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[Link] Why the kids don’t know no algebra

21 GLaDOS 04 July 2012 10:29AM

Post by fellow LW reader Razib Khan, who many here probably know from  the gnxp site or perhaps from his debate with Eliezer.

A few days ago I stumbled upon a really interesting post. And I’m wondering if my readers are at all familiar with the phenomenon outlined here (it was a total surprise to me), The myth of “they weren’t ever taught….”:

With all this I am not saying conditions which are non-hereditary are irrelevant. What I am saying is that we can’t ignore the shape of the pre-existent landscape before we attempt to reshape it to our own image. Excoriating teachers for having pupils who can’t master mid-level secondary school mathematics is in some cases like excoriating someone for the fact that their irrigation canals from the plains into the mountains are failures. You need to level the mountains before your canals can work (or, barring that design and implement a mechanical system which will move water against the grade). Easier said than done. E. O. Wilson said of Communism, “Great Idea, Wrong Species.” The reaction of Communist regimes to this reality was brutal and shocking. Obviously the modern rejection of unpalatable aspects of human nature are not so grotesque. But they have a human toll nonetheless. I’m skeptical that this generation will pass before we have to acknowledge these realities and calibrate our policies accordingly.

Stage One: I will describe this stage for algebra I teachers, but plug in reading, geometry, writing, science, any subject you choose, with the relevant details. This stage begins when teachers realize that easily half the class adds the numerators and denominators when adding fractions, doesn’t see the difference between 3-5 and 5-3, counts on fingers to add 8 and 6, and looks blank when asked what 7 times 3 is.

Ah, they think. The kids weren’t ever taught fractions and basic math facts! What the hell are these other teachers doing, then, taking a salary for showing the kids movies and playing Math Bingo? Insanity on the public penny. But hey, helping these kids, teaching them properly, is the reason they became teachers in the first place. So they push their schedule back, what, two weeks? Three? And go through fraction operations, reciprocals, negative numbers, the meaning of subtraction, a few properties of equality, and just wallow in the glories of basic arithmetic. Some use manipulatives, others use drills and games to increase engagement, but whatever the method, they’re basking in the glow of knowledge that they are Closing the Gap, that their kids are finally getting the attention that privileged suburban students get by virtue of their summer enrichment and more expensive teachers.

At first, it seems to work. The kids beam and say, “You explain it so much better than my last teacher did!” and the quizzes seem to show real progress. Phew! Now it’s possible to get on to teaching algebra, rather than the material the kids just hadn’t been taught.

But then, a few weeks later, the kids go back to ignoring the difference between 3-5 and 5-3. Furthermore, despite hours of explanation and practice, half the class seems to do no better than toss a coin to make the call on positive or negative slopes. Many students who demonstrated mastery of distributing multiplication over addition are now making a complete hash of the process in multi-step equations. And many students are still counting on their fingers.

The author is involved in education personally, so is posting their own reflections as well as what others report to them. In personal correspondence they explain that this phenomenon is common among children of average intelligence. The lowest quartile presumably would never have been able to master many of these rules in the first place. Some of the information resembles the stuff that a friend of mine experienced when he went in to do tutoring for disadvantaged students in Boston when he was getting his doctorate at MIT. At first my friend was totally taken aback at the level of ignorance (e.g., the inability to see the relationship between 1/10 and 10/100). Today he works at a major technology firm as a scientist, but continues to be involved in mentoring “at risk” kids. At some point you have to muddle on. He does his best, and does not indulge in the luxury of shock and disappointment. That helps no one.

This matters because American society is notionally obsessed with education. All this isn’t too clear or important to be frank when you aren’t a parent. It’s somewhat in the realm of the abstract. That changes when you become a parent. Suddenly you become immersed in the data of your local schools, and begin to weight various options to optimize your child’s schooling experience. Of course the real differences in school metrics have not only parental relevance, they matter in terms of national policy and attention. Both the political Left and the Right have their own pet solutions. More money, reform teachers’ unions, charter schools, vouchers, etc.

But the biggest problem at the heart of the matter is the fundamental populist drive to ignore human difference. American schools were designed to produce the citizen, and the citizen has the same rights and responsibilities from individual to individual. In some ways the public school system as it emerged in the 19th century was a project by the Protestant establishment to assimilate white ethnics, in particular Catholics (who of course created their own alternative educational system to maintain cultural separation and distinctiveness). In the 21st century the drive to produce H. Americanus seems quaint, rather, we want to citizens of the world with skills and abilities to navigate an information economy.

What American society on a deep philosophical level, no matter the political outlook, detests acknowledging is that a simple and elegant public policy solution can not abolish human difference. Some children are more athletic than others, and some children are more intelligent than others. Starting among conservatives, but now spreading to some liberals, is a rejection of this premise via blaming teachers. The premise is bewitching because it presents tractable problems with solutions on hand. Here is John B. Watson, the father of behaviorism:

Give me a dozen healthy infants, well-formed, and my own specified world to bring them up in and I’ll guarantee to take any one at random and train him to become any type of specialist I might select – doctor, lawyer, artist, merchant-chief and, yes, even beggar-man and thief, regardless of his talents, penchants, tendencies, abilities, vocations, and race of his ancestors. I am going beyond my facts and I admit it, but so have the advocates of the contrary and they have been doing it for many thousands of years

I think if Watson were alive today he’d have to admit he was wrong. Your ancestors are not destiny, but they are probability. If your father plays in the N.B.A., the probability that you will play in the N.B.A. is not high. But the probability is orders of magnitude higher than if you are a random person off the street.

