Prob­a­bil­ity the­ory is one of the more recently estab­lished fields of math­e­mat­ics, which is cur­rently a very active area of research. It has found numer­ous appli­ca­tions in math­e­mat­i­cal mod­el­ing for com­puter sci­ence, finance, sta­tis­ti­cal mechan­ics, dynam­i­cal sys­tems, bioin­for­mat­ics etc. Top­ics in prob­a­bil­ity the­ory include mar­tin­gale the­ory, sto­chas­tic processes, sto­chas­tic cal­cu­lus, ergod­ic­ity and oth­ers. Prob­a­bil­ity the­ory has drawn from many other areas of math­e­mat­ics such as mea­sure the­ory, inte­gra­tion the­ory, rep­re­sen­ta­tion the­ory, real and com­plex analy­sis and in return has pro­vided some inter­est­ing insights into alge­braic struc­tures, geom­e­try of graphs, dif­fer­en­tial geom­e­try, ergodic the­ory, PDE the­ory and sam­pling algorithms.

Prob­a­bil­ity The­ory Courses [Show]Prob­a­bil­ity The­ory Courses [Hide]
The fol­low­ing are MAT and ORF courses cov­er­ing aspects of prob­a­bil­ity the­ory.
ORF/MAT 309: Prob­a­bil­ity and Sto­chas­tic Sys­tems
The course goes over many top­ics in prob­a­bil­ity the­ory, with­out going too much in depth in any par­tic­u­lar one. It begins with the basic prop­er­ties of math­e­mat­i­cal expec­ta­tion and vari­ance, with an empha­sis on stan­dard dis­crete and con­tin­u­ous dis­tri­b­u­tions. For the sec­ond half of the semes­ter the course dis­cusses Pois­son processes, Gauss­ian processes, Brown­ian motion, and Markov chains.
The course pro­vides a firm under­stand­ing of the basics of prob­a­bil­ity the­ory and is intended pri­mar­ily for stu­dents who have not had much expe­ri­ence with the sub­ject or rig­or­ous math­e­mat­i­cal proofs. It is the basic math­e­mat­i­cal course taken by engi­neers, specif­i­cally ORFE majors, who intend to work with sto­chas­tic mod­el­ing and sim­u­la­tions. The course is extremely help­ful if one is to take courses in math­e­mat­i­cal finance, arti­fi­cial intel­li­gence, sto­chas­tic cal­cu­lus, asset/commodity pric­ing etc.
ORF/MAT 309 com­pen­sates its lack in rigor with a lot of intu­ition, so that results do not sim­ply appear out of thin air but are log­i­cally moti­vated and well sup­ported. Many stu­dents find the course to be espe­cially chal­leng­ing as the amount of mate­r­ial cov­ered is so large. ORF/MAT 309 focuses on devel­op­ing ideas and meth­ods for solv­ing prob­lems as opposed to care­fully build­ing the­ory and as such is an excep­tion­ally fun course to take.
About eighty per­cent of the course grade is based on two midterms and a final, which are all in class. In addi­tion, there are ten weekly prob­lem sets and a large “Big Home­work”, which is writ­ten dur­ing win­ter break and is about three times as long as pre­vi­ous home­work sets.
The course is extremely well orga­nized, fol­low­ing for the most part Pro­fes­sor Cinlar’s notes. Prob­lem sets focus on improv­ing the under­stand­ing of the mate­r­ial. As one of the larger courses in the ORFE depart­ment it has many TAs, which reg­u­larly host prob­lem ses­sions and pre­cepts that one can take advan­tage of.
MAT 385: Prob­a­bil­ity The­ory
The course goes over sev­eral top­ics in prob­a­bil­ity the­ory in a some­what deep man­ner. Ini­tially stu­dents are intro­duced to the basic prop­er­ties of math­e­mat­i­cal expec­ta­tion and vari­ance, dis­tri­b­u­tion func­tions and char­ac­ter­is­tic func­tions. The course then goes on to ran­dom walks on the inte­gers, reflec­tion prin­ci­ples, laws of large num­bers, cen­tral limit the­o­rems and laws of large devi­a­tions.
