tolerating perturbations that might affect the system’s functional body. In the same line ''robustness'' ca ...
in that they do not make such assumptions.


Ludic fallacy

A general criticism of decision theory based on a fixed universe of possibilities is that it considers the "known unknowns", not the " unknown unknowns": it focuses on expected variations, not on unforeseen events, which some argue have outsized impact and must be considered – significant events may be "outside model". This line of argument, called the ludic fallacy, is that there are inevitable imperfections in modeling the real world by particular models, and that unquestioning reliance on models blinds one to their limits.


See also

*
Bayesian epistemology Bayesian epistemology is a formal approach to various topics in epistemology that has its roots in Thomas Bayes' work in the field of probability theory. One advantage of its formal method in contrast to traditional epistemology is that its conc ...
*
Bayesian statistics Bayesian statistics is a theory in the field of statistics based on the Bayesian interpretation of probability where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about the event, ...
* Causal decision theory *
Choice modelling Choice modelling attempts to model the decision process of an individual or segment via revealed preferences or stated preferences made in a particular context or contexts. Typically, it attempts to use discrete choices (A over B; B over A, B & C) i ...
*
Constraint satisfaction In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution through a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for th ...
*
Daniel Kahneman Daniel Kahneman (; he, דניאל כהנמן; born March 5, 1934) is an Israeli-American psychologist and economist notable for his work on the psychology of judgment and decision-making, as well as behavioral economics, for which he was award ...
*
Decision making In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either ra ...
* Decision quality *
Emotional choice theory Emotional choice theory (also referred to as the "logic of affect") is a social scientific action model to explain human decision-making. Its foundation was laid in Robin Markwica’s monograph ''Emotional Choices'' published by Oxford University ...
* Evidential decision theory *
Game theory Game theory is the study of mathematical models of strategic interactions among rational agents. Myerson, Roger B. (1991). ''Game Theory: Analysis of Conflict,'' Harvard University Press, p.&nbs1 Chapter-preview links, ppvii–xi It has appli ...
*
Multi-criteria decision making Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings s ...
*
Newcomb's paradox In philosophy and mathematics, Newcomb's paradox, also known as Newcomb's problem, is a thought experiment involving a game between two players, one of whom is able to predict the future. Newcomb's paradox was created by William Newcomb of the ...
*
Operations research Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve decis ...
* Optimal decision *
Preference (economics) In economics and other social sciences, preference is the order that an agent gives to alternatives based on their relative utility. A process which results in an "optimal choice" (whether real or theoretical). Preferences are evaluations and con ...
*
Prospect theory Prospect theory is a theory of behavioral economics and behavioral finance that was developed by Daniel Kahneman and Amos Tversky in 1979. The theory was cited in the decision to award Kahneman the 2002 Nobel Memorial Prize in Economics. Based ...
* Quantum cognition *
Rational choice theory Rational choice theory refers to a set of guidelines that help understand economic and social behaviour. The theory originated in the eighteenth century and can be traced back to political economist and philosopher, Adam Smith. The theory postula ...
*
Rationality Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an ab ...
*
Secretary problem The secretary problem demonstrates a scenario involving optimal stopping theory For French translation, secover storyin the July issue of ''Pour la Science'' (2009). that is studied extensively in the fields of applied probability, statistics, a ...
* Signal detection theory * Small-numbers game *
Stochastic dominance Stochastic dominance is a partial order between random variables. It is a form of stochastic ordering In probability theory and statistics, a stochastic order quantifies the concept of one random variable being "bigger" than another. These are us ...
* TOTREP *
Two envelopes problem The two envelopes problem, also known as the exchange paradox, is a paradox in probability theory. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. It is a variant of an older problem known ...


References


Further reading

* * (''an overview of the philosophical foundations of key mathematical axioms in subjective expected utility theory – mainly normative'') * * * * ''(covers normative decision theory)'' * * (translation of 1931 article) * : de Finetti, Bruno. "Foresight: its Logical Laws, Its Subjective Sources," (translation of th
1937 article
in French) in H. E. Kyburg and H. E. Smokler (eds), ''Studies in Subjective Probability,'' New York: Wiley, 1964. * de Finetti, Bruno. ''Theory of Probability'', (translation by AFM Smith of 1970 book) 2 volumes, New York: Wiley, 1974-5. * De Groot, Morris, ''Optimal Statistical Decisions''. Wiley Classics Library. 2004. (Originally published 1970.) . * ''(covers both normative and descriptive theory)'' * * Khemani, Karan
Ignorance is Bliss: A study on how and why humans depend on recognition heuristics in social relationships, the equity markets and the brand market-place, thereby making successful decisions
2005. * Klebanov, Lev. B., Svetlozat T. Rachev and Frank J. Fabozzi, eds. (2009). ''Non-Robust Models in Statistics'', New York: Nova Scientific Publishers, Inc. * A rational presentation of probabilistic analysis. * * * * Reprinted in Shafer & Pearl. ''(also about normative decision theory)'' * http://psychclassics.yorku.ca/Peirce/small-diffs.htm * * * * * Ramsey, Frank Plumpton; "Truth and Probability"
PDF
, Chapter VII in ''The Foundations of Mathematics and other Logical Essays'' (1931). * * * {{Authority control Statistical inference Risk Control theory Formal sciences Epistemology of science Mathematical and quantitative methods (economics)