As an alert reader pointed out, a major mathematical prize was awarded recently. Since
\n2002, the government of Norway has been awarding a prize modeled on the Nobel, but in
\nmathematics. The prize was originally suggested by Sophus Lie, he of the Lie group, back in
\n1897, when he heard that Nobel was setting up his awards, and was not including
\nmathematics. The prize is named after Niels Abel, the Norwegian mathematician who
\ndiscovered the class of functions that are now known as Abelian functions; the same person
\nthat Abelian groups are named after, etc.<\/p>\n
Anyway, this year, the Abel <\/p>\n In basic statistics, we generally talk about things like “normal distributions”. The idea is that most of the time, when we look at random processes, we’ll see a certain Much of the basic study of probability focuses on the common cases, reasoning about what we can predict about the most likely outcome. So in our coin flipping example, basic statistics would talk about how we’d compute the expected outcome – 500 heads in a series of 1000 flips; what we’d expect from the standard deviation, etc.<\/p>\n Large deviations theory asks a very different question. It asks: what about the unlikely<\/em> outcomes? Given information about probability distributions, what can we say about outcomes that are significantly different from the expectation? In particular, what can we say about how the probability of outcomes vary as they get increasingly distant from the expectation?<\/p>\n Why is this important? Well, the field was founded by a mathematician who worked for an So for an insurance company, they need to know not just what’s the expectation for the Large deviation theory provides a way of analyzing large quantities of data, and using them to predict not just what the most likely value of some statistic is, but also what the probability is of large deviations from that value. Doing that is hard<\/em>, and Professor Varadhan derived something called the Large Deviation Principle<\/em>, which provides a way of working out probabilities of these kinds of large deviations. And as I said – it’s incredibly hard. Varadhan’s work takes nonlinear analysis, partial differential equations, and about a half-dozen other incredibly difficult fields of math – Aside from the financial applications of Professor Varadhan’s work, it’s also been a As an alert reader pointed out, a major mathematical prize was awarded recently. Since 2002, the government of Norway has been awarding a prize modeled on the Nobel, but in mathematics. The prize was originally suggested by Sophus Lie, he of the Lie group, back in 1897, when he heard that Nobel was setting up […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[24],"tags":[],"class_list":["post-380","post","type-post","status-publish","format-standard","hentry","category-goodmath"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-68","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=380"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/380\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=380"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=380"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nprize was awarded to Srinivasa Varadhan<\/a>, an Indian mathematician who is currently a professor at the NYU Courant Institute<\/a>.
\nProfessor Varadhan’s specialty is probability theory – in particular, the theory of large
\ndeviations. In honor of Professor Varadhan’s award, I thought it would be interesting to
\nvery briefly<\/em> explain what the theory of large deviations is, and why it’s so
\nimportant that it justified the award of a million dollar prize.<\/p>\n
\npattern emerge. The traditional example of this is coin flipping: if we flip a fair coin enough times, we expect to see the ratio of head-flips to tail-flips get very close to 1:1. Not just that, but we also expect that if we repeatedly flip a coin 1000 times, and plot a histogram showing how many heads we flipped each time, we’ll get a set of results that form a bell curve around 500.<\/p>\n
\ninsurance company. In general, an insurance company’s premiums are set based on the
\nstatistics of past years. For example, a flood insurance company predicts how much money it
\nwill need to pay out based on how much it had to pay to cover flood damage over a range of
\npast years. But, as we all learned after hurricane Katrina, there are years that are
\ndramatically<\/em> different from the norm – where the company has to pay out a whole lot more money than they would in a normal year. <\/p>\n
\namount that they’ll probably have to pay out in a given year – but also, what’s the
\nlikelihood of the year being one of those bizarre outliers where the damage is an order of magnitude or two outside of the norm?<\/p>\n
\nrequires a lot more information, and lot more mathematical work than basic probability theory.<\/p>\n
\nand puts them all together to derive a comprehensive theory of large deviation probability. This is probably the single most important advance in the mathematics of probability since Bayes.<\/p>\n
\nmajor contributor to modern physics. The possible behaviors of groups of particles whose
\nindividual behaviors can be predicted probabilistically can be understood better using
\nlarge deviation probability. In particular, when we look at things like thermodynamics,
\nwhat we get is a description of probabilistic behaviors of individual particles; and what we want is a description of how a huge number of those particles will behave in aggregate. Professor Varadhan has collaborated with physicists in applying his theory of large deviations to derive results in thermodynamic physics.<\/p>\n","protected":false},"excerpt":{"rendered":"