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On the Properties of Variational Approximations of Gibbs Posteriors

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2 and consultants 1st I want to thank the organisers to the opportunity to present this work actually I was the 1 to apologize to 2 or 3 people in the audience who already sold stoic exactly in the same room 2 months ago that injured 2 things actually because of when I presented this is case work for months ago there was only statisticians and don't so I made a lot of the ribald jokes about people in optimization during and after the two-day removed these jokes and on so to because that nesting nothing about a musician tourist instead I wanted to talk about it so I think the reason why and after actually to explain them that the reason for this work to be the reason why is that income from the different communities so come from the community of people working on aggregation of opportunity and in this community while there are many juridical results there are unsold people by using Monte Carlo methods and when I talk about the jury or the people I too and people know about optimization during they usually expect that a multiple to implement anything from us the middle them talking about and in some way to a strong but the object of this spoke with the proposed this talks to prove that actually some predictions that builds a navigation theory can be approximated using the binational commission so using a computer position during an even if I don't know a lot about you can imagine by yourself that actually you can use often agreeing to implement this of the church Phil I'm in 2 starts by a short introduction to aggregation during and then by communities off of violation petition soft In some ways of optimization theory for aggregation during and at 1st aggregation during and it would be a very very you know even introduction to British Teresa you might know or you're not the not buttons reacted to the slides at a level that I can understand myself and it's kind of challenge so I should not say that it is because the thought is being recorded so some of my students can come and get it in a few days so that will remove the 1st minutes for it's published online OK just the motivation for irrigation during a negative for or no injuries location so you have a sample and you want to know from the vigilant rights like the right to you what you want to do election national statistics you know examples have already assumed that these examples since 25 years on the way and so you know that you got something from these deficits but you don't want to write the likelihood
and so what you have what you usually have to do to choose you know look to deal with this problem and you have to choose a few ingredients that recurrent and old versions off supervise only you have to let you of observations 1st selected yeah we did with object levels problems so exit the object's weighs that but it would present visiting the batch alone sitting in the on-line sitting as well but In any case that we use of a to to the Senate patience and then you have to define a set of pretty told laying predictable still know about anything you want to guess what you this solution debates indexed by Palmer talked about expanding the finite or in finite eventual 6 to after golf course is meant to predict wide and then you have a return success so obviously I will define the actually notations little depending on whether we work on the back sitting or in the online sitting that but think of something like what is that 6 people tend to do things like that but that was more focus on prevention related container like for example the out of sample of prediction accuracy 7 so this would be for a classification problems is will do we have a more general problem specification regression later but you're going to be in mind so we want to protect what is a good production which would the doctor and finally we used in many cases an empirical proxy empirical approximation of ah which we will buy small R and for example you can think of the characteristics that you want to minimize of ants yeah so a lesbian all the ingredients that we need them permeable to talk about something being on injuries and
actually In aggregation of subordinated loans that and went to present right now we need 1 more ingredients it's away when acting in that you know that in some way need some assumptions on the set of pretty dull standard its complexity and not even in order to do somewhat condition but just another about 2 tool to relate in some ways this juridical reproductive of success and this thing that you cannot sell any military base this to these usually need an assumption on the sets of parameters and innovation will usually replaces the of a pretty your probability distribution of pirated space within itself it would be used in some way to to replace the complexity measure up like that and then mentioned the article states
look at end of all the bones I'm going to present to the stage of that didn't like that of history we have the bones on the operation production risk and it would be about a problem that by a bone which is a balance so you were probably used to a by S. values but trade-off in this case you have a kind of slightly different trade affected woman in the middle of a all the possible aggregation distribution and this time you have a balance between what this down which would be actually the best possible production and in some way he wouldn't want to take really a probability distribution just like they could do it must at the best possible parameter but on the other hand you have I like combines down and here it is actually the greenback indulgence between the aggregation solution and the PRI your work and so in some sense it will replace the complexity of visual that you have been like for example that the loans In the sense that what obviously others say before you want in a lot of games with them as small as possible to choose role as a solution that is very very spiked and concentrated around the best parameter but on the other hand when you concentrate rolled around a single parameter of provided us with respect for example to a uniform price your will explore and how fast when it explodes and it's related to the invention of the 62 it so he you will have the usual balance between prediction and up complexity so long as it is known is just like and in the unique thing about what went to the small wall but so was the small of long I mean the that in the common man and shown on the sample size actually precise balance but later sold just except that there will be additional guns but that's what you need to understand the moment his 2 terms again ends actually the Alderwood news is that 1 depending on the moment will hold for a large class of aggregation measure but actually very often of the 1 that so we want to this 1 because of some balls off and on the agenda for this probability distribution desegregation distribution and you can see that it looks underneath your family or related statistics unit it's really looks like a poster your focus so you the PRI Olmert applied by something that is wait small waited to the Department of make good prediction so if you see this as a kind of little likelihood then you can interpret that interpreted this as the Post said your OK but if you prefer you can see it as that kind of smooth version of empirical research immunization because obviously we give more weight to borrow it deals with a small empirical research but is clear the strongest growth in the yes it's not In a letter to admitted that there was no no sign of having go aggregation is your initial Department of Space actually sold and the thing is that depending on on the papers of 4 this year of history I would go navigation measure up any possible broke because I repeat this 1 by more convenient for later but depending on the paper retinue of many different names for this 1 invasion of the invasion statistics people where they don't really like that the because it's not like you would see the use it and go back to possible post in your mind it advisable to to get a legal battle against measure and finally with the exponential weight of arranging because the expenditure weights that you motives were also depends on paper a year later this nation there with the people the standards set out for him the other doctor into independent the result that use sometimes and then when the risk is collects all amenable to people with concessions will have to use was what I did things when it when he was a comics respectively which we knew we simply use Jensen's inequality here and actually try to lower bond and adjust our bemused by the risk of the aggregated standard or in this case Italy what you want to use to computer a post your expectations but it's all in the hands of when is not the case when it's not makes you can see that this is still abound on the procedure would be considered as a randomised procedure that will at its standards were given a new object X it will roll a a parameter opted out according to these confusion and then you protect you predict quite a spectacle X good and then this is inoperable and is run lies on from the lies procedure so independent universities ,comma Mexican religious 208 department or otherwise you have to use a kind of around the most prestigious but in both cases you have something practical and practical
