[HN Gopher] A Concrete Introduction to Probability (2018) ___________________________________________________________________ A Concrete Introduction to Probability (2018) Author : tosh Score : 438 points Date : 2021-06-03 10:50 UTC (12 hours ago) (HTM) web link (github.com) (TXT) w3m dump (github.com) | spoonjim wrote: | I've never seen Peter Norvig choose anything but the most elegant | and perfect data model for the problem at hand. I wonder what | it's like to be in his brain. | at_a_remove wrote: | I got reasonably far in stats -- up to a single grad level | course. | | My biggest problem with the teaching of math (and I can weigh in | on this a bit because I spent over a decade as a private tutor of | math) is that math is often not introduced in a concrete manner, | as well as in a way such that its utility is obvious. These are | two sides of the same coin. | | You can see this problem in _most_ Wikipedia math pages. I took | more math than most physics majors and I find quite a lot of what | I stumble across either baffling or of unknown utility. Set | theory is a great instance of this. I rarely see multiple | examples listed as it is explained, nor does anyone bother to | tell me what it is _for_ , other than to do more set theory. | | Stats at least has the benefit of having to stick closer to | applicability, I think. | anthony_r wrote: | This is a super common issue that people are raising over and | over again, but there's already a solution for this: split | teaching into technical and non-technical. In some countries | universities are split along this axis. | | Take statistics for example - most people that know something | about statistics don't exactly know what a statistical space is | (and it is a very precisely defined concept, embedded in the | set theory). And that's fine, that's for the "pure" nerds. How | to use it can be taught without defining the roots and proving | theorems from the ground up. It is also how most of software | development is done, few people out there that write code | understand how CPUs fetch and execute instructions, talk to | other perhipherials, how does malloc()_or sin() work, what is a | page fault, or how to balance a red-black tree. Just use | std::map or dict() or something, it just works :) | thaumasiotes wrote: | > most people that know something about statistics don't | exactly know what a statistical space is (and it is a very | precisely defined concept, embedded in the set theory). And | that's fine, that's for the "pure" nerds. How to use it can | be taught without defining the roots and proving theorems | from the ground up. | | It's a popular theory, but then one day you read every recent | biology paper and notice that only 2% of them are able to do | statistics in a way that isn't total nonsense. | anon_tor_12345 wrote: | >how CPUs fetch and execute instructions, talk to other | perhipherials, how does malloc()_or sin() work, what is a | page fault, or how to balance a red-black tree | | literally every single one of these things is taught in an | undergrad class and understanding of which is deemed | important by the community - curricula get lots of input from | industry partners. so you're not making a great case for why | people don't need to know what a sigma algebra is... | | >Just use std::map or dict() or something, it just works :) | | wouldn't it be swell if this is how we practiced medicine | too? patient has early stages of atherosclerosis just do a | triple-bypass or something, it just works. | anthony_r wrote: | > you're not making a great case for why people don't need | to know what a sigma algebra is | | I can assure you most people that do stats / data mining / | big data / machine learning do not know or do not remember | any more what a sigma algebra is. | | > wouldn't it be swell if this is how we practiced medicine | too? patient has early stages of atherosclerosis just do a | triple-bypass or something, it just works. | | C'mon writing websites is not surgery. There are some | people with deep knowledge required in the industry as a | whole, but you really don't know to know much to write an | app or a website, especially an internal corporate tool. | And this is where most working hours are spent. | | It's the reality of things. I mean just look at the OP | link. Do you think this is targeted at people that already | know what a sigma algebra is? | anon_tor_12345 wrote: | this is such a tired old debate. | | 1. fundamentals are important even if people forget them. | | 2. not everyone in tech does web dev. | | these things are true and self-evident. the end. | anthony_r wrote: | Let's end, indeed :) | mjreacher wrote: | I believe this is due to historical reasons, in particular due | to the Bourbaki and the French school of mathematics where | abstraction was heavily prized. On the contrary mathematics in | the Soviet Union had more of a focus on intuition and geometric | grounding for mathematics, however history played its course | and mathematics is taught more towards the Bourbaki style these | days, however it did bring out gems like [1] which give an | opposite, albeit extreme, view of how mathematics should be | taught. | | [1]: https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html | bigdict wrote: | > On the contrary mathematics in