5 ms·

Bayesian updating of Probability Distributions

45 points by darkxanthos 13y ago

spicyj 13y ago
A question for people who know more about this than I do: Why was the uniform distribution on [0, 1] chosen initially? Choosing a different distribution would give a different result. (And it doesn't make much sense to say, "Always choose the uniform distribution!" because the choice of variable affects the meaning of the distribution -- if instead we wonder about the value of p^2 and choose a uniform distribution for it on [0, 1], won't we get a completely different result?)
- christopheraden 13y ago
  We're interested in the probability of a coin flip yielding heads before we flip any coins. Uniform is just a really common prior to choose in this situation for a few reasons: -It's a special case of the beta distribution, which is the conjugate prior for binomial problems. This means that the distribution of the probability of getting heads given the coin flips is in the same family as the prior itself (ie: beta priors with binomial likelihoods yield beta posteriors). -The uniform (for this problem at least) is an "objective prior", which expresses that we don't have much information about whether the flip is biased. The example you give (modeling p^2 instead of p) is a great example of when the uniform would be a bad choice. The reason the uniform doesn't work in this case is because for binomial data (coin flips), a uniform prior is not invariant to reparametrization. If choosing priors was so simple as always going with the uniform, there'd be little reason to go with Bayes! The choice of prior sometimes makes a radical difference in the posterior (especially with small samples), and there's many things to consider when you choose priors (computational convenience, uninformative versus informative priors, hierarchical modeling, etc). http://en.wikipedia.org/wiki/Jeffreys_prior http://en.wikipedia.org/wiki/Jeffreys_prior http://en.wikipedia.org/wiki/Beta_distribution http://en.wikipedia.org/wiki/Beta_distribution http://en.wikipedia.org/wiki/Conjugate_prior http://en.wikipedia.org/wiki/Conjugate_prior
- bjterry 13y ago
  If you were holding a physical coin in your hand, you would have to be crazy to select a uniform distribution unless it was shaped like a sphere. This is sort of a really minor pet peeve of mine when people use coin-flipping as an example for these things. If it's even vaguely coin-like, even the most ridiculous distortion (maybe it's made of uranium on one side and aluminum on the other) probably couldn't bring the true probability past 70% or something.
  - spicyj 13y ago
    So what would you use as a reasonable prior here?
    - dthunt 13y ago
      For argumentative purposes (if trying to reach a consensus with someone who claims to be 95% sure that the coin is 'fair' and accepts the '45-55% heads is fair' definition), you would concentrate most of your probability mass there, and find a mutually agreeable shape. The 'it's not shaped like a sphere' thing suggests that some people would take into account visual information about the object to inform their priors, and be prone to rejecting the fitness of the uniform. That may conflict with your sensibilities if you expect very strange coins, but if the priors aren't too crazily bad, if you have enough time for a few more coin flips, and the coin isn't particularly brittle, what's an order of magnitude between friends? Edit: And for formal mathy purposes, ask someone else :)
    - bermanoid 13y ago
      If it's a normal looking coin being flipped in the air and caught before presentation, the only reasonable prior is that almost all of the probability mass should be that it's completely fair. Maybe I'd leave 1% to spread amongst the rest just in case the adversary has some seriously devious tricks up their sleeve. Because physics - it's not possible to bias a rigid body so that it rotates with non-constant angular speed when flipped, as long as air resistance can be neglected, and that means that a fair flip gives 50/50 odds as to what side you catch it on. (Edit: clarity) If other stuff is going on, like you're letting it bounce or something, then it depends on the particulars. It's rather easy to load a die, for instance.
- bluecalm 13y ago
  We would! Choosing priors is difficult and different people with different experience and views will have different (but still reasonable) priors. You may take personal (just use all your experience) or objective (try to come up with some scheme for assigning priors to types of situations) view on this problem. Work was done in both directions. It's more of a philosophical problem than math problem, although in many cases you want something specific: "I have no idea what the outcome might be but I think it's either extremely on the right or left so give me priori which converges fast to one or the other). Or in a coin example: "it's almost surely more or less fair, so I don't want to change my views after several unlucky flips".
- darkxanthos 13y ago
  Great question! With just the few updates I've given you're probably right that it would affect things significantly. However, the more data you have the less the prior matters. This is known as swamping the priors. In this case a uniform prior isn't incorrect, but you could definitely say its suboptimal and that I could make my examples much more accurate by choosing a prior that most represents my initial beliefs (like the one head one tail histogram for example.