effect you can do with this and this is what I'm going to talk about the result
would resent you actually do so more
accurate versions of these long demanding the context and the
first one activities along for baserunning so you have sample I did what I did the independence of the Baltic and removed to have some assumption in the dependence between the source and just for simplicity yeah the original
source of the virus that is on the the but and OK you
you act and I can provide menagerie later I can provide you evolve with the smaller part here but you in some sense into immediately delicate kind empirical along the 1 that you can compute on the you can know that you'll probably just is smaller than 0 . 0 cent in again here I'm more interested in this kind of iniquity where actually want to be sure that uneven you cannot to the only want to be sure that in some way would be very close to the minimize around this quantity but after both are really was the man of the exactly exactly comeback back later when I present proof which assume that you already guessed house I'm going to put the results of yet so and I
sampled from a probability distribution while the set of predator selected and anything for the moment and the risk that could be written off as the expectation of Perot's functioning and for the sake of simplicity involving it's not necessary in this kind can be generalized to wouldn't unbounded with functions but for the sake of simplicity I would presents 1 of the weakest of version of these results for abounded actually distinction OK and finally world and characteristic that will be used to define in this way and you still need the your privates for the moment I don't provide an explicit for an extension of the following results with the backbone to locate its lead with laughter liberty and say that the risk of this explosion to exponentially weighted aggregation procedure whether it's randomized or with with you knew of a comics slopes and used in Indiana people say Oh expectation that when it's a parabolic by this orchid stump I promised you look at the balance between the wood and small but for the juridical risk and then this complexity down and then the reminder that you have is well obviously a local fall over at Salem as it's with a lot of energy and then the this number of times of the squalor Over and OK we're actually reminder that this is the approval on the distinction this so extending
loans due to work when he gets in that sector is based on previous word by agents the and so that the mechanistic and so I just want to explain I mean obviously it looks like what I promise of on the produce light but on the other hand so maybe it's not very expensive so I just wanted to went to prison provide many examples littering the talk with just provides adjustment to provide 1 example but in recent bouquets were actually the product of sets sold to dump is a finite set and just to prove to check what happened in this case OK what's this stunt and establishment lacked so just
assume that you of a finite set of pretty dolls and then I'm going to do something quite simple to prioritize the uniforms tradition then obviously you have this moment just after just the same old and produce light boats if you want to I can't something too complicated you can replace actually the regional overall probability visuals but the influence over all the rock masses again and then obviously the integral off the risk in this case to the risk of of the Parliament to die and managing you can compute the crew that indulgence between direct must end the uniform prevented usual on the planet that think it's visible and then it was actually the result of the coverage of the sets and then you can see that well these long for depends on so particle number the warning that for some or all India should come back here this 1
here and did you don't know how to
use it no you of a way to
choose you can try to optimize the bones and you think that actually your aggregated up funds as well as the best pretty door press the staff square root also look em over and look without any other essential and weather and soaring optimal choice for the prime 1 that here without any assumption on the most intellectually cannot improve on this but if you don't have a reasonable assumption freedom when you use of the square the the square estimation sought to use the quantity close obviously you you you you can improve on this so they are refined version of this
bond Armenia wouldn't you up probably 156 different
national baseball some of them don't include more volatile and habits 77 particles and I want to London 77 but in the end you have bonds for example the product across with all the square root here which is once again the optimal rate again so it just needs 1 idea of it what you usually do when you have a prior
and instead of told you he was in Detroit will prolong this might just not taking the view most or probably judicial but after over a suitable sets off a probability distribution that you can actually deal with meaning that computations and you end up
with the predictions about which is but if you do the competition the calculations quite well you will usually end up with something that is not too far from being general OK
all the presents the same owns but indifferent sitting actually all nite long and so provides you the same edition but this time I don't have any assumption of duties on many of the top operatives later in this case is not very well fault because the XI where are unlikely to be run the levels deal with our own eyes sitting and any possible sequence X 1 white 1 and you of be generated by any algorithm is like a library them which knows actually which aggregation procedure you are going to use but anyway you have the same sitting a of particles in this case would focus on the regrets that his old news that at each step and when to use XT and the produce good observations to predict widened by the white to act and then I want to compare my committed loss to the Community later also really the the best placebo productive so I was of the same assumption that the loss his mother once again is something that you can remain with different assumptions bus looks I want to present once again what book the same possession of the ball so that you can compare with the produce 1 again and then at each step actually against a news you can still have a kind of proxy of the quantity of predict Dorval which is the empirical risk up to 10 2 minus-1 1 "quotation mark enterprise your but once again there will give a general direction solo essential
prior or so what they propose to do medically the same but that's the again so that DID find this exponentially weighted and OK so the price Olmert applied by exponential to minus 1 that ends the empirical research at stained and then I use as a predict or anything these days you can see that
you the complexity of the distinction sorted he added
into the runways posted your but that it could have a look at this case I use of the navigate print or working under this proposal tradition and many of
these results cannot be sure who was the 1st person to to to to ride this result you have a version of it in To an Indian was his book with discrete space boats Saturday followed a very clear explanation about this result many many balancing Sebastian deficient in this but I don't like it was the first one to ready for and of these voluntary so this time that the trade and lost this morning you still have the balance between well well the best possible aggregation but you have to keep once again these competitive canceled the distance to with respect to the PRI or so you see that in some way things with Of the almost the same I mean if you specify a price or by and the promise of 60 . then you can then choose the optimal Yukon extension the stamina to stand and then you can choose an optimal chronicle of muscle and investigation the same for example you define finite parental said so I want to do all the gate creations again OK so now vote remaining of the store would be about 12 unusual that you can
