[HN Gopher] Probability 101, the intuition behind martingales an...
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Probability 101, the intuition behind martingales and solving
problems with them
Author : cpp_frog
Score : 120 points
Date : 2023-02-23 11:30 UTC (11 hours ago)
(HTM) web link (codeforces.com)
(TXT) w3m dump (codeforces.com)
| sn41 wrote:
| Many have been asking for an intuitive explanation of
| martingales. The following is the historical context, which helps
| understand some of the motivation.
|
| suppose you have a fair coin, and you bet 1 dollar that it comes
| up heads. if it does, then you get 2 dollars, and you gain 1
| dollar and exit.
|
| suppose you lose. then put $2 on the next being head. if you win,
| you get double, for a gain of $4-($2+$1)=$4-$3=$1.
|
| on third toss, put $4. if it comes up heads, you get $8. the gain
| is $8-$7=$1.
|
| so every time you lose, double the previous bet. you can exit
| with $1 gain on your first win.
|
| this was the original, flawed, martingale strategy. what's wrong?
|
| then we realize that there is one outcome where you lose all your
| stake. On that path, your loss is exponential in the number of
| rounds!
|
| another way to state that is that the expectation of the money
| after any round is $1, your initial stake. This is where the
| insight of expected value being preserved comes from.
| kqr wrote:
| Confusingly, while this betting strategy is the original
| "Martingale", it's not really what's meant by the term in
| probability theory.
|
| An intuitive explanation of martingale in the more technical
| sense is simply that "your best guess for the future value of a
| series is its current value". I.e. on average it won't deviate
| up or down from where it is now, regardless of past values.
|
| But, crucially, that applies at any given time. It means that
| every time the value moves up and down, you should reset your
| expectation and believe that whatever value it is now is the
| new normal.
|
| Some things that are not martingales are
|
| - processes that are biased to move more up than down, or vice
| versa,
|
| - processes that are mean-reverting, i.e. when they have gone
| up for a while they are likely to go down again, and
|
| - more generally, processes whose future value can be predicted
| better by using more information about their recent past.
| sn41 wrote:
| Thank you for these nice intuitions, especially the negative
| examples. I liked the second point.
| spapas82 wrote:
| This gives a mathematical martingale. I only know the betting
| strategy martingale
| https://en.m.wikipedia.org/wiki/Martingale_(betting_system)
| hintymad wrote:
| It's pretty amusing that smart people would think that sigma
| algebra and the concept of space, let alone equivalent space and
| random variables in the language of sigma algebra, were
| "intuitive" in a Prob 101 _tutorial_.
| ajkjk wrote:
| it's crazy how unhelpful all the sigma algebra theory is for
| this. It changes none of the logic or intuition; it just
| obfuscates the hole theory and makes it inaccessible.
| thomasahle wrote:
| I agree a "Probability 101" article shouldn't use sigma
| algebras. It definitely cuts off 99.9% of the people that might
| benefit from the article.
|
| Not that sigma algebras don't have a place. If you are actually
| doing advanced things with martingales, sigma algebras are nice
| clean framework to express what you are doing.
| throwaway81523 wrote:
| It looks like a good article for those who are already
| studying theoretical probability. For laypeople, not so much.
|
| I haven't yet figured out what the author is doing with the
| sigma algebra machinery. I had thought that the event set of
| a probability space was intuitively the power set of the
| sample set, and that the machinery of sigma algebras was only
| added to avoid pathologies like non-measurable sets (think
| Banach-Tarski paradox). Maybe there is more to it?
|
| I would like to understand this better and I do plan to read
| the article, but I had found the Wikipedia article on
| martingales to be reasonably understandable a while back. The
| good thing about this article is it shows applications.
| FreeTrade wrote:
| The trouble with mathematical approaches to gambling is that
| while the expected value is taken into account, the emotional
| value of a bet is rarely considered.
|
| Take a lottery for example - from an expected value point of view
| it is of course foolish to play most of the time. But the
| emotional value to the player - he can spend a few dollars and
| spend the night imagining what he might do with his winnings.
| That's good value!
|
| With that in mind, I plan to play the martingale next time i go
| to Las Vegas. I will place a bet of a hundred dollars and rebet
| until I win a hundred dollars or lose 6300 dollars.
|
| When I win I shall enjoy dinner and drinks courtesy of the casino
| and feel very good 63 out of 64 nights. Once in a while I'll lose
| $6300 and feel terrible, but emotional value is summed quite
| differently to expected value, so it won't out weight the wins.