maaku 13y ago
A good general introduction to Bayesian updating: http://yudkowsky.net/rational/bayes/ http://yudkowsky.net/rational/bayes/
tomrod 13y ago
I love seing people figure this out for the first time. Bayesian methods just start to make more sense after working through something like this.
gtani 13y ago
for folks looking to delve into the subject, Barber and McKay's freely available content books are terrific http://web4.cs.ucl.ac.uk/staff/D.Barber/pmwiki/pmwiki.php?n=Brml.HomePage http://web4.cs.ucl.ac.uk/staff/D.Barber/pmwiki/pmwiki.php?n=... http://www.inference.phy.cam.ac.uk/itila/ http://www.inference.phy.cam.ac.uk/itila/
- liranz 13y ago
  If you want a lot of Python examples, you should check out this great book: http://camdavidsonpilon.github.io/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/ http://camdavidsonpilon.github.io/Probabilistic-Programming-...
skybrian 13y ago
This code has no memory other than the prior probability distribution. It seems like if you had previously flipped a coin a thousand times to generate it, your prior beliefs should be more strongly held than if you had just made up some numbers. Shouldn't the number of previous trials be accounted for somehow?
- dthunt 13y ago
  In fact, it is accounted for. You'll notice the negative exponent on the unlikely hypotheses is getting pretty extreme after a handful of flips.
  - bermanoid 13y ago
    In case it's not clear (that wording doesn't really click for me, since there are no exponents involved), the way this information is encoded is in the shape of the distribution itself. If there have been very few observations, it will be very wide, but after many it will narrow to a tiny spike. In this context, at least, the prior distribution encodes everything - there's no meaning to the idea that you're more or less confident in the prior, because the prior already represents your uncertainty about the outcomes. If you were 50/50 on this prior versus another one, then your actual prior would be the average of the two. There is a subfield of statistics that deals with imprecise probabilities, but that's a whole other can of worms and doesn't really relate to this problem. That said, it's fascinating stuff, and very useful in some contexts (if you're uncertain about your priors, it can be useful to do sensitivity analyses to figure out exactly how the end result depends on your prior).
    - dthunt 13y ago
      What I meant was that for h_1%, you're going to wind up with probabilities in the area of like, 1.2e-34, very quickly, if the coin is a fair one. The update process is adjusting via multiplication, and for a really bad hypothesis, that's going to bring its probability dramatically close to zero without that many trials. Even though the absolute difference between 1e-4 and 1-e34 hypothesis feels smallish when you look on a linear scale from 0-1, a 1-e34 is a lot 'stickier'. Your explanation has the benefit of being a better explanation; my goal was just to explain where the inertia was hiding.
bluecalm 13y ago
Once you figure out simple examples you slowly start thinking this way about the world. It's beautiful. Take people for example: Someone with open mind has a priori with at least slight probability assigned to unlikely (for them!) hypothesis while on the other hand very religious people for example have 0 in their priori when it comes to possibility of their religion being made up so they are forced to ignore evidence to the contrary (because bayesian updating breaks for them due to division by zero and mind's way to signal this exception is denial). In general someone with a lot of weight on given hypothesis is "stubborn" or just very convinced and someone with uniform or close distribution just doesn't know anything about given problem. Someone unable to build heavily weighted distributions is a conspiracy theorist, someone reluctant to - a sceptic and someone too much eager to a fanatic. Someone with very bad priors is un/badly educated (in given domain) or biased or maybe just stupid, someone with good priors is an expert. It's possible to combine expert with sceptic attitude or expert with fanatic or all too often stupid with fanatic (very bad and very heavily weighted priors with possible 0's on some options). Once you start thinking this way you start expressing yourself differently, you start adding those probability qualifiers to your sentences: "I am very sure it's the way to go", "My intuition tells me this but I am not really sure", "I am very convinced and it's not worth discussing" (yes, it can be rational and good attitude) or "I would do X but I need more evidence to be reasonably sure". It's all there in people's mind, language and interactions once you start thinking this way it's whole new world of perspective and understanding.
- darkxanthos 13y ago
  Yup it's true. I can seem like a Bayes nut at work as I mention updating my priors during debates and when discussing hypotheses for split tests.
- yafujifide 13y ago
  Completely agreed. Your comment is a really good advertisement for normal people to learn Bayesian probability theory. Not because it's useful in science or engineering, but because it's useful in life.
davidf18 13y ago
A good (free) explanation of Bayesian Stats using Python: Downey's Think Bayes http://www.greenteapress.com/thinkbayes/ http://www.greenteapress.com/thinkbayes/
- darkxanthos 13y ago
  This helped me the most to get started.