The course pro­vides a good intro­duc­tion to var­i­ous top­ics in prob­a­bil­ity the­ory, with a spe­cial treat­ment of con­ver­gence results. It is intended for math majors who are inter­ested in learn­ing more about prob­a­bil­ity the­ory at a deeper level. It is part of a two-course sequence, the sec­ond course focus­ing on sto­chas­tic processes. MAT 385 encour­ages stu­dents to think about prob­a­bil­ity the­ory in a more pre­cise man­ner, giv­ing a fairly care­ful treat­ment of the mate­r­ial.
Home­work sets aim to improve the student’s under­stand­ing of the the­ory and rarely go beyond the appli­ca­tion of a cou­ple of results. There are weekly prob­lem sets that account for fifty per­cent of your grade and there are prob­lem ses­sions a cou­ple of days before each prob­lem sets is due, where stu­dents go over the basic ideas behind each ques­tion.
What dis­tin­guishes this course with most oth­ers in the math­e­mat­ics depart­ment is the for­mat of the final. Sev­eral prob­lems are assigned dur­ing the last lec­ture and stu­dents have two weeks to solve them and write up their solu­tions. They show their solu­tions to the exam­in­ers, who then con­tinue to ask ques­tions regard­ing the prob­lems as well as any mate­r­ial cov­ered in the course. This verbal/written exam­i­na­tion is cer­tainly an inter­est­ing trait of the course that makes it a bit chal­leng­ing near the end. In addi­tion, Pro­fes­sor Sinai assigns a cou­ple of prob­lems, which if solved would exempt the stu­dent from tak­ing the final exam.
The course is quite unortho­dox and inter­est­ing. Dur­ing the semes­ter it is not excep­tion­ally chal­leng­ing as prob­lem sets are man­age­able and one can usu­ally ask Pro­fes­sor Sinai or go to the prob­lem ses­sions if stuck on a prob­lem.
MAT 486: Ran­dom Processes
This course is the sec­ond part of the math department’s two-semester prob­a­bil­ity sequence. It is sim­i­lar in orga­ni­za­tion and pre­sen­ta­tion to the first part, MAT 385: Prob­a­bil­ity, as it was devel­oped and is usu­ally taught by Prof. Yakov Sinai. MAT 486 cov­ers a larger num­ber of top­ics than 385, which are for most part deeper and harder than those in 385. You should con­sider tak­ing it if you liked 385, and are inter­ested in prob­a­bil­ity or a closely related field.
The con­tents of the course are cen­tered around the notion of ran­dom processes, which are col­lec­tions of ran­dom vari­ables indexed by some set, and thus include, for instance, ran­dom sequences, ran­dom func­tions etc. The course cov­ers the notion of sta­tion­ar­ity, in which case rep­re­sen­ta­tion the­o­rems (spec­tral rep­re­sen­ta­tion), and ergodic the­o­rems (Birk­hoff, von Neu­mann) are proved. Ran­dom fields, and an exam­ple of phase tran­si­tion are described. Extend­ing the notions from 385, Markov Chains with con­tin­u­ous time are intro­duced. The Wiener mea­sure and Brown­ian Motion are con­structed as the weak limit of ran­dom walks. These are applied to the solu­tion of the Dirich­let prob­lem, which you might have encoun­tered in Fourier Analy­sis. Finally, sto­chas­tic inte­grals are con­structed, and some famous for­mu­las (Ito, Feynman-Kac) are proved. The course ends with a dis­cus­sion of Sto­chas­tic Dif­fer­en­tial Equa­tions and dif­fu­sion processes.
Gen­er­ally speak­ing, the course cov­ers much more, and in more depth than 385. There are many notions and the­o­rems from analy­sis that come up, and are usu­ally taken for granted: for instance, results from func­tional analy­sis, Hilbert spaces, Fourier analy­sis are used quite lib­er­ally. Prob­a­bil­ity has very strong ties to analy­sis, and it shows. It is usu­ally pos­si­ble to under­stand the prob­a­bil­ity results with­out exten­sive knowl­edge of those fields, how­ever for a good appre­ci­a­tion, it helps to know them.
