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The principle of maximum entropy states that, subject to precisely stated prior data (such as a proposition that expresses testable information), the probability distribution which best represents the current state of knowledge is the one with largest entropy.

Another way of stating this: Take a precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. Of those, one with maximal information entropy is the proper distribution, according to this principle.

History

The principle was first expounded by E.T. Jaynes in two papers in 1957[1][2] where he emphasized a natural correspondence between statistical mechanics and information theory. In particular, Jaynes offered a new and very general rationale why the Gibbsian method of statistical mechanics works. He argued that the entropy of statistical mechanics and the information entropy of information theory are principally the same thing. Consequently, statistical mechanics should be seen just as a particular application of a general tool of logical inference and information theory.

Overview

In most practical cases, the stated prior data or testable information is given by a set of conserved quantities (average values of some moment functions), associated with the probability distribution in question. This is the way the maximum entropy principle is most often used in statistical thermodynamics. Another possibility is to prescribe some symmetries of the probability distribution. An equivalence between the conserved quantities and corresponding symmetry groups implies the same level of equivalence for both these two ways of specifying the testable information in the maximum entropy method.

The maximum entropy principle is also needed to guarantee the uniqueness and consistency of probability assignments obtained by different methods, statistical mechanics and logical inference in particular.

The maximum entropy principle makes explicit our freedom in using different forms of prior data. As a special case, a uniform prior probability density (Laplace's principle of indifference) may be adopted. Thus, the maximum entropy principle is not just an alternative to the methods of inference of classical statistics, but it is an important conceptual generalization of those methods.

In ordinary language, the principle of maximum entropy can be said to express a claim of epistemic modesty, or of maximum ignorance. The selected distribution is the one that makes the least claim to being informed beyond the stated prior data, that is to say the one that admits the most ignorance beyond the stated prior data.

Testable information

The principle of maximum entropy is useful explicitly only when applied to testable information. A piece of information is testable if it can be determined whether a given distribution is consistent with it. For example, the statements

The expectation of the variable x is 2.87

and

p2 + p3 > 0.6

are statements of testable information.

Given testable information, the maximum entropy procedure consists of seeking the probability distribution which maximizes information entropy, subject to the constraints of the information. This constrained optimization problem is typically solved using the method of Lagrange multipliers.

Entropy maximization with no testable information takes place under a single constraint: the sum of the probabilities must be one. Under this constraint, the maximum entropy discrete probability distribution is the uniform distribution,

pi=1nforalli{1,,n}.

The principle of maximum entropy can thus be seen as a generalization of the classical principle of indifference, also known as the principle of insufficient reason.

Applications

The principle of maximum entropy is commonly applied in two ways to inferential problems:

Prior probabilities

The principle of maximum entropy is often used to obtain prior probability distributions for Bayesian inference. Jaynes was a strong advocate of this approach, claiming the maximum entropy distribution represented the least informative distribution.[3] A large amount of literature is now dedicated to the elicitation of maximum entropy priors and links with channel coding.[4][5][6]

Maximum entropy models

Alternatively, the principle is often invoked for model specification: in this case the observed data itself is assumed to be the testable information. Such models are widely used in natural language processing. An example of such a model is logistic regression, which corresponds to the maximum entropy classifier for independent observations.

General solution for the maximum entropy distribution with linear constraints

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Discrete case

We have some testable information I about a quantity x taking values in {x1, x2,..., xn}. We express this information as m constraints on the expectations of the functions fk; that is, we require our probability distribution to satisfy

i=1nPr(xiI)fk(xi)=Fkk=1,,m.

Furthermore, the probabilities must sum to one, giving the constraint

i=1nPr(xiI)=1.

The probability distribution with maximum information entropy subject to these constraints is

Pr(xiI)=1Z(λ1,,λm)exp[λ1f1(xi)++λmfm(xi)].

It is sometimes called the Gibbs distribution. The normalization constant is determined by

Z(λ1,,λm)=i=1nexp[λ1f1(xi)++λmfm(xi)],

and is conventionally called the partition function. (Interestingly, the Pitman–Koopman theorem states that the necessary and sufficient condition for a sampling distribution to admit sufficient statistics of bounded dimension is that it have the general form of a maximum entropy distribution.)

The λk parameters are Lagrange multipliers whose particular values are determined by the constraints according to

Fk=λklogZ(λ1,,λm).

These m simultaneous equations do not generally possess a closed form solution, and are usually solved by numerical methods.

