Yow. I ignore the comments for weeks, and three or four interesting topics come up.
A) you may be interested in the Iowa Gambling Task experiments, one observation of which is (I'm paraphrasing here, so read some real research if you want to see what the studies really claim) that people can develop a sense of a probability from a smallish sample, _before_ they consciously recognize that they have such a sense. (Paraphrase 2: drawing from a deck of possible rewards/penalties, someone will start reacting negatively to a "bad" deck before they consciously decide it's a "bad" deck.)
B) [not actually chronological] LTG says "I am surprised any mathematician on this blog would expect children to have an intuitive grasp of probability. Adults sure don't...It is well known...that individuals tend...to treat a 99,999 to 1 chance...differently from...a 1 in 100,000 chance..."
This, and I really do not mean to offend, is because you are not a mathematician, particularly not a mathematician who has to teach non-mathematicians.
Mathematicians (especially in education) think about what is and isn't intuitive mathematics a great deal, because what _seems_ intuitive to us is clearly not to our students. And one big idea that's come out of this is the impetus to connect the language and structure of mathematics to the structures and intuition that "normal" people already have, rather than treat math as abstract and foreign.
So we talk about "number sense" and "proportional reasoning" and "intuitive probability" and part of what we mean is a non-linguistic concept of those things. The theory is that people have a sense of probability, both in the cases of extremely rare events and relatively common events, but:
a) your intuition can be misled, and a high-level, abstract argument lacks potency (e.g., because you've heard about miraculous recoveries from coma, your intuition is that is a reasonably likely event, and a doctor telling you it's a one in a million chance doesn't dissuade you.)
b) it is difficult to connect your intuition, in the many cases when it is right, to the abstract claims of probability, if they're couched in examples not relevant to your experience or in language you don't have other intuitive connections with (most people troubled by the Monty Hall problem didn't watch episode after episode of Let's Make a Deal; most people have never thought of what "1 in 100,000" means, in any concrete way, and usually don't have a reason to.)
Thus, while mathematicians cannot help but see that most adults' sense of probability is _solely_ intuitive, and unconnected to any attempts at probabilistic abstraction or language, and (therefore?) usually lacks _nuance_, we cling to the belief (well justified, really) that there is in fact an intuition there.
(I should point out that numbers are themselves language, as I'm using the phrase. I doubt that caused any confusion, but you can never be too careful on the interweb.)
C) I took Pombat's examples to just be illustrations of the way probabilities are bandied about in misleading ways, and not a claim about DNA evidence per se. For more examples of this and other grievous misquantification, John Allen Paulos' Innumeracy (and/or A Mathematician Reads the Newspaper) are excellent reading.
D) RbR's argument (and I know I shouldn't say this, but I just can't help it, so please forgive me) assumes facts not in evidence. A claim about the sensitivity of a forensic test is just that, independent of other factors. If a jury is deciding guilt or innocence, they are (or should be) weighing all the factors RbR suggests (which surely the prosecution would present) in their "reasonable doubt" analysis. But those other factors don't affect the accuracy of the test.
RbR's description also bears some resemblance to an example in Innumeracy, about a case in Los Angeles in 1964. (I'm not saying RbR is guilty of the probabilistic fallacy in that case, just that his description is similar to the set-up for the fallacy, and the story is a fun one, so I'm going to relate it.) Brutally summarized, the argument was: a pair of lawbreakers were seen with several "distinguishing characteristics" (blonde with pony tail with black man with beard in a yellow car). The defendants matched those characteristics. In this case, there was nothing else linking the defendants to the crime, but surely the chances that a couple would match all those characteristics were so small that this must be the couple in question, right?
So it was decided. Until the appeal, in which the defense pointed out that the relevant probability was not whether a couple would have those characteristics, but rather whether, out of the ~2M couples in LA, there was _another_ couple with those characteristics. (The probability that there was one such couple was actually a certainty, since the defendants were living proof.)
It turns out (and no one would suggest that this is intuitive) the likelihood of another such couple (given the prosecution's estimate of the relevant probabilities) is about 8%, which provided reasonable enough doubt for the California supreme court
to overturn the original verdict.
E) If anyone wants to talk more about math, probability, and math education, I'm very keen on it -- you could say it's my job. Email me!