The orga­ni­za­tion of the class par­al­lels that of 385, and the same pieces of advice apply. Go to the ‘office hour’, where the home­work is dis­cussed. If you have time, try the ‘hard prob­lems’ that Pro­fes­sor Sinai assigns for an auto­matic A. There are usu­ally a few stu­dents each year who solve them. But be pre­pared that the first hard part is under­stand­ing what the prob­lems really ask, since they can be less than clear. The exam is oral, as in 385: you get 5 prob­lems and a month to think about them. You are also allowed to use your notes and book. If you have solved most of the HW prob­lems by your­self, you should be able to do well. Also, be pre­pared to be asked a large num­ber of addi­tional ques­tions at the actual exam­i­na­tion. These are usu­ally rou­tine com­pu­ta­tions, def­i­n­i­tions, or proofs of the­o­rems. How­ever, in some cases it might be a lit­tle hard to under­stand what the ques­tions are.
To sum up, MAT 486 is a more chal­leng­ing con­tin­u­a­tion of 385. If you liked the lat­ter, and are seri­ously inter­ested in prob­a­bil­ity or a field where it’s applied, you should con­sider tak­ing 486.
ORF 526: Prob­a­bil­ity The­ory
The course cov­ers most of the basics of prob­a­bil­ity the­ory, from a measure-theoretic point of view. It begins with the devel­op­ment of mea­sure the­ory and Lebesgue inte­gra­tion. Most of the basic results from real analy­sis are cov­ered, includ­ing devel­op­ing inte­gra­tion from scratch, Fatou’s lemma, monot­one and dom­i­nated con­ver­gence the­o­rems etc. It then goes on to the gen­eral results in mea­sure the­ory such as the Caratheodory exten­sion the­o­rem, Radon-Nikodym the­o­rem, Fubini’s the­o­rem etc. It espe­cially focuses on dif­fer­ent types of con­ver­gence and the con­nec­tions between them. The course then goes on to Gauss­ian processes and the analy­sis of dis­crete mar­tin­gale the­ory, includ­ing Doob’s inequal­i­ties and con­ver­gence results. Finally, the course quickly goes over the main facts behind Markov chains, includ­ing the Ergodic the­o­rem and sta­tion­ar­ity.
The course is intended for first year grad­u­ate stu­dents in the ORFE depart­ment although it is an excel­lent course for peo­ple inter­ested in prob­a­bil­ity the­ory in gen­eral. The treat­ment of the mate­r­ial is extremely care­ful and well devel­oped. For the most part the course fol­lows the grad­u­ate text by Shiryaev, which is an excel­lent but chal­leng­ing read. The the­ory on Markov chains fol­lows mostly the intro­duc­tory book by Nor­ris.
There are weekly prob­lem sets that account for about half the grade in the course. Most of them aim to improve stu­dent under­stand­ing of the the­ory and can be quite chal­leng­ing. Lec­tures are usu­ally spent prov­ing results, with very lit­tle dis­cus­sion of appli­ca­tions. As such the course is extremely the­o­ret­i­cal in nature and its aim is to pro­vide stu­dents with a firm basis for upper-level prob­a­bil­ity courses such as ORF 527 (Sto­chas­tic cal­cu­lus) and ORF 551/4 (Levy processes/ Markov processes).
The course is excep­tion­ally chal­leng­ing for peo­ple who have not taken real analy­sis before, espe­cially in the first few weeks. A good under­stand­ing of the mate­r­ial and con­struc­tions is nec­es­sary for com­plet­ing prob­lem sets as well as the final exam.

Machine learn­ing (ML) is a sub-field of arti­fi­cial intel­li­gence that lies at the inter­sec­tion of com­puter sci­ence, sta­tis­tics, and prob­a­bil­ity the­ory. In broad terms, it con­cerns the design, imple­men­ta­tion, analy­sis, and appli­ca­tion of algo­rithms that can ‘learn from expe­ri­ence.’ In the last few decades, it has advanced extremely rapidly and found appli­ca­tions to a wide range of prob­lems includ­ing analy­sis of lan­guage (“nat­ural lan­guage pro­cess­ing,” or NLP) and lin­guis­tics, mod­el­ing of genetic vari­a­tion, quan­ti­ta­tive mod­el­ing in the social sci­ences, spam fil­ter­ing, auto­mated dis­cov­ery of iden­tity theft, and many more.