come to this work so I think that is
quite important I just got that this kind of thing can be important quite recently a and it's quite important that all that they have a few remarks of both these techniques to do although not to upset anyone in the audience 1
1st you have many many other version of the segregation Bonsall of everything was 1 of the national identity untypical for what the difference is that it's not exactly a prediction it's on estimation regression estimation with fixed line so you cannot really do something about certain production risk but on the other hand is very very convenient and because they have no 1 in this assumption so it's very very convenient so that was very convenient in many cities and that our although exactly the news In the paper of the bones but the 1st and then not by and by Roberts you have many many related things in aggregation during statistics that are not exactly the same kind of moment still related to this
approach answer something else there is obviously relation fulfills among you were statisticians and knows about who know about by statistics obviously there is the thing I mentioned by statistics if you see the susceptible likelihood and obviously by I according the priority in its Ackerley surprise because there is sticking with amazing statistics and that is something a few preparation fewer people motivates the use of this probability distributions thank you even when a little statistical not just when doing it by inspectors to is something I play all times of the likelihood of an getting off on an example just for to just a short Burns's useful suggestions but for example if you have a bushel observation and try to estimate the mean then he obviously you would use the Bush and post I also want to lead it would be the same but the rescue would be the credits the quad radicals OK but in practice when you have a you know that the quadratic closeness to Monrovia situation so for example here they propose to replace it by a robust if I'd lost function OK in this case you don't you feel the likelihood that still you have something that looks like this big good positional play your and selected with so you might have all the reasons even tho I believe that the woman presented before the best but you might have all the reason to
use of this kind of 2 local state where the security of the maybe even if you didn't exactly like you would vote the reason why according to those making would eventually win the likelihood you mean that you describe a world apology a parametric distribution on their observations which is not the case here vision of the system on by devoted to them that all my juridical results I choose London in a way that would not be acceptable bye bye isn't so I read not exactly the same thing even so it's plausible in just 1 last thing that I would like to mention is that so there are there is a community amazing statistics of people who like to analyze the rate of convergence because of concentration rates of as quickly because there is completely different to that in the end they usually end up with a single competition winning that when you go to
these known here you of everything that is needed just from a technical perspective involved proof is different the 2 of everything that is needed to allow the use of
force and this kind of tours to compute a concentration of people cellular your rates but OK so sorry
for the parent is that now I am coming back to the main proposal based on the stalled suddenly the program is we want to come To conclude this analyzing computer which is to compute the probability distribution I mean I want to sample from it or to come to the OK so obviously if you knew about some of competition using statistics you know that there
are the tools to do this hobbled economy that's the focus of reading the paper by unlike in special editions would be a treaty proposal from Bergamo agreement to compute their estimate or well I did the work with the job you when we use the redistribute John for agreed to compute the worst Middle suddenly there was an attempt to use more economical for this but this is where actually on the the dockside unless bright side in the forthcoming having won their presence and the in front of optimization during people believe they did move back side of the Force shouldn't do this it's too and I is not only that it's too slow because in some cases it works it works well but we don't have warranties on how far
How from these quantities of so I want to say that it's not competent and I want to mention the
this to papers of Clinton different approaches them in this former presence of concentration 4 Markov chains will start from the known as the most stationary probability distributions swings only have to prove a concentration of and recall approximation of that you get prominence insisted on the other hand depends on maintaining many assumption on the Markov chain and to me it's not clear how this assumption related for example to what the dimensions of the problem so it analysts this approach is OK if you of one-dimensional parameter space but I don't know of its case with the dimension of the parameters set and that it was a very nice paper actually where he has the exact skating with respect to the dimensions so it's for a version of the Muslim optical agreements so it's very nice I mentioned it as a preprinted but may be accepted since I will be slide so I don't know that was very that ball on the other hand you have many assumptions on me suitable study or a newspaper so he I wanted to discuss another
possible approach again sold the idea is not to use them and the government of the told
interviewers optimization became convinced me is the best thing to do so here he is well underway just for 1 side will use the addition to the usual addition pulled the poster urine by isn't statistics again and the of the 18 just that well if this is not reachable in practice they let's not try to reach its let's try to reach a symbol of objects will get selected him to do is to propose not the to pay attention to all the possible aggregation distribution but to pay attention just 2 fixed family Of aggregation distribution for example the parametric value like all the motions of distribution OK and then we are trying to minimize although the stunning the distance between what would be the truth the study or what say also our objectives and approximation but it is not our idea was this very famous amazing statistics because it sits along time so I don't want to see what with the singing of paper for this difficult by by approximation but I don't it's in the papers by Michelangelo and hold application to graphical models but I think it was 1 of the 1st 2 to use this middle unsure about initiative would you want have
to use that 1 of the 1st half of this year the system but when I say who was the 1st woman to use this and was the 1st in statistic because if we rediscover regional on-field usually don't pay attention to what was done before in all the fields so it's the local right product about you right to use in physics which with whether she later this year that now the initial at noon newspaper myself but probably because they produce provides more references to what was done before the end of things that they look you you is ever a nonparametric all parliamentary committees ladies and finite or infinite dimensional set F. Silverman field approximation would be the case of the Julian in finite dimensional said it but here are focus on parliamentary deputy so you just leave a set of probability distribution role that is indexed by apartments or any you find a mental space In any case you go if you replace your Bush post-sale something problems or yours I didn't study the problems by just under optimization problem with it so what is going on in there is some sentiment improved to present not applications I'm going to try to abuse justified his approach into pool using the prehistory that actually in some cases you don't lose adults in your prediction of but so I mean don't expect me to provide for example optimal approximation optimization algorithm here OK it's not my job and thanks to your talks yesterday and today but I'm not the woman candidate best in any case the best possible agreement here and want to to show you that acted even if using radiation during you prevent gun was known for an optimization problem which would consult so our 1st question was do we have any juridical bounties on the approximation OK so I would present result just before the fewest nations of soul this is what we get this is what they want to compute so I want to minimize the distance in terms of combating Belgians between the approximation and this little likelihood and the thing is that it is exactly what you said before her office's when you write it it was known to the aggregated version of uncharacteristic this time because the grammatical Johnson of a reminder town but that actually not depend on aid so I can forget about it when I'm doing munitions again so that's what I'm going to do is to 1st to minimize this with respect to any and then define a as my estimate or and finally my aggregation distribution if it's necessary to computed then effective role where you take the the premature and 4 aces yeah so what do you do when you want 1st make this financing he found that he used correct but it is the same suggested the U.S. attitude but the ball that I had a surreal worries the ball