Sunday, September 13, 2009
Thursday, September 10, 2009
Monday, June 29, 2009
Bipartisanship should be an effect, not a cause
Apparently, Congress and pundits favor "bipartisan" legislation, that is, bills you can get members of both parties to sign on to. Rhetorically I ask, but why? Surely we can all dispense with the naïve notion that the best solution, or the one most desired by the people, lies at the mean between the party positions. There are at least three reasons off the top of my head proving the preposterousness of that notion, and you can probably come up with more than me.
Really, bipartisanship is supposed to be a reflection of the value of a bill. The bill's not good because it's bipartisan; rather, the bill's inherently good, so good that people want to publicly support it, even if it's not what their party came up with. A bill might be bipartisan because a priori it's good.
Lack of bipartisanship is generally framed as a negative for the bill's originator: "Obama has failed to deliver on the promise of bipartisanship for his agenda." But there's a perfectly good converse view. If a bill's good, or popular -- the health care public option, for example -- opponents of the bill would be right to fear the lack of bipartisanship. If they're too dumb or stubborn to back what's good, then they're useless to their constituents.
An agenda setter should put good legislation out there, with no compromises solely to garner support from across the aisle. If it's good enough, and popular enough, then all the Congresscritters in vulnerable seats are under pressure to back it. Bipartisanship is an indicator that the bill was good in the first place; but it is only a correlated side-effect, not a necessary (or sufficient) condition for good legislation.
Really, bipartisanship is supposed to be a reflection of the value of a bill. The bill's not good because it's bipartisan; rather, the bill's inherently good, so good that people want to publicly support it, even if it's not what their party came up with. A bill might be bipartisan because a priori it's good.
Lack of bipartisanship is generally framed as a negative for the bill's originator: "Obama has failed to deliver on the promise of bipartisanship for his agenda." But there's a perfectly good converse view. If a bill's good, or popular -- the health care public option, for example -- opponents of the bill would be right to fear the lack of bipartisanship. If they're too dumb or stubborn to back what's good, then they're useless to their constituents.
An agenda setter should put good legislation out there, with no compromises solely to garner support from across the aisle. If it's good enough, and popular enough, then all the Congresscritters in vulnerable seats are under pressure to back it. Bipartisanship is an indicator that the bill was good in the first place; but it is only a correlated side-effect, not a necessary (or sufficient) condition for good legislation.
Monday, June 22, 2009
TNC cannot stop talking about slavery
If you read only 200 things Ta-Nehisi Coates writes this year, this should be one of them. Also, you will only have read about a fifth of the great stuff he writes.
Insightful, as always, but I don't think TNC is quite right here. Or perhaps he is overall, but some nuance is in order.
Specifically, I think there's a historical line that goes like TNC says "We tend to think", namely that slavery was kind of this regrettable cruelty that the world in general was tolerant of, until we got civilized and disavowed it.
But there is at least an acknowledgment, in the historical line I was taught, that America (like always) was exceptional. That line says, what with the explosion of cotton (which Eli Whitney's invention contributed to), the economy of the South became completely dependent on slavery as an economic fuel, kind of like we talk about being dependent on oil today. And that dependence shifted the culture, both toward a vigorous defense of the economic interests in slavery and toward dehumanization of slaves on a scale and to a degree beyond whatever the barbaric practices of yesteryear had been.
But that's just my impression from my history classes. I'm not sure there's a shared "American" conventional point of view on this.
Where I feel TNC has it right is that "we have never grappled with this," `this' being that most of the culture of America is formed from the fractious and unresolved shards that remain from centuries of dehumanizing black people, and the struggles both violent and nonviolent to erode and destroy them.
Again, referring to my own education, the mythology of America is the "melting pot" notion of disparate cultures coming together and peacefully sharing the opportunities of a new land. But this, it seems to me, is a gloss, a "and they all lived happily ever after" resolution of the unresolved issue that defines America. We (which of course is not really "we", since we weren't there, but the imagined "we" of this country) invented a new kind of racism, and have never been able to put that dire genie back into the bottle.
Insightful, as always, but I don't think TNC is quite right here. Or perhaps he is overall, but some nuance is in order.