the Soviet Union had more of | a focus on intuition and geometric grounding for mathematics | | A lot of it comes from mathematics always being taught in | conjunction with physics. Not sure what the roots of that | are. | nightski wrote: | The problem is sticking to applicability in probability/stats | only gets you so far. I'm in that rut having a firm grasp of | how it is applied but not being able to actually apply it in | the real world because the real world scenarios are all | different. | | It's easy to make trivial mistakes without a firm grasp of the | theoretical side in my opinion. It's important to understand | the implications of the choices being made and the theoretical | limitations of the models being created. | at_a_remove wrote: | I am not suggesting _ditching_ all but applicability, I am | suggesting that applicability _be present_. It 's not an | "either or." | nightski wrote: | Neither am I. I just feel like it's not true that | applicability is ignored. In fact I feel like books that | demonstrate applicability are far more numerous than the | theoretical ones. | | Set theory for example (since you mentioned it) is | introduced in many probability books in an applicable | fashion since probability is concerned with events which | can be modeled as sets of outcomes. So while you might not | find a book directly on "set theory applicability in the | real world" you can find many introductions to it in a | particular domain which are applicable. | nanidin wrote: | The beauty of Wikipedia is that you have the power to improve | and make changes as you see fit. | | I know it's a meme that edits to Wikipedia are fraught with | editor politics and reversion of good faith changes made by | users, but that hasn't been the case for me lately. If you have | something to add or improve, don't let anything hold you back! | analog31 wrote: | I'm the odd counterexample, where _proofs_ were what made math | come alive for me. I suppose you could say that the utility of | math is in furnishing tools for proofs, but that 's probably | not what most people mean. ;-) | | Utility was for physics, which I also majored in. | pstuart wrote: | > math is often not introduced in a concrete manner, as well as | in a way such that its utility is obvious | | This cannot be emphasized enough. Most people won't be doing | any higher order math in their lifetime -- we should be | teaching math literacy and "real world" math for the masses | rather then pretending that every student is going into | physics. | NovemberWhiskey wrote: | And we should be clear here that the bar for improvement is | verrrrry low. Take a look at the questions asked in the | National Financial Capability Study, and the results: | | https://www.usfinancialcapability.org | | The survey is composed of five really-quite-basic questions | about interest rates, inflation and risk assessment. | | As of 2018, 66% of survey participants get 3 or fewer of the | questions right. | coliveira wrote: | The role of university is exactly to teach you the language of | math, or at least enough that you can get more by yourself. If | your professors spent all the time talking only about concrete | examples, you would be completely at a loss about why math is | like it is, from the point of view of the untrained person it | would be just cargo cult. So, math is hard, but the university | has the responsibility to teach it (at least the introductory | material) to as many people as possible. | madhadron wrote: | The problem is that professional mathematicians have a common | language that they all have learned which serves them | well...and they're the ones writing the Wikipedia articles. | Imagine if all the programming documentation you worked with | was at the level of a 1980's first book on BASIC. | | Set theory _is_ the concrete starting point after that | training, and the first step of concrete problems is to | translate them into an abstract form of maps on sets. It 's a | decoupling. Instead of translating n techniques into m domains | (n*m bits of work), you develop n techniques in terms of set | theory, and translate m domains into set theory (n+m bits of | work). | | Physics majors largely use the same pieces of math on the same | domains, so this decoupling doesn't make sense for them. | Similarly, most domains carve off some piece of math and | statistics and specialize it. But if you're writing a | reference, whose specialty do you choose? | anon_tor_12345 wrote: | >Imagine if all the programming documentation you worked with | was at the level of a 1980's first book on BASIC. | | mathematical maturity is the same as "code sense". when i | started writing code a couple of years ago i would get cross- | eyed reading large blocks of code i.e. i would get lost in | the syntax and the abstractions and the idioms. at the same | time i had pretty decent "mathematical maturity" i.e. i could | read papers and textbooks pretty handily. comparing it seems | obvious that formal mathematics, with its idioms, | abstractions, and syntax is basically the same thing (without | pushing the curry-howard isomorphism too far). | aaron-santos wrote: | Most computer science students get a set theory introduction | and construction of natural numbers, rationals, and reals. | Those with an interest in