In mathematical statisticsthe Kullback—Leibler divergence also called relative entropy is a measure of how one probability distribution is different from a second, reference probability distribution. In contrast to variation of informationit is a distribution-wise asymmetric measure and thus does not qualify as a statistical metric of spread - it also does not satisfy the triangle inequality. In the simple case, a Kullback—Leibler divergence of 0 indicates that the two distributions in question are identical.
In simplified terms, it is a measure of surprise, with diverse applications such as applied statistics, fluid mechanicsneuroscience and machine learning. The Kullback—Leibler divergence was introduced by Solomon Kullback and Richard Leibler in as the directed divergence between two distributions; Kullback preferred the term discrimination information. Equivalently by the chain rulethis can be written as. Most formulas involving the Kullback—Leibler divergence hold regardless of the base of the logarithm.
Kullback  gives the following example Table 2. The Kullback—Leibler divergence is a special case of a broader class of statistical divergences called f -divergences as well as the class of Bregman divergences.
It is the only such divergence over probabilities that is a member of both classes. Although it is often intuited as a way of measuring the distance between probability distributionsthe Kullback—Leibler divergence is not a true metric. However, its infinitesimal form, specifically its Hessiangives a metric tensor known as the Fisher information metric. Arthur Hobson proved that the Kullback—Leibler divergence is the only measure of difference between probability distributions that satisfies some desired properties, which are the canonical extension to those appearing in a commonly used characterization of entropy.
Unfortunately it still isn't symmetric. There is a relation between the Kullback—Leibler divergence and the " rate function " in the theory of large deviations. The logarithm in the last term must be taken to base e since all terms apart from the last are base- e logarithms of expressions that are either factors of the density function or otherwise arise naturally.
The equation therefore gives a result measured in nats. A special case, and a common quantity in variational inferenceis the KL-divergence between a diagonal multivariate normal, and a standard normal distribution with zero mean and unit variance :. Even so, being a premetricit generates a topology on the space of probability distributions. Pinsker's inequality entails that. The Kullback—Leibler divergence is directly related to the Fisher information metric.
This can be made explicit as follows. Specifically, up to first order one has using the Einstein summation convention. More formally, as for any minimum, the first derivatives of the divergence vanish. Another information-theoretic metric is Variation of informationwhich is roughly a symmetrization of conditional entropy.
It is a metric on the set of partitions of a discrete probability space. Many of the other quantities of information theory can be interpreted as applications of the Kullback—Leibler divergence to specific cases.
The self-informationalso known as the information content of a signal, random variable, or event is defined as the negative logarithm of the probability of the given outcome occurring. When applied to a discrete random variablethe self-information can be represented as [ citation needed ]. The mutual information[ citation needed ].Cancer expression database
The Shannon entropy[ citation needed ].We recall and extend some sufficient conditions for the convex comparison of martingale measures in a one period setting, based on the elasticity of the pricing kernel. We show that the minimal entropy martingale measure MEMM and the Esscher martingale measure are comparable in the convex order, and which one is dominating depends on the sign of the risk premium on the underlying.
If it is positive, then the MEMM gives a lower price to each convex payoff. We show how the comparison result can be extended to the multiperiod i. The relationship between stochastic orders and option prices is well established in the financial literature, at least starting with the seminal paper [ 14 ].Iraq situation report: december 22, 2015
Convex ordering of martingale measures is underlying many papers related to the comparison of option prices under different models, such as [ 2310 ] in continuous time and [ 51719 ] in discrete time, where the same techniques are applied for computing option pricing bounds, which are provided by extremal martingale measures. The purpose of this note is to show that in a one period setting the minimal entropy martingale measure MEMM henceforth, introduced in [ 8 ] and the Esscher martingale measure in the original sense of [ 9 ] are always comparable in the convex order, and which one is dominating depends on the sign of the risk premium of the underlying; in the typical case of a positive risk premium the MEMM gives lower prices to each convex payoff in particular, it gives lower prices to each plain vanilla call and put.
This result can be easily extended to the multiperiod i. In Section 2 we briefly review and generalize some sufficient conditions for the convex ordering of martingale measures; in Section 3 we state our main result on the comparison of the Esscher and the MEMM measures; finally in Section 4 we show how the comparison can be generalized in the multiperiod i.
This criterion is quite useful in the applications since in the Black-Scholes case the pricing kernel has constant elasticity. In the following theorem we generalize this result and state several similar sufficient conditions for the convex ordering of densities:. Hence all pricing kernels have to cut in at least two points; each sufficient condition excludes that there are more than two cuts. The solutions of 6 and 7 may in general not exist; however we can state the following:.
Lemma 1. Proposition 1. Assuming that the solutions h 1 and h 2 of 6 and 7 exist, they are both strictly increasing as a function of the riskfree logreturn r. This enables us to state our main comparison result:.
Theorem 2. In this case the analytical expressions of the densities are identical to the one period case with Nh 1 and Nh 2 replacing h 1 and h 2and hence the two measures are ordered as in the thesis of Theorem 2.
If we try to remove the hypothesis of independent logreturns, the problem is that the representation 8 does not hold anymore; it is no more true that the density of the MEMM is a product of one period entropy minimizing densities.
In contrast, the multiperiod Esscher measure is always by construction defined as a product of one period Esscher densities. We can however state a positive general result that shows that if two densities are defined as products of one period densities, then convex ordering of the one period factors implies the convex ordering of product densities:.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. That is, what extra assumption do we need to make or is there another result that gives the below?Alex Eskin (The University of Chicago) Random Walks and Measure Rigidity I
Please give full details and steps. Then the expectation of this process is infinite at any nonzero time - just compute it explicitly using LOTUS.