that and this is actually 182 to to to the to look at national agreed the thing that is the bone of wants
far away in the 1st leg of the
moment we have on the aggregation proceed actually is proof is
based on the fact that communities along with respect to all the probability solutions while actually immunize over a power parametric states still gives you a juridical rancher you want to get in the end 2 years and is a key element in this region was the right material distance 2 Munoz about what at what he had found traditionally has given you the from the the head of Iran's security and to give the positive side the only rest and so we just fine in person at the very last minute so widely made him look at so I have to answer police and in 1st some people did
before and obviously try to replace the K L by another retreat on probability distributions things might be harder for more competition but earlier in the week .period and that Indian can contribute explicitly sold and you can have you know what you want to minimize and replace for example by that a politician measure which that we needed to thinking about how to minimize the so that is the reason why we were interested in this is that actually sorry this
minimize all quantity of goods in very easily into the analyzes that the present before OK so this fragmentation bombs that were already known when he relates production risk acted to these battles between aggregation and distance and collected over the distance with respect to the than I did you can bring peace into the analyzes and gets the juridical warranty on your approximation if you wouldn't know for sure that this is your time this season that this is exactly the way you are indicating that you just get misled the Knicks lies on a major show the outline of the
papal visit visas no makes condition was so weak that you did that you guessed it began so in some cases and in these results and I wanted to say the paper obviously but I should be proud of it and something because just weakening result that was existed before but what's on offer in some way more pulled off is what follow the fact that the treaty leads to a practical procedure for aggregation again so I urge that although soldier and forgot to mention Michael tolls at the beginning of a little slower than reciprocal that ruled with the production of engines which way Jensen was beautiful and of course I can say that now is doing postdoc in in Bristol at and so desert that actually if you use the special petition so will I liked some of the Rojas that they had before so we will demand that
victory will pay lot of these 1 the 1 1 the analyzer obviously again
and the application that you have then instead of having the meaning of the minimize the threat the best balance will OK for between risk and the and for all the problems traditional than what it for 20 in the final year of a book on the UPN parametric approximation of Uganda which form arbitrageur approximation likening the allegedly we did it in the paper but as this is only a prehistoric I just wanted to present the filtration so the thing is that is the work is not done then in view of all its roots in that transformed the question is I mean do you have an actor obviously should take care of the report and a it's possible that as long as the result of good so the thing is now it doesn't give you a way to prove that Savimbi virtual May the petition always works it provides a way to checking when it works and you have is not you go to the long-dominant if you get something that is good then indeed it means that to use did you will not use lose anything in terms of accuracy when used production on the other hand difficult is normally find something large just as you nothing so just to try to which show that the European Commission accessories in France where the people whom we also yet the idea of what some waiting to get you to radical warranties at least that at in some cities it should work so and I print down the proof of these results so I'm magician should 1st present the applications and then go back to the proof that reversal of focus on to apply the Soo Line our classification problem again so In the back certificate so I come back to this to this sitting against all you someone II from
the history then the time classify spot manioc also consider just to the skull of products between the public and apartments a and then hitchhiked on what side of the hype applying EU rejects the released is actually even case the classification recent probability maker prediction error and I want to approximately it's by the empirical risk so as thanks to you I know a lot about half of the musician during I can try to minimize these of sums for just up to compute the gradient of this quantity incident 0 I think it's always working all to make a petition and to make optimization "quotation mark in this case it was not able to compute a sensible residents of soul actually I decided to use approach with abbreviations all users Gaussian priority of just because it was simple foregone conclusion that would explain what can be changed if you replaced by another prior 5 for the moment just stick to Wishon and then actually visible so what is going to be posted on the poster or is the priority played by to the -minus this again it's not very nice and yet he was just what proximity by abortion distribution look at which was made but this year and in some sense then you just have to much respect you and with respect to signal and you hope that in the end we get something sensible so 1st what is your accumulation protection in this case we use we wanted to write that don't explicitly and in the end what you this sort 35 is the CDF of the Russian information and you obtain something that obviously was acting this looks like a a slightly modified version of the empirical risk was small specialty empirical risk is due to the integration with respect to the posterior to the aggregation solution then of new square dance was likely to reach the 90 and then you of facilities that depending on the certain underlines metrics but so the premier that was in some ways things better and this 1 in the sense that this is a smooth In addition run when this 1 was to on on the other hand clearly did not very good in the sense that it is not complex and I heard some in the port today that you have to use the world collects a lot when you talk about the conditions this is another thing about optimization that so it's not so much in the way we tried to deny the time in small mentioned against 2 weeks using for example right in the sense but at different scales using deterministic and eating
and 1st they should show you the results may be before the juridical entities but OK so what we need here with 2 Klosterman net assets on the national repository reused signature on the gallows which I mean it's among the government told of that works well and is when the dimensional too-large you know let's we between you kind of benchmark results we go about our emotional approach in this case and then just as we use knowing our has been compelled to are underdogs just said you know what I mean I don't know what we did that's exactly but there is a lot was that wanted to use another middle and check for the deficit with the it's sensible in some way to analyze the choosing of dynamical and for example in this case is that the 2 men on the books they do what they don't perform so well in this case it's clear that you shouldn't use learning on it on the other hand the letter said when it seemed that the among parliamentary at middle is magazine the lowering you on Malta when compared to the number killed in this case and then they are that's traditionally sensible anatomy examples actually but not in all the examples but here for them but in many examples we have a slightly better performance which is due to the fact yes this is the signature musical here with which which the following sense among all yell missing wildernesses also resorted
