loss function

Risk is the potential harm that may arise from some present process or from some future event. In everyday usage, "risk" is often used synonymously with "probability", but in professional risk assessments, risk combines the probability of a negative event occurring with how harmful that event would be. Thus in many engineering applications (see 2.1 below) Risk = probability of an accident/'event' (eg events per year) times its consequence (eg lost money, ... or deaths, per event). Risk - Formal ...

Including:

• Risk - Formal Definitions
• Risk - Background
• Risk - Risk in business
• Risk - Risk-sensitive industries
• Risk - Risk in finance
• Risk - Psychology of risk
• Risk - Regret
• Risk - Framing
• Risk - Fear as intuitive risk assessment?
• Risk - Two widely used antidotes for high risk

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In statistics, the use of Bayes factors is a Bayesian alternative to classical hypothesis testing. Given a model selection problem in which we have to choose between two models M1 and M2, on the basis of a data vector x. The Bayes factor K is given by This is similar to a likelihood-ratio test, but instead of maximising the likelihood Bayesians average it over the parameters. Generally, the models M1 and M2 will be parametrised by vectors of parameters θ1 and θ2Including:

• Bayes factor - Example

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Bayesian inference is a statistical inference in which probabilities are interpreted not as frequencies or proportions or the like, but rather as degrees of belief. The name comes from the frequent use of Bayes' theorem in this discipline. Bayes' theorem is named after the Reverend Thomas Bayes. However, it is not clear that Bayes would endorse the very broad interpretation of probability now called "Bayesian". Bayesian inference - Evidence and the scientific method. Bayesian statisticians claim that ...

Including:

• Bayesian inference - Evidence and the scientific method
• Bayesian inference - Simple examples of Bayesian inference
• Bayesian inference - From which bowl is the cookie?
• Bayesian inference - False positives in a medical test
• Bayesian inference - In the courtroom
• Bayesian inference - Search theory
• Bayesian inference - More mathematical examples
• Bayesian inference - Naive Bayes classifier
• Bayesian inference - Posterior distribution of the binomial parameter
• Bayesian inference - Computer applications
• Bayesian inference - Bayesian point estimation

Read more here: » Bayesian inference: Encyclopedia - Bayesian inference

The goal of supervised learning of a global model is to find a function g, given a set of points of the form (x, g(x)). It is assumed that the set of points for which the behavior of g is known is an i.i.d. sample drawn according to an unknown probability distribution p of a larger, possibly infinite, population. Furthermore, one assumes the existence of ...

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Supervised learning, Supervised learning - Empirical risk minimization, Supervised learning - Approaches and algorithms, Supervised learning - Applications, Supervised learning - General issues

Read more here: » Supervised learning: Encyclopedia II - Supervised learning - Empirical risk minimization

Risk is often mapped to the probability of some event which is seen as undesirable. Usually the probability of that event and some assessment of its expected harm must be combined into a believable scenario (an outcome) which combines the set of risk, regret and reward probabilities into an expected value for that outcome. Thus in statistical decision theory, the risk function of an estimator δ(x) for a parameter θ, calculated from some observables x; is defined as the expectation value of the l ...

See also:

Risk, Risk - Formal Definitions, Risk - Background, Risk - Risk vs. Uncertainty, Risk - Risk in business, Risk - Risk-sensitive industries, Risk - Risk in finance, Risk - Psychology of risk, Risk - Regret, Risk - Framing, Risk - Fear as intuitive risk assessment?, Risk - Two widely used antidotes for high risk

Read more here: » Risk: Encyclopedia II - Risk - Formal Definitions

Main articles: decision theory, prospect theory Risk - Regret. Main article: regret theory In decision theory, regret (and anticipation of regret) can play a significant part in decision-making, distinct from risk aversion (preferring the status quo in case one gets worse off). Risk - Framing. Framing is a fundamental problem with all forms of risk assessment. In particular, because of bounded rationality (our brains get overloaded, so we take mental ...

See also:

Risk, Risk - Formal Definitions, Risk - Background, Risk - Risk in business, Risk - Risk-sensitive industries, Risk - Risk in finance, Risk - Psychology of risk, Risk - Regret, Risk - Framing, Risk - Fear as intuitive risk assessment?, Risk - Two widely used antidotes for high risk

Read more here: » Risk: Encyclopedia II - Risk - Psychology of risk

A minimax algorithm is a recursive algorithm for choosing the next move in a two-player game. A value is associated with each position or state of the game. This value is computed by means of a position evaluation function and it indicates how good it would be for a player to reach that position. The player then makes the move that maximises the minimum value of the position resulting from the opponent's possible following moves. If it is A' ...