statistics get a foundation laid in | measure theory. For other domains, what are good places to | look for set-theoretic approaches? A set-theoretic approach | to calculus has me intrigued. | contravariant wrote: | If you define the probability function you'll need set theory | within about 5 minutes. | | If you want to define the probability measure you'll need to be | pretty comfortable with set theory. | skipants wrote: | I suppose it's tough to balance detail with complexity when it | comes to a general wiki like Wikipedia. Especially with math | and its domain language. | | I believe that the issue you're outlining was the precursor to | simple.wikipedia.org. Sorry... I couldn't come up with a math | example on the spot but here's a good CompSci example: | | https://simple.wikipedia.org/wiki/Dijkstra's_algorithm | segmondy wrote: | It's difficult tho, the idea is the student should want to | learn that they learn no matter how boring because the | teacher/institution says so. I remember when I got introduced | to matrix in high school, it felt pointless and stupid. Why | would I want to transform matrix? I hated the entire thing, | sure I could solve it but for what purpose? Then I learned I | could use it for 3D engine, and I could load my object in a | matrix and transform to move things around, that got my | attention. However, a student that has no interest in 3D | engines won't care. How then do you make it exciting? It's not | enough to make the problem concrete, but the student will also | need to be interested in the concrete application of the | subject. Teaching is hard! | laichzeit0 wrote: | Maybe it was taught in the wrong order. Matrix multiplication | makes perfect sense if you see (a) that linear | transformations can be represented as matrices, I.e A = f, B | = g and (b) you define matrix multiplication so that the | composition of linear transformations I.e. f(g(x)) = ABx | gives the same result. That's a pretty cool idea, that you | can represent a function by a matrix and that you can compose | those functions and the composition is the same as | multiplying the matrices together. So let's say f(x) computes | the derivative of a polynomial, and you have a matrix | representation for that, say A. Now what if I want the second | derivative? That would be f(f(x)) or just AAx. | | I suppose I would tell a student that asks "why would I want | to remember the stupid rule for multiplying matrices together | in that way?" to try and figure out a rule for multiplying | two matrices together such that you could represent | functional composition by it, and they would self-discover | the matrix multiplication rule and see why it would be useful | to do it that way. | segmondy wrote: | It's not that it didn't make sense, it's that it was | boring! Of what use is it? Sure, you have composition of | linear transformations, but of what use is it again? If | there's no use, it's boring. Pure math is like a puzzle, | some people really love to solve the puzzle. The joy is in | the solving the puzzle, and then for some people the joy | only manifests if it's applied. Point of the original post | being that math is harder for students to accept in its | pure form if not applied. | zoomablemind wrote: | Matrix calc organically fits with systems of linear | equations. Hard not to appreciate its expessiveness and | beauty. | | If one was already exposed to the need of solving the | linear systems, then the matrix calc becomes of a direct | utility. | at_a_remove wrote: | Absolutely. As a private tutor, finding applicability and | motivation for students was quite a challenge. I spent a lot | of time reformulating problems in terms of things they might | care about. | Tainnor wrote: | > You can see this problem in most Wikipedia math pages. | | This gets mentioned so often that I feel it hints at a deep | misunderstanding at what Wikipedia is or is supposed to be. | | Wikipedia is an _encyclopedia_. It 's not a teaching resource, | and it shouldn't be. (Nor, for that matter, is it a primary | source, which makes it useless as a reference if you're writing | a paper, unless you're specifically writing about Wikipedia.) | | Mathematics is such a ubiquitous language. Different people | will require different kinds of mathematics, for vastly | different purposes and in various levels of depth and | formality, there's no way this can be unified into a single- | size-fits-all resource. If you want to learn mathematics, now | more than ever there is such an amazing breadth of text books, | blog posts, lecture videos, online communities and so on that | you can use depending on your very specific needs. I don't know | why people go and try reading up on mathematics from Wikipedia, | out of all places. | state_less wrote: | I've found it helpful to use probabilities throughout my life. | When picking a career, it's helpful to know the graduation rate | and the likely salary if you do graduate to calculate the | expected value of pursuing any particular career. I started | playing poker and in order to be profitable, I learned the odds | of various situations. | | Sometimes folks will be afraid of things, even though the odds | are they'll be okay. Knowing the odds can give you courage | where others find it difficult to go beyond fear. | | I've been