to the approximation of the study or using it OK so doing we just try to do what we have known before we don't about conditions in the it's something seems
certainly since seats is set of iron because it meets the centering on whether it generates .period from from the priority in just 18 minutes or that have a porpoise the ones that have for the last 4 study or you duplicate them and then applied to each of them exactly wasn't exactly exactly it's did so the following are some of the same the same vicinity made-to-order computed in 2 different ways and the 1 that I'm trying to sell to the this 1 in his office that and the severity of the I'm not saying that it is superior to use their activities OK I I say that on Friday as it (left parenthesis soldiers the and so that the statement that the states that prove that I'm right and I just had this 1 together I know that you have relations you don't have to be writes of 1 of the possible that the money was used in the desert in here every in year and not and yes you're right that's morning and that its use is laying us of pretty tossed in the air as long as you're right OK but even even my objective in the end and using this moment represent later and I might the much much better than the best thing after the fire but actually my juridical analyzes just say that I do as well as the best thing I classify your the system so that means only this was married to marriages when the finding was set off later also was too as well as the best on it's true that we were going to get out much better than this but involved it is not what we tried to do in these analyzes focus and entering these days obviously and these improvements might be due to the plea Noel energy of this but I mean for example in this case where clearly I think that there is a very good may not the to have improvement and which was used for what we observe that you to the fact that's what obviously we stopped as an essential sometimes and you much electrical managed well actually we stopped the residents agree that when the Williams was exactly equal to and so we were sure that the convention is that they will get the ejection of to 0 . 5 dB yesterday he said that in front of the animal you right and it is collects and I will discuss it promised and that I want
to be felt exactly well it's not
makes as alleges at point this out and we don't have warranties on the minimize also there would come back to the center of an overblown literature sold but 1st
OK before this so we have these results that seems well on his promising even tho we have to see more yet this is the founder of Hamas refused to provide you with me and the use of the little confidence among Asians but he problem this meeting know I will and as a surreal about the books here anyway but Sonnier said you right in wanting a weakened that collected the theater when we discuss the approved just 1 thing now I think we knew was food render the present also what sort of worst
here this 1 and replace the said a bite what is said
that the thing up until then we staying here this result will this
result so what is it that you when there is only an assumption that we collected 1 in the 1st interview that the risk off your aggregated data energy you that she used the randomized national console you right and can be if you look can be better than but anyway what they know at least for sure that there was when as the best possible light they are pretty on because dissuades square root of the over and which is actually a wandering justification domestic and whether there is an additional terms which can be removed at the cost of a very very exhaustive analyzes that in some cases it can be removed actually losing by but In this in knew of the slogan don't book at ends up in yes I just wanted to mention the sleeve of a kind of regular adjustments and some sense which they you that's when you change at the top of the real risk doesn't jump lots which is obviously sensible in the sense that you replace .period estimate or by an aggregated history that this neighborhood and obviously cannot work and infuse abortion and radiation it cannot work of cannot have just the kind of review regularity so you of this assumption and know that the sessions and not maturity about what happens again I'll mention that this assumption is not necessary for the primaries and analyzes of the year and your great aggregator shall not doing here In this case we have this assumption but under this assumption we know that the middle and actually this resulting in the serial port injected usually 1 of the application of the of the trade-off between risk and the anatomical selected
what we do is that we just apply deterrent sold the risk of the aggregated pretty door it is actually as good as the instrument over all the possible Gaussian distributions and I did it for the sake of simplicity is just the ones with the ones with a diagonal combines metrics but actually just and you can change would improve on the Constance but what is make life worse than what you must seemed OK and then getting I don't arrive all the calculations but 1st you know the correct indulgence between 2 wishes so in this case it's the stamp and see that you knew of the that is the very mention sold out during periods and ending the previous slide was the
that is the same and so you have this
understand him and then the all the time actually this is where the scandal smoothness comes into focus on selected the aggregate agrees is as almost as good as the risk of the meaning of the distribution process some reminder you get and just optimized with respect to everything was back to his best square here and then with respect to London and you get to the end of the optimal but the
console this is how it works but can
as you have 15 minutes as part of the family of 4 with views of the mountains by allegedly sold the most of the beeper that they know when people wanted to use the the butt of the conquered the for the food exports dialog that is here that the money and the 1st step of the was to replace the injured by the overall problem is worsened by the reasonable parametric set anyway and so on and the ball is the same in all you can call this a exactly so this is what you have is that frustrating in some way to use the approximation involved but not for the estimate or an activist you can do so this is what people running in the beginning there was choosing interleague ocean but rather of something like that uniform distribution on the ball around the best barometer up with that which actually is what because it allows you to use a being mounted to bonds on the empirical research Missouri as well but Israeli army as long as you have as it's possible to change the meaning to the political wanting to change the scale of the distribution of income as get watches look at it
and get it so I said to you before
it works well but the problem is that obviously it's not scumbags ends so as not collects you'll not convinced it's 1 of the jokes that prevailed sorry OK so far the only thing you notice now during classification you go and you want to replace the 0 1 lost by a conviction regained sole for example can use the paper items on the way in which you have many many Duran's about all the possible replacements for those who 1 also in hot what using them before the colleges so for example here we wanted to use the huge at sold the 1 that use support the Commission's console fine now or risk in this way and we have this empirical risk and was to use a bushel approximation for which at this time and this is important for the energy sector this we this time we things there was approximation with just if it's not against I mean all the coordinates of this in my hands and the violence the glands metrics is 1 In this connection in the
calculations ends I calculations on lottery difficult but what is not made you use when you see it is that this is a complex of cricket Arkansas you have to do this in the air for the which institution here you of the density function of the distribution of these distribution so well here in the end you have to tell that students like a kind of thing just to collect Surrogate's it's it's it's a new comics or even sooner .period of the stimulus than at the Justin convicts Gates of the you 1 function but if you win 1 for which you have the warranty OK so this will be the empirical research is to have the scandal-free each downhill canceled ability that is that the squalor of the apparent ingenuity to the wish and priority and then there's a tumor the violence and what not abuse for example we tried for fun to
