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Empirical risk minimization for a classification problem with a 0-1 loss function is known to be an NP-hard problem even for a relatively simple class of functions such as linear classifiers. [5] Nevertheless, it can be solved efficiently when the minimal empirical risk is zero, i.e., data is linearly separable .
The parameter space has just two elements and each point on the graph corresponds to the risk of a decision rule: the x-coordinate is the risk when the parameter is and the y-coordinate is the risk when the parameter is . In this decision problem, the minimax estimator lies on a line segment connecting two deterministic estimators.
Statistical risk is a quantification of a situation's risk using statistical methods.These methods can be used to estimate a probability distribution for the outcome of a specific variable, or at least one or more key parameters of that distribution, and from that estimated distribution a risk function can be used to obtain a single non-negative number representing a particular conception of ...
Sawilowsky [56] distinguishes between a simulation, a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality, a Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical ...
The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled).
For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to someone of a known age to be assessed more accurately by conditioning it relative to their age, rather than assuming that the person is typical of the population as a whole.
In a finite decision problem, the risk point of an admissible decision rule has either lower x-coordinates or y-coordinates than all other risk points or, more formally, it is the set of rules with risk points of the form (,) such that {(,):,} = (,). Thus the left side of the lower boundary of the risk set is the set of admissible decision rules.
The Bayes risk of ^ is defined as ((, ^)), where the expectation is taken over the probability distribution of : this defines the risk function as a function of ^. An estimator θ ^ {\displaystyle {\widehat {\theta }}} is said to be a Bayes estimator if it minimizes the Bayes risk among all estimators.