a fan of Peter's AI book(s) since my university | coursework. Glad he's introduce so many of us to these helpful | ideas. Basic ideas can go a long way if you learn where they | apply to your life. | agumonkey wrote: | AFAIK logo (the old turtle language) was born as a way to | address this. Papert and his buddies wanted to turn thinking | into semi tangible forms. More interactions, more inputs to | feed your brain to think more. | | Also, maybe it's only me, but mathematicians often forgot their | own culture. Or maybe it's due to the iconic status of the | field during early education, making people never explain why | they do the things they do. When you read about history of | mathematics you see that problems were 1) very practical ones | 2) first tricks were very natural. It is not a thing from the | gods.. at least it didn't start like this.. it condensed into a | diamond over centuries of refinements. But if you don't show | that to the crowd, you lose 90% of the audience. It's a pity. | Topgamer7 wrote: | I graduated with a computer science major, and I've taking | courses on linear algebra. It didn't really hit me about | practical applications of linear algebra until I watched a | video on youtube by Stuff Made Here[1]. | | Which is funny because I totally recognize now how it can be | applied, but moving from system of equations on paper to real | world applications never really clicked in usefulness until it | was explained in that video. | | 1: https://www.youtube.com/watch?v=myO8fxhDRW0. | joppy wrote: | I agree with you on the incomprehensibility of mathematics | Wikipedia pages, even as a working mathematician I can really | only read the pages about content I already know in-depth. But | Wikipedia (and many other reference-type sources) are not the | place I would go to learn a new topic in maths, I would always | prefer to read a short book aimed at the correct level of | knowledge, or a set of course notes (many of which are | available online). I don't think Wikipedia is representative of | the teaching of maths at all. | | Many of the courses on very abstract content I've taken (from | other people, in person) has been very concretely introduced, | or at least the course has been run with a 50/50 split on the | abstract (general theory) and the concrete (solving problems). | screye wrote: | Shoutout for another great online MOOC : | https://www.edx.org/course/probability-the-science-of-uncert... | (It is the same as MIT OCW's 6.0.41) | | I did it as preparation for my Masters and it was genuinely | helpful. Would recommend it to everyone looking to do a prob 201 | before taking advanced-ish courses. | beforeolives wrote: | I would recommend Harvard Stat 110 over MIT's probability | courses - https://www.edx.org/course/introduction-to- | probability (you can find full lectures on youtube and the book | online) | jackallis wrote: | do you the link to youtube. Cant' find it; that why i gave up | on the class. | beforeolives wrote: | Youtube playlist - https://www.youtube.com/playlist?list=PL | 2SOU6wwxB0uwwH80KTQ6... | SeaWhales1000 wrote: | I found the EdX course too rushed (assuming you don't pay for | the verification to get lifetime access). I like Bliztstein, | so I instead used his Youtube playlist [0]. It has his full | lectures. Also, his book is free to view [1]. | | [0] https://youtube.com/playlist?list=PL2SOU6wwxB0uwwH80KTQ6h | t66... | | [1] https://drive.google.com/file/d/1VmkAAGOYCTORq1wxSQqy255q | LJj... | someguy101010 wrote: | How come? | conjectures wrote: | Blitzstein is a good lecturer. I think I did this course | back in the day. | jypepin wrote: | Everything I see from Peter Norvig is just always so incredibly | well written and coded. I'd love to know what it's like to work | for him. | | Every year looking at his solutions for advent of code [0] brings | just so much learnings. Strongly recommend. | | [0] | https://github.com/norvig/pytudes/blob/master/ipynb/Advent%2... | mikevin wrote: | > Everything I see from Peter Norvig is just always so | incredibly well written and coded. .... | | I feel his skill of dividing a problem into small pieces and | expressing them in code in a natural way is unparalleled. Most | books/blogs/articles I see often focus on one of two patterns. | | The most frequent one is pulling in some dependencies and using | a high level API, essentially skipping any real problem | solving. Great when you just need a problem solved and are | familiar with some framework/library but not that great for | learning to program or problem solving | | The other one is a deep dive into a data structure, algorithm | or performance tuning. This is great when studying theory or | optimizing. These articles are more interesting but I haven't | encountered many people who are in a position where this is | relevant to day to day work. | | The missing pattern is one where Peters' work shines. The parts | in between. All the libraries that are used in the first | example I described are the result of someone taking the | building blocks that result from the second example and | applying them to a real world problem. Peter Norvig is my go to | recommendation when someone is interested in becoming better at | solving day to day problems because of this. | the_decider wrote: | The following data