replace here are those you want was by all the committee 5 dysfunction and then when integrate with respect to the post office you not necessarily gets a comic critized local Energywatch candy and usually comments with respect to you but what can we bring food is part of it In this case the delegates comics contestants here you can have just Windows 7 derivative we did the computation indeed works will get you that you read the paper OK I know it's a sort of use and once again is not something that it's not because bigger comics lost and then you integrate that you will you will of comics engine but then in this case it works with and then
in this case so this is where I told you that I don't mean I took many provision of all that and another specialist of musician during was initiated there's not a lot
but you can't because it believes that it's not Renault Convex you don't have many many good properties for example you can change you make some and some assumptions and of that freedoms to Princeton not from going to crawl to the rule then you can make little things but in this case we just like to Kabul on the parameter sets of UN sigma
and then use the Bradenton and with them that's what we were told AFP is that even tho we don't know a lot about condition during where able to write the story which as you basically under the same assumption that as the previous 1 that's you have I mean at this step will you compute at a simple bread and everything you compute I'm
aggravated distributions 11 new acts Kate Anderson much game and then the
risk according to this personal tools sold the risk according to what comes out of the computer and not what comes out of the paper is actually as good as the best possible risk for Anania failure breasts vote next ball for the classification problems will interview over and and blow something that depends what providing badly for people who are within the condition during this is something that don't explicitly on the number of steps that you have look so obviously I mean what I wanted to present to you and that is why I went into the details more than into good the end is that's obviously you can play with these you
can change the equities is you can change the class of those stories pretty top-class class you can change the price or and in
the end you can change reason and I'm sure that you will be able to improve on his bald butts still now you see that using batteries and involves possible to provide warranties on the prediction that you have for Vashon approximations any possible to reduce the problem to a convex optimization problem in some cases but I just wondered whether
something that's an inactive proclaimed that the because of minor duties just after all and we submitted this paper jams rejoice 1 of my cordial decided to write a package which after a very very long purely due process is now available on the Horowitz laughed and said Well I mean I don't want to on and it works like the investigators went on to which went all the details but just that and you just can't obviously the cemeteries of levels and the metaphysical that's but the thing where they want to show is that even tho we didn't make the brought into you can get involved OK so for example in this case you know that's the probability with probability at his 99 % the probability of Aurora swallowed by 0 . 79 which is not so much as it is involved in this case gestures as if this is the moment of 1 variable but yeah you're right edition empirical the national ball on the huge costs in jail OK rights even tho it's not a very very good and we know that the smaller opportunistic and especially when you use it in a problem when clearly hobbling after the failure of then the square root of Deuba and regional Jones can replace invited you over and when you have imagined conditional something so the money is usually not to move forward but I think we have happy with the fact that the United known provides this little book delay and in some points it's possible Indians to get something smaller than one-half you have to wait for time that is possible have less time for the post I it's you because so I go back to the mainland results the truth quite
obvious that I want to I mean it will be the opportunity to mention this intricate which all involved in the plot but OK so low so we have to
put this torrent so the fact that your approximation multimedia petition he performs as well as the best possible approximation the parametric 8 and then we
started the inequality OK so you just have a moment exponential on the exclusion women Capital are minors were are and then what I just wrote teaching introducing my probability signed on OK and you integrated so this is the change mean with respect to what extent boundaries that you integrate with respect to the priority which doesn't change anything for its own pocket because it's a constant and then so this is very strong on the then
you gets actually to this point OK and then you use this may not mean that the fight which cannot use that can compute the complex conjugates about half of the year k and indulgence and not tomorrow whether we use that little union although that actually you know the Commission ruled that reaches the minimum here OK and of those who promote mystery here so genuine use this landmark you get this kind of uniform national uniform eyes additional funding iniquity but as an integrated with respect to the priority difference with what you have and that makes a type analyzes is that nothing analysis rather soup with respect to the permit us to come up with hear you understood with respect to all the problems which initials on them on the current on the 6 again and then finally use markets-inequality you this empirical and I wanted to insist on this because of market forces mentioned it many many times but here you know that it's not only for the minimize for all possible probability distribution role opposite relations solution then the risk of those aggregated procedure is small island something that is completely empirical and intelligence on you the risk the Eucharist so dependent on the sample and then the Promised Land added you choose and then the prevented evidence with respect to the PRI all the teachers as well so that you can compute small miracles and this is the ball that is provided as well by the package OK so at any hour I mentioned in the beginning of the told the origins of batteries and mounds of McGinest work focused mainly on the scandal because the thing that's what's important and to be sure that your classify you're in with probability 19 and pursuant hasn't state that is more of an 0 . 1 OK employ few want to prove something of a juridical bold something that depends on the troop reduction risk you have to to to to direct this hazard to anyway so this is quite important yes you have have around them know you'll not allowed to do as well although not in these bombs rockets on the other hand the euro to we wrote to optimize or .period boats for a number that doesn't depend on the sample will get on there and what you can do is to use the union anonymously for unknown agreed and then actually using the fact that the is increasing increasing respect remember you can even optimizing of Nunavut and of all and this is what people do actually well and usually they provide a more sophisticated doesn't only whether in the with spectrum that it was obvious the thing that watching practice and practice condition and that's the main problem that we have to forget because the minimization in human unionized is known with respect to number usually don't have the best possible underhand across the nation for example much better but obviously it's much much expensive
OK so you want these empirical long and actually the thing is that the reason why people use this exponentially weighted the brigade that introduced in the beginning is that actually this is the 1 that minimizes the right inside pocket and as it minimizes the right inside you have your and the money and then what you do
love 1st to you you have these empirical won't get that before but as you minimize the right inside what you can do is meant the reveals Balkan would replace you start to
gain all the process which will replace ominous more are quite the opposite of this and then actually what that's you