Or, since you are using discrete time, you can use the random walk with iid normal 0,1 steps as the martingale, and the same convex function. By the way - you don't need the bound on L1 to be uniform in time. For example, you definitely want a simple random walk to be a martingale, but the L1 norms are increasing as time goes on. Not sure if that's what you intended or not. Sign up to join this community.
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Convex ordering of Esscher and minimal entropy martingale measures for discrete time models
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Find valuable angles through data analysis, not touts. Get instant access to hundreds of systems made by the community and the pros at Bet Labs. Building your systems is as easy as point and click. Follow on TwitterLike on FacebookWatch on YouTube. Morris May 6, 2017 The Kentucky Derby is set to kick off in mere hoursat 6:34 p. Eastern, to be preciseand will be all over in about two minutes. Other horse betting sites include TVG and Bovada. Bovada uses a Latvian domain and also takes bets on sports other than horseracing, which are less likely to be legal in the U.
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Fortune may receive compensation for some links to products and services on this website. The Hollywood Sign is a trademark and intellectual property of Hollywood Chamber of Commerce.All the reviews we have ever published have the available (free) bonuses listed too so before you pick your desired bookmaker, have a browse through them and see which one fits you best. Online bookmakers are advanced machines designed to make online betting quick and easy, so do not take them at face-value only.
They have a lot of different betting markets, tools, tips, strategies, apps and more for you to use and learn from. Keep your emotions out of it. This way you can be sure you are betting responsibly and with a clear head. Do not let the emotions of wanting your team to win help you place the wrong bet. Simply load up an Excel sheet and list your chosen bets, the odds, and any other information you can put in. Another good read would be our recently published round-up post where we asked 10 experts on their biggest win and how they managed to pull it off.
Thank you for reading our post. We hope these 10 betting expert tips and hacks have given you a clear idea of what it takes to be successful at sports betting. If you want to read more, be sure to visit our sports betting tips, strategies and reviews using the links provided. Does this mean you should set out to reading them all.
Bet On Sports You Know And Understand With the rise in popularity of online bookmakers, it has given punters worldwide the ability to bet on any sporting event, in any country happening at any time. As we recently shared in a previous post, with so much power and flexibility comes the risk of possibly over-reaching with your betting, and this is one of them: Know what you are betting on.
Only Bet While Sober We all know how fun and exciting it is to be at the local bar out and about and after a few drinks, you decide to place a bet to make the match on the telly a little bit more interesting. Use Promo Codes (Bonuses) To Your Advantage A huge benefit of having so many online bookmakers with an online presence is the many, many bonuses they offer to all new sign ups and VIP customers. Your Favourite Team Is Not The Best Team Another tip that is often overlooked is to not bet on your personal favourite team (if possible).
Thank You For Reading Thank you for reading our post. We look forward to hearing from you. ALPHA Latest In Sports Betting How To Bet On Super Bowl 2018 December 8, 2017 Our World Cup 2018 Predictions December 7, 2017 Bet On World Cup Outright Winner: Who Will Win WC 2018.
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PINNACLE - accept very large stakes. BET365: Simply the best bookie for livebetting. PINNACLE: BA's all time favourite bookmaker. Med andre ord betaler man pr. Jeg bryder mig ikke om ideen. Jeg mener ikke at det er en god afregningform. The guerilla plant How the world's oldest clove tree defied an empire Walking away Why Royal Ballet principal Sergei Polunin quit.
With campaigns that ran before, during, and after the tournament, we were able to utilise the tournament to achieve coverage and promote our clients internationally.Teams play only 16 regular season games. Compare that with Premier League football at 38, or Major League Baseball at 162. This means the handicap line (spread) is so important when placing bets. A lot of people will miss -2. Be sure to browse the NFL markets on Matchbook and check out the alternative point spreads.
Led by the irrepressible Tom Brady the Patriots are almost always a well backed Favourite. One trend I do believe in is backing big underdogs in divisional games, where the familiarity between the two teams helps coaches to game plan and nullify talent disparity.
Todd Furhman: Like any sport set aside money you can afford to lose. Spend some time familiarizing yourself with US geography and the teams before you take big swings in the market. I personally went through a three year learning curve when I started studying the EPL and the NFL, aside from no relegation, offers similar challenges to new bettors.
Nick Goff: Learn the Quarterbacks and gradually develop opinions on all 32 of them. I remember going an entire season betting on JaMarcus Russell when there was an ongoing discussion as to whether his throwing motion was the worst in the history of the game.
This is actually the greatest angle in the history of NFL betting.
Tony Dungy is an idiot. He usually tips up games on ESPN. The more he likes a team, the more you should bet the other way. While you should be very wary of any long-term trends in NFL betting there are some reliable signals you can follow to find some decent value bets and our panel have a few to get you going.
Although Nick Goff offers a word of caution. Brad Allen: LA Chargers QB Philip Rivers is one to back as a dog: His record against the spread as of October last year: Underdog: 41-24 (63.
Brad Allen loves to back the Chargers and their QB Philip Rivers as underdogs. The main point appears to be look to the skies, in more ways than one. Travel and time zone factors are also a huge consideration. For exampleif the 49ers are playing a 1pm game in the Eastern Time Zone, their bodies are on a 9am time clock and this is generally worth about 2pts to the home team. Brad Allen: I like the weather angles. Wind is a big one that is underrated by the market.
Per a 2014 study, wind speeds of 10 mph are estimated to reduce quarterback ratings by 1. According to Pinnacle research, in the 50 games in recent years when average wind speed was 20 mph or greater, the average total was 38.
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