| throwaway81523 wrote:
| You want to read about the Kelly criterion for that. Russ
| O'Connor has a good article explaining why the lottery is a bad
| deal on that basis: http://r6.ca/blog/20090522T015739Z.html
| Keegs wrote:
| "The expected value is negative so playing the lottery is
| silly" also misses that someone looking for a way out of
| poverty and seeing a lack of alternatives might judge the
| opportunity cost of his 2 dollars and 5 minutes worth the
| vanishingly small chance of getting a payout. The choice gets
| harder to defend the more you play, but I think it's in bad
| taste to say lotteries are a tax on people bad at math.
| IIAOPSW wrote:
| Being in poverty and lacking in numeracy and other
| educational basics correlate. What you point out is no
| contradiction.
| arcticfox wrote:
| > Once in a while I'll lose $6300 and feel terrible, but
| emotional value is summed quite differently to expected value,
| so it won't out weight the wins.
|
| I'm not sure I'd trade two months of "free" dinners for one day
| of getting smacked $6300...! To each their own I guess.
| MrPatan wrote:
| One of my first programs calculated the odds of losing everything
| with a martingale strategy for a desired outcome and starting
| capital.
|
| It needed so much capital to get to a safe enough point that it'd
| be better to just put the money in the bank (but thinking back,
| this was back before 0 interest rates, so there's another lesson
| there for you....)
| outside1234 wrote:
| Is there a slightly less dense version of this for those of us a
| bit more removed from uni maths somewhere? Willing to read three
| times as much text to get the same insight. :)
| hansvm wrote:
| Martingales are a representation of a class of "do a random
| thing 'forever'" problems, with a self-similarity constraint
| and also the restriction that your reward is finite, both of
| which prevent them from being applied to most problems but make
| them vastly more computationally tractable. That they're well
| studied means we know some of those features.
|
| The author then mentions a few of those computational
| properties (like "Azuma's inequality") and gives a smattering
| of problems where directly applying them and ignoring most of
| the rest of the article would help build an intuition.
|
| The rest of the article is useful mostly because it puts all of
| the above in a more rigorous context and because if you're
| closer to uni maths it'll help provide some sort of intuition
| for why the restrictions were necessary and what sort of things
| might or might not be achievable with them. There's probably a
| less dense version somewhere, but I'm not quite sure what it
| would be. Hopefully this short summary helps you know what to
| look for when you find it.
| nerdponx wrote:
| The article posted here is kind of a crash course in
| mathematical probability, so don't feel bad for being
| intimidated or overwhelmed.
|
| The Wikipedia page gives a succinct explanation: https://en.m.w
| ikipedia.org/wiki/Martingale_(probability_theo....
|
| > ... the conditional expected value of the next observation,
| given all the past observations, is equal to the most recent
| observation.
|
| The book _Introduction to Probability Models_ by Sheldon Ross
| is a solid undergraduate-level introduction to this material.
| Martingales are covered in chapter 10.
|
| It does use plenty of math notation, but none of the abstract
| stuff here about triples, filtrations, etc. It might be a bit
| slow at first as you "reactivate" the mathy parts of your
| brain, but as long as you read with a pencil and notepad handy
| you should be able to work through it.
|
| There's actually a full upload of the 11th edition (current is
| the 12th I believe) on some university course webpage:
| http://mitran-lab.amath.unc.edu/courses/MATH768/biblio/intro...
| dataflow wrote:
| The one thing I've never seen any "introduction" to martingale
| cover is why you should (intuitively) be able to deduce anything
| useful about them at all, given the expectation is defined not to
| change (to clarify, I meant expected not to change). Especially
| when the first example is often the stock market, and everyone
| knows you can't predict movements of the stock market...
| CrazyStat wrote:
| >given the expectation is defined not to change.
|
| To be clear, in a martingale the expectation is equal to the
| current value. It changes as the current value changes (i.e. as
| time passes).
|
| >Especially when the first example is often the stock market,
| and everyone knows you can't predict movements of the stock
| market...
|
| Not being able to predict movements is exactly what "the
| expectation is equal to the current value" looks like. If you
| had information that changed your expectation to be different
| from the current value, that would be predicting a movement in
| one direction or the other.
| chinaman425 wrote:
| [dead]
| JustFinishedBSG wrote:
| > given the expectation is defined not to change.
