**TrueSkill** is a skill-based ranking system developed by Microsoft for use with video game matchmaking on Xbox Live. Unlike the popular Elo rating system, which was initially designed for chess, TrueSkill is designed to support games with more than two players.^{ [1] }^{ [2] } In 2018, Microsoft published details about an extended version of TrueSkill, named TrueSkill2. ^{ [3] }

A player's skill is represented as a normal distribution characterized by a mean value of (**mu**, representing perceived skill) and a variance of (**sigma**, representing how "unconfident" the system is in the player's value).^{ [1] }^{ [2] } As such can be interpreted as the probability that the player's "true" skill is .^{ [1] }^{ [2] }

On Xbox Live, players start with and ; always increases after a win and always decreases after a loss. The extent of actual updates depends on each player's and on how "surprising" the outcome is to the system. Unbalanced games, for example, result in either negligible updates when the favorite wins, or huge updates when the favorite loses surprisingly.

Factor graphs and expectation propagation via moment matching are used to compute the message passing equations which in turn compute the skills for the players.^{ [1] }^{ [2] }

Player ranks are displayed as the conservative estimate of their skill, . This is conservative, because the system is 99% sure that the player's skill is actually higher than what is displayed as their rank.

The system can be used with arbitrary scales, but Microsoft uses a scale from 0 to 50 for Xbox Live. Hence, players start with a rank of . This means that a new player's defeat results in a large sigma loss, which partially or completely compensates their mu loss. This explains why people may gain ranks from losses.

TrueSkill is patented,^{ [4] } and the name is trademarked,^{ [5] } so it is limited to Microsoft projects and commercial projects that obtain a license to use the algorithm.

A **normal distribution** is a probability distribution used to model phenomena that have a default behaviour and cumulative possible deviations from that behaviour. For instance, a proficient archer's arrows are expected to land around the bull's eye of the target; however, due to aggregating imperfections in the archer's technique, most arrows will miss the bull's eye by some distance. The average of this distance is known in archery as *accuracy*, while the amount of variation in the distances as *precision*. In the context of a normal distribution, accuracy and precision are referred to as the *mean* and the *standard deviation*, respectively. Thus, a narrow measure of an archer's proficiency can be expressed with two values: a mean and a standard deviation. In a normal distribution, these two values mean: there is a ~68% probability that an arrow will land within one standard deviation of the archer's average accuracy; a ~95% probability that an arrow will land within two standard deviations of the archer's average accuracy; ~99.7% within three; and so on, slowly increasing towards 100%.

In mathematical analysis and in probability theory, a **σ-algebra** on a set *X* is a collection Σ of subsets of *X*, is closed under complement, and is closed under countable unions and countable intersections. The pair is called a measurable space.

A **measure space** is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring and the method that is used for measuring. One important example of a measure space is a probability space.

In probability theory and statistics, the **multivariate normal distribution**, **multivariate Gaussian distribution**, or **joint normal distribution** is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be *k*-variate normally distributed if every linear combination of its *k* components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.

**Noether's theorem** or **Noether's first theorem** states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

The **Einstein–Hilbert action** in general relativity is the action that yields the Einstein field equations through the principle of least action. With the (− + + +) metric signature, the gravitational part of the action is given as

In Bayesian statistics, a **maximum a posteriori probability** (**MAP**) **estimate** is an estimate of an unknown quantity, that equals the mode of the posterior distribution. The MAP can be used to obtain a point estimate of an unobserved quantity on the basis of empirical data. It is closely related to the method of maximum likelihood (ML) estimation, but employs an augmented optimization objective which incorporates a prior distribution over the quantity one wants to estimate. MAP estimation can therefore be seen as a regularization of maximum likelihood estimation.

In measure theory, **Carathéodory's extension theorem** states that any pre-measure defined on a given ring *R* of subsets of a given set *Ω* can be extended to a measure on the σ-algebra generated by *R*, and this extension is unique if the pre-measure is σ-finite. Consequently, any pre-measure on a ring containing all intervals of real numbers can be extended to the Borel algebra of the set of real numbers. This is an extremely powerful result of measure theory, and leads, for example, to the Lebesgue measure.