science book also does a great job of | balancing problem solving with underlying theory. | https://www.manning.com/books/data-science-bookcamp And it | starts with sample-space probability problems in Python, much | like Peter's tutorial. | abecedarius wrote: | The times I had a peek behind the curtain, he didn't expect to | stop with the first version, even though it'd be good. | | (Maybe those Advent of Code solutions are the first working | draft, I don't know.) | xxdd8378yjk wrote: | thanks | da39a3ee wrote: | This is some extremely stylish expository python (as most of the | other comments are saying). | javier10e6 wrote: | I read portability and rush to the link waiting to hear about pip | and all other mayhems and I landed on a math page. Boy I need my | coffee. | sillysaurusx wrote: | Similarly, I thought "from fractions import Fraction" required | a `pip install fractions`. But nope, turns out fractions is a | built-in module I've never heard of. Neat! | beforeolives wrote: | Similarly, one of the most suprising things for me was that | complex numbers are built into the language (no imports at | all). | sillysaurusx wrote: | Be sure not to miss the other two probability notebooks: | | - Probability, Paradox, and the Reasonable Person Principle | https://github.com/norvig/pytudes/blob/master/ipynb/Probabil... | | - Estimating Probabilities with Simulations | https://github.com/norvig/pytudes/blob/master/ipynb/Probabil... | | There are dozens of other notebooks on a variety of topics in the | 'ipynb' folder: | https://github.com/norvig/pytudes/tree/master/ipynb | cousin_it wrote: | Nice! I had fun solving this problem in my head: | | > _I have two children. At least one of them is a boy born on | Tuesday. What is the probability that both children are boys?_ | | An interesting thing about this problem is the unspoken | assumption of what happens in other counterfactual worlds. If | the person always answers the question "is one of your kids a | boy born on Tuesday?" then the problem is solvable. But if a | different family history would've caused the person to answer a | different question ("born on a Monday" instead of Tuesday), | then the answer would depend on the person's algorithm. Eliezer | gave a dramatized explanation here: | https://www.lesswrong.com/posts/Ti3Z7eZtud32LhGZT/my-bayesia... | | Further on this path, there are seemingly basic questions that | cause disagreement among actual statisticians. For example, see | the voltmeter story in | https://en.wikipedia.org/wiki/Likelihood_principle: | | > _An engineer draws a random sample of electron tubes and | measures their voltages. The measurements range from 75 to 99 | Volts. A statistician computes the sample mean and a confidence | interval for the true mean. Later the statistician discovers | that the voltmeter reads only as far as 100 Volts, so | technically, the population appears to be "censored". If the | statistician is orthodox this necessitates a new analysis. | However, the engineer says he has another meter reading to 1000 | Volts, which he would have used if any voltage had been over | 100. This is a relief to the statistician, because it means the | population was effectively uncensored after all. But later, the | statistician ascertains that the second meter was not working | at the time of the measurements. The engineer informs the | statistician that he would not have held up the original | measurements until the second meter was fixed, and the | statistician informs him that new measurements are required. | The engineer is astounded: "Next you'll be asking about my | oscilloscope!"_ | vlovich123 wrote: | > In the correct version of this story, the mathematician | says "I have two children", and you ask, "Is at least one a | boy?", and she answers "Yes". Then the probability is 1/3 | that they are both boys. | | I don't understand this reasoning. If at least one is a boy, | the only configurations I can think of is 1 boy 1 girl or 2 | boys. Where does the 1/3 come from? | tijsvd wrote: | With 2 children, there are 4 configurations of equal | probability. The one with 1 boy 1 girl occurs twice. Take | away the 2 girl case, then 2 boys is 1 in 3. | vlovich123 wrote: | Yeah, the way the problem is formulated though there's | absolutely no indication that order matters so how are | there two configurations within which there's 1 boy and 1 | girl? | da39a3ee wrote: | Order doesn't matter in the sense that the observed data | set is unordered (just counts of girls and boys). What | matters is how many ways there are that the universe can | give rise to those unordered data sets. And in fact, | there are more ways that the universe can give rise to | the unordered state 1 boy 1 girl, than to the unordered | state 2 boys. For similar reasons , there are more ways | in which your papers can be in a mess across your desk | than ways in which your papers can be neatly piled up. | | And to count how many ways the universe can give rise to | the unordered data sets, the usual technique is to expand | the unordered data sets into all the equivalent ordered | data sets, and count the latter. | kgwgk wrote: | Because the order exists even if it doesn't matter (at | least for two children, maybe not for two quantum | particles). | | With the risk of being accused of