TurnTrout comments on Judgment Day: Insights from ‘Judgment in Managerial Decision Making’

• I don’t think about my self-study as “know a bunch of math things”. Rather, it’s a) con­tinu­ally im­prove at math­e­mat­i­cal rea­son­ing, and b) ac­cu­mu­late a bunch of prob­lem-solv­ing strate­gies and ways-of-look­ing-at-the-world. I can

• con­sider the an­a­lytic prop­er­ties of a real polyno­mial (how quickly does the out­put change with the in­put? all polyno­mi­als are an­a­lytic. polyno­mial ap­prox­i­ma­tion the­o­rem)

• put on my group the­ory hat (what kinds of op­er­a­tions does the space of polyno­mi­als ad­mit?)

• con­sider it from a lin­ear alge­braic stand­point ( is se­cretly )

• put on a num­ber-the­o­retic hat (con­sider the dis­tri­bu­tional be­hav­ior of in­te­ger polyno­mi­als in a mod­u­lar con­text)

• think about how polyno­mi­als might be used in ML (hy­poth­e­sis classes, com­plex­ity-of-fit, what’s the search space be­hav­ior like for SGD if the model space is just the co­effi­cients for de­gree-50 polyno­mi­als?)

• think about com­plex­ity the­ory (the nice clo­sure prop­er­ties of polyno­mial com­po­si­tion)

• or even con­sider the com­putabil­ity prop­er­ties of polyno­mi­als (this is a bit of a stretch, but… for ev­ery de­ci­sion prob­lem con­tain­ing a finite amount of YES an­swers, there ex­ists a polyno­mial with roots on the base-ten en­cod­ings of those YES in­puts).

Then, when I ac­tu­ally need to do stuff with polyno­mi­als for my re­search, I can see how the differ­ence of re­turns from two poli­cies can be rep­re­sented by a polyno­mial in the agent’s dis­count rate, which is a nice re­sult (lemma 32). In­sights build on in­sights.

So, I’m not try­ing to mem­o­rize ev­ery­thing. Leafing through Lin­ear Alge­bra Done Right, I don’t re­mem­ber much about what self-ad­joint­ness means, or Jor­dan nor­mal form, or what­ever. How­ever, I don’t think that re­ally mat­ters. If I need to use the ex­tra­ne­ous stuff, I know it ex­ists, and could just pick it back up.

I am, how­ever, able to re­gen­er­ate fun­da­men­tal things I ac­tu­ally use /​ run into in later stud­ies. I have a men­tal habit of mak­ing my­self re­gen­er­ate ran­dom claims in ev­ery proof I con­sider. If we’re rely­ing on the com­mu­ta­tivity of ad­di­tion on the re­als, I re­flex­ively sup­ply a proof of that prop­erty. I came up with: use the Cauchy se­quence limit no­tion of re­als, then rely on the com­mu­ta­tivity of ra­tio­nals un­der ad­di­tion for each mem­ber of the se­quence.

It’s not like I’ve perfectly re­tained ev­ery­thing I need. I can tell I’m a lit­tle rusty on some things. How­ever, I never do con­scious re­view­ing (be­yond build­ing on past in­sights through fur­ther study, us­ing the in­sights in my pro­fes­sional re­search, and red­eriv­ing things pe­ri­od­i­cally). I have em­piri­cal feed­back that this works pretty well. My PhD qual­ifier exam in­volved a ma­trix anal­y­sis ques­tion; with­out hav­ing even taken the class, I was able to get the right an­swer by rea­son­ing us­ing the skills and knowl­edge I got from self-study.

ETA: FWIW, when I talk with math un­der­grads at my uni­ver­sity about ar­eas we’ve both stud­ied /​ solve prob­lems with them, my im­pres­sion is that my com­pre­hen­sion is of­ten bet­ter.