Continuous case

For continuous distributions, the simple definition of Shannon entropy ceases to be so useful (see differential entropy). Instead Edwin Jaynes (1963, 1968, 2003) gave the following formula, which is closely related to the relative entropy.

Hc=p(x)logp(x)m(x)dx

where m(x), which Jaynes called the "invariant measure", is proportional to the limiting density of discrete points. For now, we shall assume that it is known; we will discuss it further after the solution equations are given.

A closely related quantity, the relative entropy, is usually defined as the Kullback-Leibler divergence of m from p (although it is sometimes, confusingly, defined as the negative of this). The inference principle of minimizing this, due to Kullback, is known as the Principle of Minimum Discrimination Information.

We have some testable information I about a quantity x which takes values in some interval of the real numbers (all integrals below are over this interval). We express this information as m constraints on the expectations of the functions fk, i.e. we require our probability density function to satisfy

p(xI)fk(x)dx=Fkk=1,,m.

And of course, the probability density must integrate to one, giving the constraint

p(xI)dx=1.

The probability density function with maximum Hc subject to these constraints is

p(xI)=1Z(λ1,,λm)m(x)exp[λ1f1(x)++λmfm(x)]

with the partition function determined by

Z(λ1,,λm)=m(x)exp[λ1f1(x)++λmfm(x)]dx.

As in the discrete case, the values of the λk parameters are determined by the constraints according to

Fk=λklogZ(λ1,,λm).

The invariant measure function m(x) can be best understood by supposing that x is known to take values only in the bounded interval (a, b), and that no other information is given. Then the maximum entropy probability density function is

p(xI)=Am(x),a<x<b

where A is a normalization constant. The invariant measure function is actually the prior density function encoding 'lack of relevant information'. It cannot be determined by the principle of maximum entropy, and must be determined by some other logical method, such as the principle of transformation groups or marginalization theory.

Examples

For several examples of maximum entropy distributions, see the article on maximum entropy probability distributions.

Justifications for the principle of maximum entropy

Proponents of the principle of maximum entropy justify its use in assigning probabilities in several ways, including the following two arguments. These arguments take the use of Bayesian probability as given, and are thus subject to the same postulates.

Information entropy as a measure of 'uninformativeness'

Consider a discrete probability distribution among m mutually exclusive propositions. The most informative distribution would occur when one of the propositions was known to be true. In that case, the information entropy would be equal to zero. The least informative distribution would occur when there is no reason to favor any one of the propositions over the others. In that case, the only reasonable probability distribution would be uniform, and then the information entropy would be equal to its maximum possible value, log m. The information entropy can therefore be seen as a numerical measure which describes how uninformative a particular probability distribution is, ranging from zero (completely informative) to log m (completely uninformative).

By choosing to use the distribution with the maximum entropy allowed by our information, the argument goes, we are choosing the most uninformative distribution possible. To choose a distribution with lower entropy would be to assume information we do not possess; to choose one with a higher entropy would violate the constraints of the information we do possess. Thus the maximum entropy distribution is the only reasonable distribution.

The Wallis derivation

The following argument is the result of a suggestion made by Graham Wallis to E. T. Jaynes in 1962.[7] It is essentially the same mathematical argument used for the Maxwell-Boltzmann statistics in statistical mechanics, although the conceptual emphasis is quite different. It has the advantage of being strictly combinatorial in nature, making no reference to information entropy as a measure of 'uncertainty', 'uninformativeness', or any other imprecisely defined concept. The information entropy function is not assumed a priori, but rather is found in the course of the argument; and the argument leads naturally to the procedure of maximizing the information entropy, rather than treating it in some other way.

Suppose an individual wishes to make a probability assignment among m mutually exclusive propositions. She has some testable information, but is not sure how to go about including this information in her probability assessment. She therefore conceives of the following random experiment. She will distribute N quanta of probability (each worth 1/N) at random among the m possibilities. (One might imagine that she will throw N balls into m buckets while blindfolded. In order to be as fair as possible, each throw is to be independent of any other, and every bucket is to be the same size.) Once the experiment is done, she will check if the probability assignment thus obtained is consistent with her information. If not, she will reject it and try again. Otherwise, her assessment will be

pi=niN

where pi is the probability of the ith proposition, while ni is the number of quanta that were assigned to the ith proposition (if the individual in our experiment carries out the ball throwing experiment, then ni is the number of balls that ended up in bucket i).