of this time for the Raczynski was lesson but indicated interval of them .period research smaller than that general all the true risk of something but you can't put it in the
previous analysis OK so what we had before the interval of the the story of the year In miser the smaller here than the empirical knowledge but the empirical belonging to take it's in the moment and then using these results you can replace the graced by the poorest in the OK so here's what the Turin so this quite a stand the century that's the only .period was that remark that treaty she up and if you don't
realize respect to all the probability distribution but
just over your approximation funny then it still works look at the
west-central England so I
have just 1 slide good conclusion and think yes and so there are other things to do and some of them are already known in the paper we always been we also provide for example complete analysis of fronting models so it's embarrassing obviously the classification because we still use they are still functions but we also have the sincerity that we are able to pull replace and since mid goes by optimization the and even in some cases by complex musician problems we have had to have a sketch of the amenities in matrix factorization in a sketch of the United States in the sense that there are many problems there 7 In the sense that you have just
1 moment we've gone back to the
some of these results right yes this is a
danger that the Barcelona-based approximation performs as when I was in some sense as the exponentially weighted the brigades that wanted in the meeting if the system is not too large such what we did was to prove in this case that these loans without computed exponentially weighted average at all for the much-loved tradition that is using practice by people by Beijing statistician when they do much except a revision of the law the same but the mean that's in some way the band magnesium mom would hold for the exponentially weighted they need hold for the abortion approximation as well on the other hand of dilapidation do not know nobody was able to pull that it holds for the exponentially weighted aggregate so there is an issue part indiscriminate and already there is the although missing parts that is that's the predating used anyway for matrix factorization on complex and so we don't know whether we
another to a proper anymore even to put the number at all
that's involved I mean it
seems that there is something to be done
there and will not able to complete the analysis of its all the Tories local it's important we have all the works in progress at reason James is Corinthian in the package for the movement of 1 of some of the red this which is not so much in society the most efficient ones to the currently writing all the functions to use all their of optimization and what I'm interested in correctly and this will be my last lap my my last legs to question .period can you help me on this but remember that I presented a new beginning to policing bonds 1 for the about sitting in the Oval 1 for the online sitting ends of the 1 for the road back sitting he told the extension waited readmitted as a horse for the vital approximation a single single bouts of musicians and about the online known and the reason why it is that we were not able to perform the same analyzes so I can show you what we have to remember that in this case at each step the you whilst Wallstreet computes the mean according to this probability distribution and what we could do with a ministry the Jedi approximations and Buffalo it would be already mall costs for actually done a proper on language and everything because he just said he would have to perform online communication but even tho we will not really sure that this 1 works because if you use this offical the main according to vote the introduction according to the suit Of the approximated posted soaring rather than the 1 that you
have these loans look at it so we have committed those that use moral than decreed that the same protection as the 1 that would have 14 W but on the class but in the Senate in this case you have an approximation down but instead of having it only once but manager you of it's at each step of the grievance it so that the cost is much much higher so maybe actually that this is what comes out of this town of 2 4 0 9 4 0 9 W and maybe there's a better way to analyze it but until now OK
will not be able to To improve on the fact that we don't pay the price for the approximation once we paid at each step of the grievance and so I mean if there is an uneasy with distance between them and the way that the 2 of them have a new approximation funnier than you might think you pricier so will not able to generalize this analysis for in all lined up production but would like to salute to also made use of it and you want to write a paper for me that the woman was gone will be sold in the sports world jokes the talk that livable but it would have been for your
attention in the future and I know that there are things that I was wondering about election and the situation in the entire world to be aware of it and so on "quotation mark over the development of the maximum what he called the humanities nations so that security services needed to know what we just the fall formulation where he would you use it's an an expected value of the currency and you you have a to history of the the last thing that hurt the the distribution of writers are and services prior to the invariably this is of work and the no longer have the interests of other fronts obviously was released before start tonight at a little over a year already interested but that you know it is different from 1970 connections with questions about is not as a way to work with 1 of the judges there and in the form of the violation involved only after the fire was believed he will be sworn in on it :colon elimination it wasn't a thing that we have different objectives because usually they are more interested in you In the end the recognition of the and sold something like this when you related news provider lists involved for the for that and it's true in some way by reason moments in Europe to minimize respect finite dimensional objects and people already had the idea to true to unionize just over a small set of partner also so it just I mean I really the thing was to make a connection with the but for the way he has handled his With so we had already existing in the region in which he was 1 of the dead and the inclusion of the year the growth of the year award any I don't think that any of the
agreement he on the they was just to provide analyzes of existing ovaries lets you you right in most of the original review existed before anyone can we win all this work by so here he is he version the army in version on experiments that he had the kind of player he that yes it will a follow a religious did and this is the version that is implementing the package years so ideally I hope it will be a story I removed the slide actually because it you can I thought I had it's somewhere eradicated sought to workers
intimacy yeah we had this and we have way in the paper with the Regional Special sojourn in Italy and in most cases obscene and underestimated 506 cases it improves on the this and that is going to much much faster but there is 1 case where we have a very very surprising in like accident I don't know why we cannot understand it but in which case it's much these works with much much better than this 1 this was much better the difference between the interest so exactly but there is always in some way the idea was to win I mean this is a pastoral care issuing talks was much better than this and that in future we were under the algorithm for some time and in the end we have a better store in some sense the idea of these simulations to the was aware that we don't want to dispute between the optimization and the test in orbit that wanted end the would test or whatever you have in the sense that you can have a complicated modern we've poor optimization procedure or a simple model with good optimization procedure in the end we to compete in this world and the improvement that we have the test or Indian but obviously you too literal translation that here this of the fare with the original of the various situation you the local news among most agree is yes and you say if you compare that with the airlines which is now using that there is really very well the legalization of the situation was not that yes and I think we to he gave this saying that there was much faster they gave this even the a slight bit of history or in some cases I mean and enlivened actually write it in mind that the hacks unless we are legally accountable for which it promised for this 1 I think which is very easy Zalantun and ineligibility action and this and that is the worst and may be due to the fact that you have to restricts our attention today alone the vines metrics I don't know but yet in this case we don't have a traditional yeah what I will there is the potential of that tester or on them half of the sample In the only measure that's the integration of the compressed to compress functional necessary contracts that 2 of the original strongly committed to function I don't know what it does something useful for the I think the American novelist exactly adding that the premier in the that way when you integrate local banks instead of the ones you almost went away to collect suspension with respect to 2 OK but the problem is that every
