[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...
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