Prediction of events with some component of randomness is necessarily uncertain—to think otherwise is to misunderstand the whole endeavor. The fault here is not with the scientists, for predicting incorrectly, but with the public for believing this prediction came with certainty. To suppress predictors of random events for predicting incorrectly is to court ignorance and block progress, as it discourages the very research which will one day improve our ability to predict such events.
Showing posts with label probability. Show all posts
Showing posts with label probability. Show all posts
Tuesday, October 23, 2012
step backward for uncertainty
Reuters reports that six Italian earthquake scientists have been jailed for manslaughter for "failing to give adequate warning of the 2009 earthquake in the city of L'Aquila."
Thursday, February 23, 2012
dutch book by dutch architect
Finally, a "Dutch book" style argument (against bulk discounts, no less) from an honest-to-God Dutchman, architect and cartoonist Joost Swarte. Finally available in English for the first time.
Thursday, November 25, 2010
probability in popular culture
"So you're telling me it's a matter of probability and odds; I was worried there wasn't chance involved."
"Well only if you assume the player with the best hand wins."
"So that would be what you call bluffing?"
"You've heard the term. Then you'll also know that in poker you never play your hand; you play the man across from you."
~ Casino Royale (2006)
James Bond discussing poker with his government sponsored financier. Blind faith that skill can overcome the roll of the die (or deal of the cards, as the case may be).
Monday, November 22, 2010
jacob bernoulli on luck
The mathematics of probability began in the 1650s with the correspondence between Fermat and Pascal on the problem of points. The crucial question, first proposed to Pascal by the Chevalier de Méré, concerned the question of how to divide the pot if a gambling game had to be interrupted before it could be completed, e.g. suppose $x is the pot, 7 rounds are needed to win, player A has won 6 rounds, while player B has won 5 rounds: if they are forced to interrupt their game at this stage, what constitutes a fair division of the pot?
All early treatises (and even most modern ones) addressed the problem of points in their presentation of the probability calculus. For example, Christiaan Huygens places a discussion of the problem of points near the very start of one of the first treatises on probability, his De ratiociniis in ludo aleae (1657).
In 1713, Jacob Bernoulli's treatise, Ars Conjectandi was published posthumously. This was the first work to prove a limit theorem about probability, as well as extending probability theory to cases where probabilities where unequal, but could be derived from underlying equipossibility. The first of Ars Conjectandi's four section reprints Huygens' treatise with extensive commentary and solutions to all problems. When Huygens presents the problem of points, he points out that only the remaining games need be considered (and not the number of games already played), a remark which Huygens supplements with the comment:
All early treatises (and even most modern ones) addressed the problem of points in their presentation of the probability calculus. For example, Christiaan Huygens places a discussion of the problem of points near the very start of one of the first treatises on probability, his De ratiociniis in ludo aleae (1657).
In 1713, Jacob Bernoulli's treatise, Ars Conjectandi was published posthumously. This was the first work to prove a limit theorem about probability, as well as extending probability theory to cases where probabilities where unequal, but could be derived from underlying equipossibility. The first of Ars Conjectandi's four section reprints Huygens' treatise with extensive commentary and solutions to all problems. When Huygens presents the problem of points, he points out that only the remaining games need be considered (and not the number of games already played), a remark which Huygens supplements with the comment:
In general, we should take no account of past games when we compute the lots for games that are all in the future. For in any new game the probability that fortune will continue to favor those that it has favored before is no great than the probability that it will favor those who have been the most unfortunate. I observe this in opposition to the ridiculous opinion of the many who think of fortune as some kind of habit, which remains in a person for a while and somehow gives him almost a right to expect similar fortune to continue.
Of course, Bernoulli is absolutely right, a point which can even be empirically determined, e.g. as in the hot hand fallacy.
An interesting question, though, is the balance of opinion on the question of streaks of luck vs. the gambler's fallacy, which expects streaks to reverse, or be balanced by future outcomes. Of course, from both the mathematical and the empirical standpoints, probabilities are not affected by streaks or runs in the outcomes. But, from a sociological standpoint, is belief in luck (good or bad) more common than belief in the gambler's fallacy? Given that they are mutually contradictory, how frequently can both beliefs be found in the same individual? If belief in one statistically dominates belief in the other, why? Underlying optimism (pessimism)? If our beliefs about such matters are derived from our experience with the world, why the tendency to distort rather than simply probability match (especially given that humans can be quite good at probability matching under other circumstances)?
