Large deviations theory

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Large Deviations Theory concerns the remote tail of a probability distribution. For example, consider a sequence of independent tosses of a fair coin. The possible outcomes could be head or tail. Let us denote the possible outcome of the i-th trial by $X i$ , where we encode head as -1 and tail as 1. Now let $M N$ denote the mean value after $N$ trials, i.e

$M_N = \frac{1}{N}\Sigma_{i=1}^{N} X_i$

Then $M N$ lies between -1 and 1; but from the weak or strong law of large numbers (and also from our experience) we know that as N become very large (tends to infinity) $M N$ becomes increasingly likely to lie increasingly close to $0$ . Now let us make the preceding statement more precise : For a given value $x > 0$ what is the probability that $M N$ is greater than $x$ ? Let us denote this probability by $P (M N > x)$ . It is known that $P (M N > x) < e x p ( - x 2 N / 2)$ , and that in fact this upper bound is rather sharp, in a certain technical sense. In other words the probability $P (M N > x)$ is decaying exponentially rapidly as N grows large, at a rate depending on x. Roughly speaking, large deviation theory concerns itself with the exponential decay of the probability of certain kinds of extreme or "tail" events, as the number of observations grows arbitrarily large.

1 Basic statement
2 Brief History
3 Large Deviation and Entropy
4 Further reading
5 See also

[edit] Basic statement

In the above mentioned example of coin-tossing we tacitly assumed that each toss is an independent trial. And for each toss, the probability of getting head or tail is always the same. This makes the random number $X i$ Independent and Identically Distributed (I.I.D.). For I.I.D. variables whose common distribution satisfies a certain growth condition, large deviation theory states that the following limit exists:

$\lim_{N\to \infty} \frac{1}{N} \log P(M_N > x) = - I(x)$

The function $I (x)$ is called the "rate function" or "Cramer function" or sometime "entropy function". Roughly speaking, the existence of this limit is what establishes the above mentioned exponential decay and allows us to conclude that for large $N$ , $P (M N > x)$ takes the form:

$P(M_N >x) \approx \exp[-NI(x) ].$ (The inequality given in the first paragraph, as opposed to the asymptotic formula presented here, requires an additional argument.) Also $I (x)$ is a monotonically increasing convex function. This is the basic result of Large Deviations Theory.

If we know the probability distribution of $X i$ , an explicit expression for the rate function can be obtained. This is given by a Legendre transform

$I(x) = \sup_{\theta > 0} [\theta x - \lambda(\theta)]$

where the function $λ(θ)$ is called the "Cumulant Generating Function (CGF)", given by

$λ(θ) = log E [exp(θ X)]$

Here $E []$ denotes expectation value with respect to the probability distribution function of $X i$ and $X$ is any one of $X i$ s. If $X i$ follows a Gaussian distribution, the rate function becomes a parabola with its apex at the mean of the Gaussian distribution.

If the condition of Independent Identical Distribution is relaxed, particularly if the numbers $X i$ are not independent but nevertheless satisfies Markov Property, the basic large deviations result stated above can be generalized.

[edit] Brief History

Large deviations theory was perhaps discovered by Swedish mathematician Harald Cramér who applied it to model the insurance business. From the point of view of an insurance company, the earning is at a constant rate per month (the monthly premium) but the claims come randomly. For the company to be successful over a certain period of time (preferably many months), the total earning should exceed the total claim. Thus to estimate the premium you have to ask the following question : "What should we choose as the premium $q$ such that over $N$ months the total claim $C = Σ X i$ should be less than $N q$ ? " This is clearly the same question asked by the large deviations theory. Cramer gave a solution to this question for I.I.D. Gaussian random variables, where the rate function is expressed as a power series. The results we have quoted above were later obtained by H. Chernoff, among other people. A very incomplete list of mathematicians who have made important advances would include S.R.S. Varadhan, D. Ruelle and O.E. Landford.

Abel Prize 2007 - to S.R.S. Varadhan

From the Abel Commitee's citation:

"for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations."
"Over the last four decades, the theory of large deviations has become a cornerstone of modern probability, both pure and applied."

[edit] Large Deviation and Entropy

The rate function is related to the entropy is statistical mechanics. This can be heuristically seen in the following way. In statistical mechanics the entropy of a particular macro-state is related to the number of micro-states which corresponds to this macro-state. In our coin tossing example the mean value $M N$ could designate a particular macro-state. And the particular sequence of heads and tails which gives rise to a particular value of $M N$ constitutes a particular micro-state. Loosely speaking a macro-state having more number of micro-states giving rise to it, has higher entropy. And a state with higher entropy has more chance of being realised in actual experiments. The macro-state with mean value of half has the highest number micro-states giving rise to it and it is indeed the state with the highest entropy. And in most practical situation we shall indeed obtain this macro-state for large number of trials. The "rate function" on the other hand measures the probability of appearance of a particular macro-state. The smaller the rate function the higher is the chance of a macro-state appearing. In our coin-tossing the value of the "rate function" for mean value equal to half is zero. In this way one can see the "rate function" as the negative of the "entropy".