See also:

Minimax, Minimax - Minimax criterion in statistical decision theory, Minimax - Minimax algorithm with alternate moves, Minimax - Minimax theorem with simultaneous moves, Minimax - Minimax in the face of uncertainty, Minimax - Minimax in non-zero-sum games, Minimax - External link

Read more here: » Minimax: Encyclopedia II - Minimax - Minimax algorithm with alternate moves

Bayesian inference - From which bowl is the cookie?. To illustrate, suppose there are two bowls full of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns o ...

See also:

Bayesian inference, Bayesian inference - Evidence and the scientific method, Bayesian inference - Simple examples of Bayesian inference, Bayesian inference - From which bowl is the cookie?, Bayesian inference - False positives in a medical test, Bayesian inference - In the courtroom, Bayesian inference - Search theory, Bayesian inference - More mathematical examples, Bayesian inference - Naive Bayes classifier, Bayesian inference - Posterior distribution of the binomial parameter, Bayesian inference - Computer applications, Bayesian inference - Bayesian point estimation

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Suppose we have a random variable which produces either a success or a failure. We want to consider a model M1 where the probability of success is q=½, and another model M2 where q is completely unknown and we take a prior distribution for q which is uniform on [0,1]. We take a sample of 200, and find 115 success and 85 failures. The likelihood is: So we have but The ratio is then 1.197..., which is "barely worth mentioning" even if it points ver ...

See also:

Bayes factor, Bayes factor - Example

Read more here: » Bayes factor: Encyclopedia II - Bayes factor - Example

Bayesian statisticians claim that methods of Bayesian inference are a formalisation of the scientific method involving collecting evidence that points towards or away from a given hypothesis. There can never be certainty, but as evidence accumulates, the degree of belief in a hypothesis changes; with enough evidence it will often become very high (almost 1) or very low (near 0). As an example, this reasoning might be The sun has risen and set for billions of years. The sun has set tonight. With ...

See also:

Bayesian inference, Bayesian inference - Evidence and the scientific method, Bayesian inference - Simple examples of Bayesian inference, Bayesian inference - From which bowl is the cookie?, Bayesian inference - False positives in a medical test, Bayesian inference - In the courtroom, Bayesian inference - Search theory, Bayesian inference - More mathematical examples, Bayesian inference - Naive Bayes classifier, Bayesian inference - Posterior distribution of the binomial parameter, Bayesian inference - Computer applications, Bayesian inference - Bayesian point estimation

Read more here: » Bayesian inference: Encyclopedia II - Bayesian inference - Evidence and the scientific method

Main articles: decision theory, prospect theory Risk - Regret. Main article: regret theory In decision theory, regret (and anticipation of regret) can play a significant part in decision-making, distinct from risk aversion (preferring the status quo in case one gets worse off). Risk - Framing. Framing is a fundamental problem with all forms of risk assessment. In particular, because of bounded rationality (our brains get overloaded, so we take mental ...

See also:

Risk, Risk - Formal Definitions, Risk - Background, Risk - Risk vs. Uncertainty, Risk - Risk in business, Risk - Risk-sensitive industries, Risk - Risk in finance, Risk - Psychology of risk, Risk - Regret, Risk - Framing, Risk - Fear as intuitive risk assessment?, Risk - Two widely used antidotes for high risk

Read more here: » Risk: Encyclopedia II - Risk - Psychology of risk

See also insurance industry Means of measuring and assessing risk vary widely across different professions--indeed, means of doing so may define different professions, e.g. a doctor manages medical risk, a civil engineer manages risk of structural failure, etc. A professional code of ethics is usually focused on risk assessment and mitigation (by the professional on behalf of client, public, society or life in general).

See also:

Risk, Risk - Formal Definitions, Risk - Background, Risk - Risk vs. Uncertainty, Risk - Risk in business, Risk - Risk-sensitive industries, Risk - Risk in finance, Risk - Psychology of risk, Risk - Regret, Risk - Framing, Risk - Fear as intuitive risk assessment?, Risk - Two widely used antidotes for high risk

Read more here: » Risk: Encyclopedia II - Risk - Risk in business

Bayesian inference - Naive Bayes classifier. See: naive Bayes classifier. Bayesian inference - Posterior distribution of the binomial parameter. In this example we consider the computation of the posterior distribution for the binomial parameter. This is the same problem considered by Bayes in Proposition 9 of his essay. We are given m observed successes and n observed failures in a binomial experiment. The experiment may be tossing a coin, drawing a ...