While ML is a com­plex and wide field, it can be roughly split in two parts: (1) pre­dic­tion (or, more cor­rectly, “super­vised” ML); and (2) mod­el­ing (or, more cor­rectly, “unsu­per­vised” ML). Pre­dic­tion prob­lems require that one pre­dict the value of some vari­able in the future given many obser­va­tions of it in the past; fore­cast­ing the value of a stock given its value (and mar­ket con­di­tions) over the course of a year is an exam­ple. One way to think about mod­el­ing prob­lems is as pre­dic­tion prob­lems in which the vari­able to be pre­dicted has not been observed at all; fore­cast­ing the value of a stock just from mar­ket con­di­tions, with­out ever see­ing its pre­vi­ous val­ues, would be an exam­ple of a mod­el­ing prob­lem. In gen­eral, unsu­per­vised prob­lems con­cern the dis­cov­ery and exploita­tion of hid­den pat­terns in data: the canon­i­cal exam­ple is the prob­lem of col­lect­ing together sim­i­lar images in a large data­base (“clustering”).

From a math major’s per­spec­tive, either or both of the the­o­ret­i­cal and applications-oriented sides of ML might be very inter­est­ing. The­o­ret­i­cal prob­lems can involve the analy­sis of spe­cific pre­dic­tion algo­rithms, either from a com­puter sci­ence or sta­tis­ti­cal per­spec­tive; the analy­sis and design of meth­ods of “infer­ence,” the key algo­rith­mic step in mod­el­ing that esti­mates impor­tant unob­served quan­ti­ties from observed data; or con­nec­tion of machine learn­ing prob­lems with ideas from infor­ma­tion the­ory, prob­a­bil­ity the­ory, or other theoretical—and some­times application—domains. Applications-oriented prob­lems, a catch-all cat­e­gory that includes any prob­lem that ulti­mately con­cerns the analy­sis of real data, show a sim­i­lar degree of vari­ety. They can involve the deploy­ment of exist­ing meth­ods to inter­est­ing data analy­sis prob­lems; the design of novel method­olo­gies that allow for new kinds of analy­ses; the use of unsu­per­vised meth­ods to solve tra­di­tion­ally super­vised prob­lems; and a range of other research activ­i­ties. It should be noted that ML research enjoys a rich and fruit­ful inter­play between the­ory and prac­tice, so that the­o­ret­i­cal projects can often include an empir­i­cal com­po­nent and applied projects often require under­stand­ing rel­e­vant theory—and some­times extend­ing it!

Machine Learn­ing at Prince­ton [Show]Machine Learn­ing at Prince­ton [Hide]
At Prince­ton, ML research occurs prin­ci­pally in the Depart­ments of Com­puter Sci­ence (COS) and of Oper­a­tions Research and Finan­cial Engi­neer­ing (ORF). In COS, the two main fac­ulty mem­bers asso­ci­ated with ML are: Pro­fes­sor David Blei, whose research focuses on unsu­per­vised prob­lems and encom­passes both the­o­ret­i­cal work, par­tic­u­larly in infer­ence, and applied work, par­tic­u­larly in text analy­sis prob­lems; and Pro­fes­sor Rob Schapire, whose research focuses on algo­rithms for super­vised prob­lems, par­tic­u­larly “boost­ing meth­ods,” an impor­tant class of algo­rithms he co-invented. While read­ing these brief sum­maries of their research areas, keep in mind that they are only approx­i­ma­tions: both have wider inter­ests and have worked on a wide range of prob­lems. There is also increas­ing inter­est in ML by the the­o­ret­i­cal com­puter sci­ence com­mu­nity; at Prince­ton, this line of research is being led by Pro­fes­sor San­jeev Arora, whose research gen­er­ally focuses on com­plex­ity the­ory and the the­ory of algo­rithms. In ORF, there are sev­eral fac­ulty mem­bers whose research can be classed as ML. For details, you should con­sult the ORF web­site. A good start­ing list of fac­ulty is: Pro­fes­sors Sébastien Bubeck, Jian­qing Fan, War­ren Pow­ell, Phillipe Rigol­let, and Ramon van Handel.