Specifically, I think there's a historical line that goes like TNC says "We tend to think", namely that slavery was kind of this regrettable cruelty that the world in general was tolerant of, until we got civilized and disavowed it.
But there is at least an acknowledgment, in the historical line I was taught, that America (like always) was exceptional. That line says, what with the explosion of cotton (which Eli Whitney's invention contributed to), the economy of the South became completely dependent on slavery as an economic fuel, kind of like we talk about being dependent on oil today. And that dependence shifted the culture, both toward a vigorous defense of the economic interests in slavery and toward dehumanization of slaves on a scale and to a degree beyond whatever the barbaric practices of yesteryear had been.
But that's just my impression from my history classes. I'm not sure there's a shared "American" conventional point of view on this.
Where I feel TNC has it right is that "we have never grappled with this," `this' being that most of the culture of America is formed from the fractious and unresolved shards that remain from centuries of dehumanizing black people, and the struggles both violent and nonviolent to erode and destroy them.
Again, referring to my own education, the mythology of America is the "melting pot" notion of disparate cultures coming together and peacefully sharing the opportunities of a new land. But this, it seems to me, is a gloss, a "and they all lived happily ever after" resolution of the unresolved issue that defines America. We (which of course is not really "we", since we weren't there, but the imagined "we" of this country) invented a new kind of racism, and have never been able to put that dire genie back into the bottle.
Thursday, May 28, 2009
Economicomics
Regarding this:
I worry a little bit that enough research has been done about this:
I don't follow X-Men religiously anymore, and they sneak things like Spidey's organic webbing past me, so this may have changed, but historically (i.e., in the 80's-90's) it was explicitly established that Storm moves humidity around, but doesn't create it. If she irrigates the midwest, she does it by exacerbating the drought in California. In fact, she was essentially doing this as a local rain goddess when Prof. X recruited her.
My geeky trivium aside, I think it's weird when people complain about an amusing theoretical like this as being tired, overdone, or silly. Superheroes are cartoons -- superhero economics is a cartoon of economics. Most of us aren't economists, and thinking through simplified illustrations (including their shortcomings) makes key concepts clearer. Also, it's fun.
I worry a little bit that enough research has been done about this:
Storm could irrigate the crops of all the suffering farmers in the midwest and California when the droughts of summer are destroying their crops.
I don't follow X-Men religiously anymore, and they sneak things like Spidey's organic webbing past me, so this may have changed, but historically (i.e., in the 80's-90's) it was explicitly established that Storm moves humidity around, but doesn't create it. If she irrigates the midwest, she does it by exacerbating the drought in California. In fact, she was essentially doing this as a local rain goddess when Prof. X recruited her.
My geeky trivium aside, I think it's weird when people complain about an amusing theoretical like this as being tired, overdone, or silly. Superheroes are cartoons -- superhero economics is a cartoon of economics. Most of us aren't economists, and thinking through simplified illustrations (including their shortcomings) makes key concepts clearer. Also, it's fun.
Thursday, May 07, 2009
on Yglesias re future of newspapers
Matthew Yglesias » Is Anyone Reading the News?
I echo the sentiments that local bloggers fill the niche of local reporting pretty well. I pooh-pooh the sentiments that local bloggers mostly have axes to grind and newspapers provide “disinterested” reporting. That’s an overgeneralization — in the cities I’ve lived, the local papers have had axes to grind on most local reporting, sometimes obscured by false disinterest.
I’d also like to echo the value of narrowly read news as providing a record, available for later review when someone realizes they _do_ care about some local issue. Local blogs do this in the short term, but there’s a high risk of blogs eventually dying (due to the interested people moving, for example), and small-scale locally focused blogs are less likely to come up on Google searches, even if they’re most relevant.
People have said the newspapers crib from weeklies — might there not be a business model of collecting/linking/rating local blogs, a sort of “preserving for posterity” spin on the “plagiarizing from the alt-weeklies” concept? You know who might have the skillset and market positioning for something like that? I’m thinking the local newspaper, that’s who.
Lastly, I’d also point out that local reporting from locals (newspapers, blogs, whoever) can raise money from local advertising, which is a more stable revenue model than the world-famous dailies. The new sushi place down the street needs to advertise, but its ad dollar would be wasted on anything more widely distributed than the local paper (or local blog.)
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