|
| Sure the unconditional expectation doesn't change, but that's
| kinda useless because it's the expectation given that you know
| nothing. The interesting part is studying what is next given
| what I know right now i.e conditional expectations. And the
| martingale assumption i.e. "my best guest for tomorrow is the
| same as right now" is honestly a pretty sensible assumption for
| many things.
|
| If I tell you $TSLA is at 200 right now, it's not unreasonable
| to assume it will be around 200 tomorrow.
|
| If it's raining right now, it doesn't seem too far fetched to
| guess it will probably be raining in 1 minute.
|
| etc.
|
| And because you can prove so many things on martingales, it is
| often very very useful and powerful when you have something
| that isn't quite a martingale to think of a way to make it a
| martingale, prove _whatever_ and then go back to the original
| object.
| kqr wrote:
| > If it's raining right now, it doesn't seem too far fetched
| to guess it will probably be raining in 1 minute.
|
| That's a bit of an unfair example, though. If the Tesla stock
| is at 200 right now, the martingale property implies that I
| should expect it to be at 200 not just next minute or
| tomorrow, but also next week, two years from now, next
| decade, and so on. A martingale is not restricted in its time
| scale.
|
| (This is using clearly expectation in the technical sense.
| The stock price may well go up, or go down, but we can't tell
| which or how much, so in the grand scheme of things, we're
| better off assuming it won't move at all.)
| mizzlr_ wrote:
| To expect a value of 200 means to have the average of 200
| from this point in time onwards, assuming stock price is
| random walk. Not that the value tomorrow will be exactly
| 200. It could be 200, 201, 199, 202, 198 etc. the average
| expected is 200. If you possess no external knowledge such
| as insider information, then random walk is a sensible and
| obvious choice for stock price.
| ChainOfFools wrote:
| yeah there's a built in assumption that the behavior of the
| function we're estimating with martigale is continuous near
| the limit of the guess, and thus predictable over the
| interval of the guess and the prediction.
| time_to_smile wrote:
| > everyone knows you can't predict movements of the stock
| market...
|
| That's why you use a distribution to model the distribution of
| future possible prices given the information you have today.
|
| The entire point of modeling the market as a martingale: you
| _don 't_ know what the future price is, but you do know a
| pretty good deal about where the price might go at various
| point in the future. Perhaps the single most important thing
| you know is that the future price is _expected_ to be the same
| as the current price (ignoring the risk-free rate).
|
| The Martingale property is very important to understand about
| stocks because if you are certain that the expected price of a
| stock is less than its current value you should sell, if you
| believe it will be more you should buy. An entire market
| thinking like this means that the current price should be equal
| to the expected future price.
|
| Additionally these models don't "predict" future prices, but
| rather represent what the _market believes_ about future prices
| given the current price and other information. This is
| essential to properly modeling risk and pricing assets.
| nerdponx wrote:
| I like to see good math topics, but I really can't stand when the
| example problems are either totally fanciful or totally abstract.
| I'm too stupid for that. Give me an example of a practical
| problem in business, finance, engineering, social science, etc.
| that I can solve with this tool. Does the "dance party" problem
| have an analogue in real-world allocation and planning problems?
| Does a gambler gambling an _infinite_ number of times have
| practical applications in finance? Etc.
| Swizec wrote:
| > Does a gambler gambling an infinite number of times have
| practical applications in finance?
|
| If you're running a sovereign investment fund (or Softbank?),
| that's a lot like gambling with an infinite runway. Hopefully
| you're using part of that infinite runway to hire people with
| actual chops in probability and don't need this article.
| chinaman425 wrote:
| [dead]
| cjohnson318 wrote:
| I agree, I kept looking for an example of solving the problem
| that the post opened with: "how much longer until XYZ happens".
| Maybe I missed it?
| jrumbut wrote:
| I was expecting survival analysis to show up at some point.
|
| Perhaps it did, at some point I started to skim.
| behnamoh wrote:
| I wouldn't call this "intuition", more like a detailed and
| thorough explanation of the topic.
| chasebank wrote:
| Let's say I'm applying martingale on an index like SPY or QQQ,
| which I plan to hold otherwise in a tax free retirement account.
| If I run out of cash to double down, I'm in the same position as
| I would be otherwise, except I've averaged down my entry price.
| When the index eventually comes back up, you profit, and resume
| martingale. How could this be a losing strategy on an asset you
| plan to hold otherwise?
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