binarism, there are | four distinct possibilities with (close to) equal a | priory probability of 25%: older boy/younger boy, older | boy/younger girl, older girl/younger boy, and older | girl/younger girl. | | Discarding the girl/girl case leaves three equally | probable cases. | dalmo3 wrote: | I immediately modelled the problem like you did, then I | thought of this interesting variation: | | "I have two children, Michael and Alex. Michael is a boy. | What's the probability of both being boys?" | | If you make a truth table with names as columns, you | clearly have only two possibilities for Michael=1. | | However if you pick older/younger again you're back to 3 | possible states. | | I think the answer is still 1/3, but it's a trickier one | to reason about immediately. | | It seems the question adds information by naming the | children, but there's a hidden statement in the form "at | least one of them is Michael", which invalidates a truth | table with names as columns. | | I can only conclude that birth order is an underlying | property of the entity. A strict, real differentiator as | much as sex is. Names aren't, so names don't add | information in this case. | | Is there a term for that? Or am I just wrong? | maxov wrote: | Another way to think about it is counting the probability | of getting k boys out of 2 children. 0 | boys - 1/4 1 boy - 1/2 2 boys - 1/4 | | There's a half chance of getting exactly one boy, and one | way to calculate this is by noticing there are two | different ways to get one boy if we take order in | account. You are right that the orderings don't matter in | this case, so we could also e.g. model this with a | binomial distribution. Once you know there are >= 1 boys, | the chance you have two is 0.25/(0.25+0.5) = 1/3. | benlivengood wrote: | Maybe I'm missing something about the voltmeter example. My | assumption is that the 100-volt-maximum meter can distinguish | between 100 volts and more than 100 volts, in which case | there's no problem. If the voltmeter doesn't accurately | indicate whether or not a measurement is outside of its range | then the statistician is correct that everything should be | re-measured. | | Do some people think that the possibility of not being able | to take accurate measurements is the same as not having taken | accurate measurements? | | EDIT: Maybe the ambiguity is in what the engineer would have | recorded if finding a voltage >100 volts while the other | meter was broken? It's like undefined behavior in | programming; if you know your software will have undefined | behavior when encountering certain data then you can't trust | whether the output is valid unless there's independent | confirmation that the data won't cause undefined behavior. If | the statistician doesn't have certainty that the engineer | will have defined behavior (e.g. say "I couldn't complete the | measurements" vs. undefined behavior like writing down "99" | or exploding) then they of course want to re-measure. | senthil_rajasek wrote: | This is a cute introduction to probability. However, I would've | loved to see some mention of dependent events and continuos | probability. | cesarosum wrote: | I do think that there's some merit in sticking with probability | on discrete spaces for a while. Once you start dealing with | continuous spaces, soon you're talking measure theory and you | can wade deep into the technical details and miss some | understanding of what's going on. I go back and forth on this | as I think it's largely down to the reader to figure out what | works for them, but I think probability is one of those fields | where developing intuition early on is a must if you want to go | further. | beforeolives wrote: | The actual requirement for measure theory is overblown. As | long as you've taken single and multivariable calculus, you | can study continuous probability without any problems and | without even knowing what measure theory is. | cesarosum wrote: | Agreed, not knowing measure theory never stopped me from | computing a conditional expectation. Some courses and books | overemphasize rigor in probability and, while it obviously | has its place, I've seen newcomers to the field become | obsessed with doing everything via measure theory. Further | to your point, volume two of Feller is pretty light on | measure theory IIRC. | scribu wrote: | It does have a section on continuous probabilities at the end. | senthil_rajasek wrote: | Yes, In the appendix thanks. | khazhoux wrote: | I'm always amazed at how Peter Norvig continues to create the | kind of content that a top grad student would do, even as his | career and rank in the industry has skyrocketed. | | Back in university and grad school I would write tutorials and | post them online (and still get thanks from random people many | many years later). I would explore random interesting subjects | and dive deep. I would constantly publish code and demos, etc. As | my career grew, one of one these started to fall off. I look at | my peers and it's the same story: they were all vibrant hot-shots | in their early-mid 20s, and now are just weighed down by the | teams and projects they manage. | | Peter is an inspiration. I will ponder this... ___________________________________________________________________ (page generated 2021-06-03 23:00 UTC)