45 points by darkxanthos 13y ago

A question for people who know more about this than I do: Why was the uniform distribution on [0, 1] chosen initially? Choosing a different distribution would give a different result. (And it doesn't make much sense to say, "Always choose the uniform distribution!" because the choice of variable affects the meaning of the distribution -- if instead we wonder about the value of p^2 and choose a uniform distribution for it on [0, 1], won't we get a completely different result?)

christopheraden 13y ago
We're interested in the probability of a coin flip yielding heads before we flip any coins. Uniform is just a really common prior to choose in this situation for a few reasons: -It's a special case of the beta distribution, which is the conjugate prior for binomial problems. This means that the distribution of the probability of getting heads given the coin flips is in the same family as the prior itself (ie: beta priors with binomial likelihoods yield beta posteriors). -The uniform (for this problem at least) is an "objective prior", which expresses that we don't have much information about whether the flip is biased. The example you give (modeling p^2 instead of p) is a great example of when the uniform would be a bad choice. The reason the uniform doesn't work in this case is because for binomial data (coin flips), a uniform prior is not invariant to reparametrization. If choosing priors was so simple as always going with the uniform, there'd be little reason to go with Bayes! The choice of prior sometimes makes a radical difference in the posterior (especially with small samples), and there's many things to consider when you choose priors (computational convenience, uninformative versus informative priors, hierarchical modeling, etc). http://en.wikipedia.org/wiki/Jeffreys_prior http://en.wikipedia.org/wiki/Jeffreys_prior http://en.wikipedia.org/wiki/Beta_distribution http://en.wikipedia.org/wiki/Beta_distribution http://en.wikipedia.org/wiki/Conjugate_prior http://en.wikipedia.org/wiki/Conjugate_prior
bjterry 13y ago
If you were holding a physical coin in your hand, you would have to be crazy to select a uniform distribution unless it was shaped like a sphere. This is sort of a really minor pet peeve of mine when people use coin-flipping as an example for these things. If it's even vaguely coin-like, even the most ridiculous distortion (maybe it's made of uranium on one side and aluminum on the other) probably couldn't bring the true probability past 70% or something.
- spicyj 13y ago
  So what would you use as a reasonable prior here?
  - dthunt 13y ago
    For argumentative purposes (if trying to reach a consensus with someone who claims to be 95% sure that the coin is 'fair' and accepts the '45-55% heads is fair' definition), you would concentrate most of your probability mass there, and find a mutually agreeable shape. The 'it's not shaped like a sphere' thing suggests that some people would take into account visual information about the object to inform their priors, and be prone to rejecting the fitness of the uniform. That may conflict with your sensibilities if you expect very strange coins, but if the priors aren't too crazily bad, if you have enough time for a few more coin flips, and the coin isn't particularly brittle, what's an order of magnitude between friends? Edit: And for formal mathy purposes, ask someone else :)
  - bermanoid 13y ago
    If it's a normal looking coin being flipped in the air and caught before presentation, the only reasonable prior is that almost all of the probability mass should be that it's completely fair. Maybe I'd leave 1% to spread amongst the rest just in case the adversary has some seriously devious tricks up their sleeve. Because physics - it's not possible to bias a rigid body so that it rotates with non-constant angular speed when flipped, as long as air resistance can be neglected, and that means that a fair flip gives 50/50 odds as to what side you catch it on. (Edit: clarity) If other stuff is going on, like you're letting it bounce or something, then it depends on the particulars. It's rather easy to load a die, for instance.
bluecalm 13y ago
We would! Choosing priors is difficult and different people with different experience and views will have different (but still reasonable) priors. You may take personal (just use all your experience) or objective (try to come up with some scheme for assigning priors to types of situations) view on this problem. Work was done in both directions. It's more of a philosophical problem than math problem, although in many cases you want something specific: "I have no idea what the outcome might be but I think it's either extremely on the right or left so give me priori which converges fast to one or the other). Or in a coin example: "it's almost surely more or less fair, so I don't want to change my views after several unlucky flips".
darkxanthos 13y ago
Great question! With just the few updates I've given you're probably right that it would affect things significantly. However, the more data you have the less the prior matters. This is known as swamping the priors. In this case a uniform prior isn't incorrect, but you could definitely say its suboptimal and that I could make my examples much more accurate by choosing a prior that most represents my initial beliefs (like the one head one tail histogram for example.