Now, in order to reduce the 'graininess' of the probability assignment, it will be necessary to use quite a large number of quanta of probability. Rather than actually carry out, and possibly have to repeat, the rather long random experiment, the protagonist decides to simply calculate and use the most probable result. The probability of any particular result is the multinomial distribution,

Pr(p)=WmN

where

W=N!n1!n2!nm!

is sometimes known as the multiplicity of the outcome.

The most probable result is the one which maximizes the multiplicity W. Rather than maximizing W directly, the protagonist could equivalently maximize any monotonic increasing function of W. She decides to maximize

1NlogW=1NlogN!n1!n2!nm!=1NlogN!(Np1)!(Np2)!(Npm)!=1N(logN!i=1mlog((Npi)!)).

At this point, in order to simplify the expression, the protagonist takes the limit as N, i.e. as the probability levels go from grainy discrete values to smooth continuous values. Using Stirling's approximation, she finds

limN(1NlogW)=1N(NlogNi=1mNpilog(Npi))=logNi=1mpilog(Npi)=logNlogNi=1mpii=1mpilogpi=(1i=1mpi)logNi=1mpilogpi=i=1mpilogpi=H(p).

All that remains for the protagonist to do is to maximize entropy under the constraints of her testable information. She has found that the maximum entropy distribution is the most probable of all "fair" random distributions, in the limit as the probability levels go from discrete to continuous.

Compatibility with Bayes' theorem

Giffin et al. (2007) state that Bayes' theorem and the Principle of Maximum Entropy (MaxEnt) are completely compatible and can be seen as special cases of the Method of Maximum (relative) Entropy. They state that this method reproduces every aspect of orthodox Bayesian inference methods. In addition this new method opens the door to tackling problems that could not be addressed by either the MaxEnt or orthodox Bayesian methods individually. Moreover, recent contributions (Lazar 2003, and Schennach 2005) show that frequentist relative-entropy-based inference approaches (such as empirical likelihood and exponentially tilted empirical likelihood - see e.g. Owen 2001 and Kitamura 2006) can be combined with prior information to perform Bayesian posterior analysis.

Jaynes stated Bayes' theorem was a way to calculate a probability, while maximum entropy was a way to assign a prior probability distribution.[8]

It is however, possible in concept to solve for a posterior distribution directly from a stated prior distribution using the Principle of Minimum Cross Entropy (or the Principle of Maximum Entropy being a special case of using a uniform distribution as the given prior), independently of any Bayesian considerations by treating the problem formally as a constrained optimisation problem, the Entropy functional being the objective function. For the case of given average values as testable information (averaged over the sought after probability distribution), the sought after distribution is formally the Gibbs (or Boltzmann) distribution the parameters of which must be solved for in order to achieve minimum cross entropy and satisfy the given testable information.

See also

Notes

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References

  • 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  • Jaynes, E. T., 1986 (new version online 1996), 'Monkeys, kangaroos and N', in Maximum-Entropy and Bayesian Methods in Applied Statistics, J. H. Justice (ed.), Cambridge University Press, Cambridge, p. 26.
  • Bajkova, A. T., 1992, The generalization of maximum entropy method for reconstruction of complex functions. Astronomical and Astrophysical Transactions, V.1, issue 4, p. 313-320.
  • Giffin, A. and Caticha, A., 2007, Updating Probabilities with Data and Moments
  • Guiasu, S. and Shenitzer, A., 1985, 'The principle of maximum entropy', The Mathematical Intelligencer, 7(1), 42-48.
  • Harremoës P. and Topsøe F., 2001, Maximum Entropy Fundamentals, Entropy, 3(3), 191-226.
  • Kapur, J. N.; and Kesevan, H. K., 1992, Entropy optimization principles with applications, Boston: Academic Press. ISBN 0-12-397670-7
  • Kitamura, Y., 2006, Empirical Likelihood Methods in Econometrics: Theory and Practice, Cowles Foundation Discussion Papers 1569, Cowles Foundation, Yale University.
  • Lazar, N., 2003, "Bayesian Empirical Likelihood", Biometrika, 90, 319-326.
  • Owen, A. B., Empirical Likelihood, Chapman and Hall.
  • Schennach, S. M., 2005, "Bayesian Exponentially Tilted Empirical Likelihood", Biometrika, 92(1), 31-46.
  • Uffink, Jos, 1995, 'Can the Maximum Entropy Principle be explained as a consistency requirement?', Studies in History and Philosophy of Modern Physics 26B, 223-261.

Further reading

  • Template:Cite PMID Open access article containing pointers to various papers and software implementations of Maximum Entropy Model on the net.

External links

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    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  5. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  6. One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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