day and I wasn't even on the greenback Dungeons and the prevented emergence of a new or renewal of the independent republics will this is going on in the production of soaring
when you minimize this OK if article makes you integrate with respect to a low probability distribution in an Indonesian beagle mix as well but the question is is this bottlenecks and with respect to the nonstop with the case this year the International of racial comments about Bloomberg but I just want the thet
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Statistik
Ortsoperator
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Minimierung
t-Test
Physikalische Theorie
Übergang
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Stab
Fächer <Mathematik>
Wurzel <Mathematik>
Gleichmäßige Konvergenz
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Computeranimation
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Richtung
Unendlichkeit
Prognoseverfahren
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Diskrete Wahrscheinlichkeitsverteilung
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Gruppe <Mathematik>
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Permutation
Komplex <Algebra>
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Computeranimation
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Distributionstheorie
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Statistik
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Teilmenge
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Viereck
Radikal <Mathematik>
Konvergenzgeschwindigkeit
Aggregatzustand
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Theorem
Statistik
Prozess <Physik>
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Physikalisches System
Konzentrizität
Forcing
Perspektive
Beweistheorie
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Ideal <Mathematik>
Markov-Kette
Beobachtungsstudie
Diskrete Wahrscheinlichkeitsverteilung
Parametersystem
Subtraktion
Approximation
Hausdorff-Dimension
Natürliche Zahl
Stichprobe
Kette <Mathematik>
Technische Optik
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Rechenschieber
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Ideal <Mathematik>
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Distributionstheorie
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Distributionstheorie
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Machsches Prinzip
Familie <Mathematik>
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Euler-Winkel
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Term
Analysis
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Temperaturstrahlung
Numerisches Modell
Globale Optimierung
Abstand
Diskrete Wahrscheinlichkeitsverteilung
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Distributionstheorie
Approximation
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Likelihood-Funktion
Nichtkommutative Jordan-Algebra
Optimierungsproblem
Physikalisches System
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Diskrete Wahrscheinlichkeitsverteilung
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Approximation
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Güte der Anpassung
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Fokalpunkt
Analysis
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Approximation
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Diskrete-Elemente-Methode
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Wurzel <Mathematik>
Gravitationsgesetz
Explosion <Stochastik>
Parametrische Erregung
Gerade
Resultante
Lineare Abbildung
Distributionstheorie
Subtraktion
Gewichtete Summe
Gewicht <Mathematik>
Momentenproblem
Minimierung
A-posteriori-Wahrscheinlichkeit
Zahlenbereich
Inzidenzalgebra
Stichprobenfehler
Computeranimation
Gradient
Dynamisches System
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Prognoseverfahren
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A-posteriori-Wahrscheinlichkeit
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Singularität <Mathematik>
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Integral
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Konditionszahl
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Beobachtungsstudie
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Relativitätstheorie
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Physikalisches System
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Objekt <Kategorie>
Energiedichte
Statistischer Test
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Ablöseblase
Tabelle
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Theorem
Punkt
Singularität <Mathematik>
Besprechung/Interview
Gradient
Kombinatorische Gruppentheorie
Statistischer Test
Bereichsschätzung
Verbandstheorie
Sortierte Logik
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Nachbarschaft <Mathematik>
Lineare Abbildung
Theorem
Logarithmus
Kartesische Koordinaten
Extrempunkt
Knoten <Statik>
Term
Computeranimation
Approximation
Energiedichte
Temperaturstrahlung
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Diskrete-Elemente-Methode
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Wurzel <Mathematik>
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Distributionstheorie
Theorem
Glatte Funktion
Linienelement
Logarithmus
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Rechnen
Frequenz
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Rechenschieber
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Approximation
Logarithmus
Familie <Mathematik>
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Mereologie
Parametrische Erregung
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Derivation <Algebra>
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Komplex <Algebra>
Computeranimation
Approximation
Dichtefunktional
Sortierte Logik
Mereologie
Blase
Distributionstheorie
Theorem
Statistik
A-posteriori-Wahrscheinlichkeit
Gradient
Extrempunkt
Sigma-Algebra
Analysis
Computeranimation
Spieltheorie
Verweildauer
Radius
Parametersystem
Gruppe <Mathematik>
Kategorie <Mathematik>
Logarithmus
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Schlussregel
Widerspruchsfreiheit
Approximation
Menge
Funktion <Mathematik>
Ebene
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Metadaten

Formale Metadaten

Titel On the Properties of Variational Approximations of Gibbs Posteriors
Serientitel Computational and statistical trade-offs in learning
Teil 07
Anzahl der Teile 10
Autor Alquier, Pierre
Lizenz CC-Namensnennung 3.0 Unported:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/20250
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract PAC-Bayesian bounds are useful tools to control the prediction risk of aggregated estimators. When dealing with the exponentially weightedaggregate (EWA), these bounds lead in some settings to the proof that the predictions are minimax-optimal. EWA is usually computed through Monte Carlo methods. However, in many practical applications, the computational cost of Monte Carlo methods is prohibitive. It is thus tempting to replace these by (faster) optimization algorithms that aim at approximating EWA: we will refer to these methods as variational Bayes (VB) methods. In this talk I will show, thanks to a PAC-Bayesian theorem, that VB approximations are well founded, in the sense that the loss incurred in terms of prevision risk is negligible in some classical settings such as linear classification, ranking... These approximations are implemented in the R package pac-vb (written by James Ridgway) that I will briefly introduce. I will especially insist on the the proof of the PAC-Bayesian theorem in order to explain how this result can be extended to other settings. Joint work with James Ridgway (Bristol) and Nicolas Chopin (ENSAE).

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