maaku 13y ago

A good general introduction to Bayesian updating: http://yudkowsky.net/rational/bayes/ http://yudkowsky.net/rational/bayes/

tomrod 13y ago

I love seing people figure this out for the first time. Bayesian methods just start to make more sense after working through something like this.

gtani 13y ago

for folks looking to delve into the subject, Barber and McKay's freely available content books are terrific http://web4.cs.ucl.ac.uk/staff/D.Barber/pmwiki/pmwiki.php?n=Brml.HomePage http://web4.cs.ucl.ac.uk/staff/D.Barber/pmwiki/pmwiki.php?n=... http://www.inference.phy.cam.ac.uk/itila/ http://www.inference.phy.cam.ac.uk/itila/

liranz 13y ago
If you want a lot of Python examples, you should check out this great book: http://camdavidsonpilon.github.io/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/ http://camdavidsonpilon.github.io/Probabilistic-Programming-...

skybrian 13y ago

This code has no memory other than the prior probability distribution. It seems like if you had previously flipped a coin a thousand times to generate it, your prior beliefs should be more strongly held than if you had just made up some numbers. Shouldn't the number of previous trials be accounted for somehow?

dthunt 13y ago
In fact, it is accounted for. You'll notice the negative exponent on the unlikely hypotheses is getting pretty extreme after a handful of flips.
- bermanoid 13y ago
  In case it's not clear (that wording doesn't really click for me, since there are no exponents involved), the way this information is encoded is in the shape of the distribution itself. If there have been very few observations, it will be very wide, but after many it will narrow to a tiny spike. In this context, at least, the prior distribution encodes everything - there's no meaning to the idea that you're more or less confident in the prior, because the prior already represents your uncertainty about the outcomes. If you were 50/50 on this prior versus another one, then your actual prior would be the average of the two. There is a subfield of statistics that deals with imprecise probabilities, but that's a whole other can of worms and doesn't really relate to this problem. That said, it's fascinating stuff, and very useful in some contexts (if you're uncertain about your priors, it can be useful to do sensitivity analyses to figure out exactly how the end result depends on your prior).
  - dthunt 13y ago
    What I meant was that for h_1%, you're going to wind up with probabilities in the area of like, 1.2e-34, very quickly, if the coin is a fair one. The update process is adjusting via multiplication, and for a really bad hypothesis, that's going to bring its probability dramatically close to zero without that many trials. Even though the absolute difference between 1e-4 and 1-e34 hypothesis feels smallish when you look on a linear scale from 0-1, a 1-e34 is a lot 'stickier'. Your explanation has the benefit of being a better explanation; my goal was just to explain where the inertia was hiding.

bluecalm 13y ago

Once you figure out simple examples you slowly start thinking this way about the world. It's beautiful. Take people for example: Someone with open mind has a priori with at least slight probability assigned to unlikely (for them!) hypothesis while on the other hand very religious people for example have 0 in their priori when it comes to possibility of their religion being made up so they are forced to ignore evidence to the contrary (because bayesian updating breaks for them due to division by zero and mind's way to signal this exception is denial). In general someone with a lot of weight on given hypothesis is "stubborn" or just very convinced and someone with uniform or close distribution just doesn't know anything about given problem. Someone unable to build heavily weighted distributions is a conspiracy theorist, someone reluctant to - a sceptic and someone too much eager to a fanatic. Someone with very bad priors is un/badly educated (in given domain) or biased or maybe just stupid, someone with good priors is an expert. It's possible to combine expert with sceptic attitude or expert with fanatic or all too often stupid with fanatic (very bad and very heavily weighted priors with possible 0's on some options). Once you start thinking this way you start expressing yourself differently, you start adding those probability qualifiers to your sentences: "I am very sure it's the way to go", "My intuition tells me this but I am not really sure", "I am very convinced and it's not worth discussing" (yes, it can be rational and good attitude) or "I would do X but I need more evidence to be reasonably sure". It's all there in people's mind, language and interactions once you start thinking this way it's whole new world of perspective and understanding.

darkxanthos 13y ago
Yup it's true. I can seem like a Bayes nut at work as I mention updating my priors during debates and when discussing hypotheses for split tests.
yafujifide 13y ago
Completely agreed. Your comment is a really good advertisement for normal people to learn Bayesian probability theory. Not because it's useful in science or engineering, but because it's useful in life.

davidf18 13y ago

A good (free) explanation of Bayesian Stats using Python: Downey's Think Bayes http://www.greenteapress.com/thinkbayes/ http://www.greenteapress.com/thinkbayes/

darkxanthos 13y ago
This helped me the most to get started.