Monday, April 19, 2010
"just another statistic"
This expression is used ironically to indicate an event, often an instance of human tragedy, or the subject of that event, was treated as merely a number in discussions of that event (i.e. tragedy) type in the public forum.
Strictly speaking the usage is incorrect: a "statistic" is a function on data. The complaint is not that the person or event has been treated as a statistic, but as a data point.
According to the Bayesian foundations of statistics, we can legitimately apply the standard methods of statistics (e.g. fitting the distribution of data to a normal curve) if we can treat data points as exchangeable. This follows from de Finetti's Theorem.
If we can't treat data points as exchangeable, however, if we know more relevant features of one data point than another, then in principle standard techniques and Bayesian techniques (which take into account prior knowledge) can produce radically different analyses.
So, when someone tells a tragic anecdote and bemoans that it is "just another statistic", they tell us more about that data point, one which had perhaps been the object of "statistics" which delivered an incorrect analysis without the benefit of the prior knowledge provided by the particulars of its case.
Strictly speaking the usage is incorrect: a "statistic" is a function on data. The complaint is not that the person or event has been treated as a statistic, but as a data point.
According to the Bayesian foundations of statistics, we can legitimately apply the standard methods of statistics (e.g. fitting the distribution of data to a normal curve) if we can treat data points as exchangeable. This follows from de Finetti's Theorem.
If we can't treat data points as exchangeable, however, if we know more relevant features of one data point than another, then in principle standard techniques and Bayesian techniques (which take into account prior knowledge) can produce radically different analyses.
So, when someone tells a tragic anecdote and bemoans that it is "just another statistic", they tell us more about that data point, one which had perhaps been the object of "statistics" which delivered an incorrect analysis without the benefit of the prior knowledge provided by the particulars of its case.
Thursday, July 23, 2009
information i
Suppose we observe an event e — how much information do we acquire?
We'd like a function I such that for any e, I(e) tells us how much information is contained in the occurrence of e.
Intuitively, if the probability of e occurring is very large, then I(e) should be very high. We learn a lot when something unlikely occurs. If your math teacher stands on one leg, barks like a dog, turns around three times, then leaps through the window, you learn a lot about him.
If the probability of e occurring is very high, though, I(e) should be very small. In the limit, as the probability of e approaches 1, i.e. certainty, I(e) should approach zero — we don't learn very much when we see the sun rise in the morning (at least, not if we've been paying attention!).
In the limit in the other direction, as the probability of e approaches zero, the information we gain from observing e approaches infinity.
Finally, if two unconnected (i.e. independent) events occur together then the information we receive about them is just the sum of the information we would have received had each of them occurred separately. (Remember that two events are independent if the probability of both of them occurring equals the probability of one times the probability of the other.) If my math teacher puts a banana on his head, and your english teacher, in a different state, smothers himself in jello, then we learn everything I would have learned separately about my math teacher, plus everything you would have learned separately about your english teacher.
So, we now have several constraints on the function I:
1. P(e1) < P(e2) implies I(e1) > I(e2)
2. P(e) = 1 implies I(e) = 0
3. P(e) = 0 implies I(e) = ∞
4. P(e1 ∧ e2) = P(e1) × P(e2) implies I(e1 ∧ e2) = I(e1) + I(e2)
One convenient analysis of information which satisfies these four constraints is
I(e) = − log P(e)
We'd like a function I such that for any e, I(e) tells us how much information is contained in the occurrence of e.
Intuitively, if the probability of e occurring is very large, then I(e) should be very high. We learn a lot when something unlikely occurs. If your math teacher stands on one leg, barks like a dog, turns around three times, then leaps through the window, you learn a lot about him.
If the probability of e occurring is very high, though, I(e) should be very small. In the limit, as the probability of e approaches 1, i.e. certainty, I(e) should approach zero — we don't learn very much when we see the sun rise in the morning (at least, not if we've been paying attention!).
In the limit in the other direction, as the probability of e approaches zero, the information we gain from observing e approaches infinity.