In mathematics, a **π-system** on a set is a collection of certain subsets of such that

In mathematics, a positive measure *μ* defined on a *σ*-algebra Σ of subsets of a set *X* is called a finite measure if *μ*(*X*) is a finite real number, and a set *A* in Σ is of finite measure if *μ*(*A*) < ∞*.* The measure *μ* is called **σ-finite** if *X* is a countable union of measurable sets with finite measure. A set in a measure space is said to have ** σ-finite measure** if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite.

The **Glicko rating system** and **Glicko-2 rating system** are methods of assessing a player's strength in games of skill, such as chess and Go. It was invented by Mark Glickman as an improvement on the Elo rating system, and initially intended for the primary use as a chess rating system. Glickman's principal contribution to measurement is "ratings reliability", called RD, for *ratings deviation*.

In estimation theory and decision theory, a **Bayes estimator** or a **Bayes action** is an estimator or decision rule that minimizes the posterior expected value of a loss function. Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator within Bayesian statistics is maximum a posteriori estimation.

In statistics, **Bayesian linear regression** is an approach to linear regression in which the statistical analysis is undertaken within the context of Bayesian inference. When the regression model has errors that have a normal distribution, and if a particular form of prior distribution is assumed, explicit results are available for the posterior probability distributions of the model's parameters.

In statistics, the **bias** of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called **unbiased**. In statistics, "bias" is an **objective** property of an estimator. Bias can also be measured with respect to the median, rather than the mean, in which case one distinguishes *median*-unbiased from the usual *mean*-unbiasedness property. Bias is a distinct concept from consistency. Consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased; see bias versus consistency for more.

**Expectation propagation (EP)** is a technique in Bayesian machine learning.

In statistics, **identifiability** is a property which a model must satisfy for precise inference to be possible. A model is **identifiable** if it is theoretically possible to learn the true values of this model's underlying parameters after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameters must generate different probability distributions of the observable variables. Usually the model is identifiable only under certain technical restrictions, in which case the set of these requirements is called the **identification conditions**.

In probability theory, the **rectified Gaussian distribution** is a modification of the Gaussian distribution when its negative elements are reset to 0. It is essentially a mixture of a discrete distribution and a continuous distribution as a result of censoring.

In probability theory and statistics, the **normal-inverse-Wishart distribution** is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and covariance matrix.

**Lagrangian field theory** is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees of freedom. Lagrangian field theory applies to continua and fields, which have an infinite number of degrees of freedom.

In statistics and machine learning, **Gaussian process approximation** is a computational method that accelerates inference tasks in the context of a Gaussian process model, most commonly likelihood evaluation and prediction. Like approximations of other models, they can often be expressed as additional assumptions imposed on the model, which do not correspond to any actual feature, but which retain its key properties while simplifying calculations. Many of these approximation methods can be expressed in purely linear algebraic or functional analytic terms as matrix or function approximations. Others are purely algorithmic and cannot easily be rephrased as a modification of a statistical model.

- 1 2 3 4 Murphy, Kevin (2012).
*Machine Learning: A Probabilistic Perspective*. MIT Press. ISBN 978-0262018029. - 1 2 3 4 Herbrich, Ralf; Minka, Tom; Graepel, Thore (2007), Schölkopf, B.; Platt, J. C.; Hoffman, T. (eds.), "TrueSkill : A Bayesian Skill Rating System" (PDF),
*Advances in Neural Information Processing Systems 19*, MIT Press, pp. 569–576, retrieved 2018-10-11 - ↑ Minka, Tom; Cleven, Ryan; Zaykov, Yordan (2018-03-22). "TrueSkill 2: An improved Bayesian skill rating system".
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(help) - ↑ "United States Patent Application 20090227313: Determining Relative Skills of Players".
*USPTO*. Retrieved 2014-02-16. - ↑ "Trademark Electronic Search System (TESS)".
*tmsearch.uspto.gov*. Retrieved 2020-01-16.

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