See also:

Bayesian inference, Bayesian inference - Evidence and the scientific method, Bayesian inference - Simple examples of Bayesian inference, Bayesian inference - From which bowl is the cookie?, Bayesian inference - False positives in a medical test, Bayesian inference - In the courtroom, Bayesian inference - Search theory, Bayesian inference - More mathematical examples, Bayesian inference - Naive Bayes classifier, Bayesian inference - Posterior distribution of the binomial parameter, Bayesian inference - Computer applications, Bayesian inference - Bayesian point estimation

Read more here: » Bayesian inference: Encyclopedia II - Bayesian inference - More mathematical examples

See also insurance industry Means of measuring and assessing risk vary widely across different professions--indeed, means of doing so may define different professions, e.g. a doctor manages medical risk, a civil engineer manages risk of structural failure, etc. A professional code of ethics is usually focused on risk assessment and mitigation (by the professional on behalf of client, public, society or life in general).

See also:

Risk, Risk - Formal Definitions, Risk - Background, Risk - Risk in business, Risk - Risk-sensitive industries, Risk - Risk in finance, Risk - Psychology of risk, Risk - Regret, Risk - Framing, Risk - Fear as intuitive risk assessment?, Risk - Two widely used antidotes for high risk

Read more here: » Risk: Encyclopedia II - Risk - Risk in business

The following example of a zero-sum game, where A and B make simultaneous moves, illustrates the minimax algorithm. If each player has three choices and the payoff matrix for A is: and B has the same payoff matrix with the signs reversed (i.e. if the choices are A1 and B1 then B pays 3 to A) then the simple minimax choice for A is A2 since the worst possible result is then having to pay 1, while the simple minimax choice for B is B2 since the worst possible result is then ...

See also:

Minimax, Minimax - Minimax criterion in statistical decision theory, Minimax - Minimax algorithm with alternate moves, Minimax - Minimax theorem with simultaneous moves, Minimax - Minimax in the face of uncertainty, Minimax - Minimax in non-zero-sum games, Minimax - External link

Read more here: » Minimax: Encyclopedia II - Minimax - Minimax theorem with simultaneous moves

Minimax theory has been extended to decisions where there is no other player, but where the consequences of decisions depend on unknown facts. For example, deciding to prospect for minerals entails a cost which will be wasted if the minerals are not present, but will bring major rewards if they are. One approach is to treat this as a game against Nature, and using a similar mindset as Murphy's law, take an approach which minimizes the maximum expected loss, using the same techniques as in the two-person zero-sum games. In addition, expectiminimax trees have been developed, for two ...

See also:

Minimax, Minimax - Minimax criterion in statistical decision theory, Minimax - Minimax algorithm with alternate moves, Minimax - Minimax theorem with simultaneous moves, Minimax - Minimax in the face of uncertainty, Minimax - Minimax in non-zero-sum games, Minimax - External link

Read more here: » Minimax: Encyclopedia II - Minimax - Minimax in the face of uncertainty

Risk is often mapped to the probability of some event which is seen as undesirable. Usually the probability of that event and some assessment of its expected harm must be combined into a believable scenario (an outcome) which combines the set of risk, regret and reward probabilities into an expected value for that outcome. Thus in statistical decision theory, the risk function of an estimator δ(x) for a parameter θ, calculated from some observables x; is defined as the expectation value of the l ...

See also:

Risk, Risk - Formal Definitions, Risk - Background, Risk - Risk in business, Risk - Risk-sensitive industries, Risk - Risk in finance, Risk - Psychology of risk, Risk - Regret, Risk - Framing, Risk - Fear as intuitive risk assessment?, Risk - Two widely used antidotes for high risk

Read more here: » Risk: Encyclopedia II - Risk - Formal Definitions

In classical statistical decision theory, we have an estimator δ that is used to estimate a parameter . We also assume a risk function R(θ,δ), usually specified as the integral of a loss function. In this framework, is called minimax if it satisfies . An alternative criterion in the decision theoretic framework is the Bayes estimator in the presence of a prior distribution Π. An esti ...

See also:

Minimax, Minimax - Minimax criterion in statistical decision theory, Minimax - Minimax algorithm with alternate moves, Minimax - Minimax theorem with simultaneous moves, Minimax - Minimax in the face of uncertainty, Minimax - Minimax in non-zero-sum games, Minimax - External link

Read more here: » Minimax: Encyclopedia II - Minimax - Minimax criterion in statistical decision theory