Finally, if two unconnected (i.e. independent) events occur together then the information we receive about them is just the sum of the information we would have received had each of them occurred separately. (Remember that two events are independent if the probability of both of them occurring equals the probability of one times the probability of the other.) If my math teacher puts a banana on his head, and your english teacher, in a different state, smothers himself in jello, then we learn everything I would have learned separately about my math teacher, plus everything you would have learned separately about your english teacher.
So, we now have several constraints on the function I:
1. P(e1) < P(e2) implies I(e1) > I(e2)
2. P(e) = 1 implies I(e) = 0
3. P(e) = 0 implies I(e) = ∞
4. P(e1 ∧ e2) = P(e1) × P(e2) implies I(e1 ∧ e2) = I(e1) + I(e2)
One convenient analysis of information which satisfies these four constraints is
I(e) = − log P(e)
Sunday, November 25, 2007
probability and public policy II: subjective bayes and the "fair coin"
[part one of this series here]
Subjective Bayes is the view that probabilistic statements reflect our state of uncertainty; as such, they are subjective, a mere reflection of a particular human's internal state of belief.
for example: Consider a hypothetical "fair coin": what does it mean to say this coin is "fair"? As it turns out, if we know the precise conditions under which a coin is flipped, we can confidently predict the outcome of a coin toss. The physics itself is relatively simple (comparatively), and if one knows the initial conditions one can predict the outcome (i.e. if one knows the physical properties of the coin and the details of the force which set it spinning (as well as air resistence, wind velocity, etc.), one can calculate precisely which side will land face up).
Consider briefly a simplified model where we assume that air resistance and wind play no role and that the mass of the coin is uniformly distributed. We can graph the outcome of a coin toss against the upward velocity at the time of release (V, in feet per second) and the angular velocity (ω, in revolutions per second). Assuming the coin starts heads up, we can treat grey as heads and white as tails:
Persi Diaconis, "A Place for Philosophy? The Rise of Modeling in Statistical Science," 1998.
The above analysis is from work by Persi Diaconis. He discusses where in the above graph the "typical" coin toss falls:
Subjective Bayes is the view that probabilistic statements reflect our state of uncertainty; as such, they are subjective, a mere reflection of a particular human's internal state of belief.
for example: Consider a hypothetical "fair coin": what does it mean to say this coin is "fair"? As it turns out, if we know the precise conditions under which a coin is flipped, we can confidently predict the outcome of a coin toss. The physics itself is relatively simple (comparatively), and if one knows the initial conditions one can predict the outcome (i.e. if one knows the physical properties of the coin and the details of the force which set it spinning (as well as air resistence, wind velocity, etc.), one can calculate precisely which side will land face up).
Consider briefly a simplified model where we assume that air resistance and wind play no role and that the mass of the coin is uniformly distributed. We can graph the outcome of a coin toss against the upward velocity at the time of release (V, in feet per second) and the angular velocity (ω, in revolutions per second). Assuming the coin starts heads up, we can treat grey as heads and white as tails:
Persi Diaconis, "A Place for Philosophy? The Rise of Modeling in Statistical Science," 1998.The above analysis is from work by Persi Diaconis. He discusses where in the above graph the "typical" coin toss falls:
For a typical one-foot toss, experiments show that coins go up at about 5 m.p.h. and turn over 35-40 revolutions per second. In the units of the picture the velocity is concentrated at about 0.2 on the velocity scale. This is close to zero in the picture. Fortunately, the spin is concentrated at about 40 units up on the ω axis.
For additional details see "The Probability of Heads" by Joseph Keller.
The point here is just that the "probability," or fairness, in a coin toss is not a property of the physical setup, but rather of the observer's belief state about the outcome. Once we are privy to the initial conditions of the coin toss, there is no more uncertainty and thus no more "probability" or "chance" in the event.
Another way of thinking about subjective Bayes is in terms of one's willingness to bet upon an event the outcome of which is as yet undetermined. de Finetti used this thought experiment as the basis of his theory of probability and demonstrated that any set of odds which cannot be subjected to a Dutch Book is a valid probability distribution (a "Dutch Book" is a set of bets devised by the bookie such that you lose no matter what the outcome of the event is).
If probability is just a measure of one's uncertainty, how can we use it to interact with the world? To put it another way: how can I get my subjective probability assignments to match up with the "actual" proportion of outcomes due to the underlying mechanism at work? Or, if the coin isn't "fair, if I'm being cheated, how do I figure that out? The answers to these questions in our next installment.
next: Bayes' Rule
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