Machine Learn­ing Courses [Show]Machine Learn­ing Courses [Hide]
Course offer­ings in ML gen­er­ally par­al­lel research, with a mix of rel­e­vant COS and ORF classes.
The rel­e­vant COS courses are:
COS 402: Arti­fi­cial Intel­li­gence
This is Pro­fes­sor Rob Schapire’s sur­vey course on mod­ern arti­fi­cial intel­li­gence. ML is a major com­po­nent of the course. Two units in the course exam­ine in some detail algo­rithms for “prob­a­bilis­tic infer­ence,” a key part of ML that allows quan­ti­ties that are never directly observed to be esti­mated from observed data. This is only done for a few fun­da­men­tal mod­els, but the mate­r­ial pre­sented is foun­da­tional and use­ful. Addi­tion­ally, a long unit at the end of the course focuses on the pre­dic­tion prob­lems and intro­duces numer­ous pre­dic­tion algo­rithms, includ­ing Pro­fes­sor Schapire’s own AdaBoost.
COS 424: Inter­act­ing with Data
This is Pro­fes­sor David Blei’s sur­vey course on sta­tis­ti­cal machine learn­ing. The course builds up the edi­fice of ML from the ground up, start­ing with a brief review of basic prob­a­bil­ity the­ory and ulti­mately ana­lyz­ing sev­eral impor­tant classes of mod­el­ing prob­lems, as well as their appli­ca­tions to text, image, and other data analy­sis. The course cul­mi­nates with a month-long group data analy­sis project that gives stu­dents hands-on expe­ri­ence with ML.
COS 513: Foun­da­tions of Prob­a­bilis­tic Mod­el­ing
This is Pro­fes­sor David Blei’s advanced under­grad­u­ate– or graduate-level sur­vey course on prob­a­bilis­tic mod­el­ing, an approach to ML prob­lems that artic­u­lates pre­cise assump­tions about the prob­a­bilis­tic rela­tion­ships between vari­ables of inter­est and exploits these (via infer­ence) to dis­cover and make pre­dic­tions about struc­ture in data. The courses cov­ers the foun­da­tional the­ory of prob­a­bilis­tic mod­els, the most impor­tant algo­rithms for infer­ence, and the prop­er­ties of sev­eral rep­re­sen­ta­tive classes of prob­a­bilis­tic mod­els, along with their appli­ca­tions.
COS 597X: Advanced Top­ics in Com­puter Sci­ence
In some years, this fam­ily of graduate-level courses includes one or more on ML. Recent exam­ples include Pro­fes­sor David Blei’s course on Non­para­met­ric Bayesian Meth­ods, Pro­fes­sor Rob Schapire’s course on Boost­ing, and Pro­fes­sor San­jeev Arora’s course on prov­able meth­ods in ML (Is Learn­ing Easy?).
Some of the rel­e­vant ORF courses are:
ORF 418: Opti­mal Learn­ing
ORF 505/FIN 505: Mod­ern Regres­sion and Time Series
ORF 524: Sta­tis­ti­cal The­ory and Meth­ods
ORF 525: Gen­er­al­ized Regres­sion Mod­els
ORF 525: Sta­tis­ti­cal Learn­ing and Non­para­met­ric Esti­ma­tion

Con­tacts [Show]Contacts[Hide]
Evgeni Dim­itrov ’13 (edimitro[at]princeton[dot]edu) for prob­a­bil­ity the­ory
Edgar Dobriban ’12 (dobribanedgar[at]gmail[dot]com) for prob­a­bil­ity the­ory
Max Rabi­novich ’13 (mrabinov[at]princeton[dot]edu) for machine learning