HOME

TheInfoList



Click Here for Items Related To - Laplace Principle (large Deviations Theory)
OR:
Laplace Principle (large Deviations Theory) on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, Laplace's principle is a basic
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...
in
large deviations theory In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...
which is similar to
Varadhan's lemma In mathematics, Varadhan's lemma is a result from large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic ''φ''(''Z'ε'') of a family of random variables ''Z'� ...
. It gives an
asymptotic expression In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing Limit (mathematics), limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very larg ...
for the
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
of exp(−''θφ''(''x'')) over a fixed set ''A'' as ''θ'' becomes large. Such expressions can be used, for example, in statistical mechanics to determining the limiting behaviour of a system as the temperature tends to
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibrati ...
.


Statement of the result

Let ''A'' be a Lebesgue-measurable
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
of ''d''-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
al
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
R''d'' and let ''φ'' : R''d'' → R be a
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is i ...
with :\int_A e^ \,dx < \infty. Then :\lim_ \frac1 \log \int_A e^ \, dx = - \mathop_ \varphi(x), where ess inf denotes the
essential infimum In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory and functional analysis, where one often deals with statements that are not valid for ''al ...
. Heuristically, this may be read as saying that for large ''θ'', :\int_A e^ \, dx \approx \exp \left(-\theta \mathop_ \varphi(x) \right).


Application

The Laplace principle can be applied to the family of
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
s P''θ'' given by :\mathbf_\theta (A) = \left( \int_A e^ \, dx \right) \bigg/ \left( \int_ e^ \, dy \right) to give an asymptotic expression for the probability of some event ''A'' as ''θ'' becomes large. For example, if ''X'' is a standard
normally distributed In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu is ...
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
on R, then :\lim_ \varepsilon \log \mathbf \big \sqrt X \in A \big= - \mathop_ \frac for every measurable set ''A''.


See also

*
Laplace's method In mathematics, Laplace's method, named after Pierre-Simon Laplace, is a technique used to approximate integrals of the form :\int_a^b e^ \, dx, where f(x) is a twice- differentiable function, ''M'' is a large number, and the endpoints ''a'' ...


References

* Asymptotic analysis Large deviations theory Probability theorems Statistical mechanics Mathematical principles Theorems in analysis {{statisticalmechanics-stub