Why Bayes? A Wise Ruling

13 Vaniver 25 February 2013 03:52PM

Why is Bayes' Rule useful? Most explanations of Bayes explain the how of Bayes: they take a well-posed mathematical problem and convert given numbers to desired numbers. While Bayes is useful for calculating hard-to-estimate numbers from easy-to-estimate numbers, the quantitative use of Bayes requires the qualitative use of Bayes, which is noticing that such a problem exists. When you have a hard-to-estimate number that you could figure out from easy-to-estimate numbers, then you want to use Bayes. This mental process of testing beliefs and searching for easy experiments is the heart of practical Bayesian thinking. As an example, let us examine 1 Kings 3:16-28:

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The noncentral fallacy - the worst argument in the world?

157 Yvain 27 August 2012 03:36AM

Related to: Leaky Generalizations, Replace the Symbol With The Substance, Sneaking In Connotations

David Stove once ran a contest to find the Worst Argument In The World, but he awarded the prize to his own entry, and one that shored up his politics to boot. It hardly seems like an objective process.

If he can unilaterally declare a Worst Argument, then so can I. I declare the Worst Argument In The World to be this: "X is in a category whose archetypal member gives us a certain emotional reaction. Therefore, we should apply that emotional reaction to X, even though it is not a central category member."

Call it the Noncentral Fallacy. It sounds dumb when you put it like that. Who even does that, anyway?

It sounds dumb only because we are talking soberly of categories and features. As soon as the argument gets framed in terms of words, it becomes so powerful that somewhere between many and most of the bad arguments in politics, philosophy and culture take some form of the noncentral fallacy. Before we get to those, let's look at a simpler example.

Suppose someone wants to build a statue honoring Martin Luther King Jr. for his nonviolent resistance to racism. An opponent of the statue objects: "But Martin Luther King was a criminal!"

Any historian can confirm this is correct. A criminal is technically someone who breaks the law, and King knowingly broke a law against peaceful anti-segregation protest - hence his famous Letter from Birmingham Jail.

But in this case calling Martin Luther King a criminal is the noncentral. The archetypal criminal is a mugger or bank robber. He is driven only by greed, preys on the innocent, and weakens the fabric of society. Since we don't like these things, calling someone a "criminal" naturally lowers our opinion of them.

The opponent is saying "Because you don't like criminals, and Martin Luther King is a criminal, you should stop liking Martin Luther King." But King doesn't share the important criminal features of being driven by greed, preying on the innocent, or weakening the fabric of society that made us dislike criminals in the first place. Therefore, even though he is a criminal, there is no reason to dislike King.

This all seems so nice and logical when it's presented in this format. Unfortunately, it's also one hundred percent contrary to instinct: the urge is to respond "Martin Luther King? A criminal? No he wasn't! You take that back!" This is why the noncentral is so successful. As soon as you do that you've fallen into their trap. Your argument is no longer about whether you should build a statue, it's about whether King was a criminal. Since he was, you have now lost the argument.

Ideally, you should just be able to say "Well, King was the good kind of criminal." But that seems pretty tough as a debating maneuver, and it may be even harder in some of the cases where the noncentral Fallacy is commonly used.

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The Conjunction Fallacy Does Not Exist

-38 curi 10 April 2011 10:35PM

The conjunction fallacy says that people attribute higher probability to X&Y than to Y.

This is false and misleading. It is based on bad pseudo-scientific research designed to prove that people are biased idiots. One of the intended implications, which the research does nothing to address, is that this is caused by genetics and isn't something people can change except by being aware of the bias and compensating for it when it will happen.

In order to achieve these results, the researchers choose X, Y, and the question they ask in a special way. Here's what they don't ask:

What's more likely this week, both a cure for cancer and a flood, or a flood?

Instead they do it like this:

http://en.wikipedia.org/wiki/Conjunction_fallacy

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

Linda is a bank teller.

Linda is a bank teller and is active in the feminist movement.

Or like this:

http://lesswrong.com/lw/ji/conjunction_fallacy/

"A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983."

"A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983."

These use different tricks. But both are biased in a way that biases the results.

By the way, this is a case of the general phenomenon that bad research often gets more impressive results, which is explained in _The Beginning of Infinity_ by David Deutsch. If they weren't bad researchers and didn't bias their research, they would have gotten a negative result and not had anything impressive to publish.

The trick with the first one is that the second answer is more evidence based than the first one. The first answer choice has nothing to do with the provided context. The second answer choice has something to do with the provided context: it is partially evidence based. Instead of taking the question really literally as to be about the mathematics of probability, they are deciding which answer makes more sense and saying that. The first one makes no sense (having nothing to do with the provided information). The second one partially makes sense, so they say it's better.

A more literally minded person would catch on to the trick. But so what? Why should people learn to split hairs so that they can give literally correct answers to bad and pointless questions? That's not a useful skill so most people don't learn it.

The trick with the second one is that the second answer is a better explanation. The first part provides a reason for the second part to happen. Claims that have explanatory reasons are better than claims that don't. People are helpfully reading "and" as expressing a relationship -- just as they would do if their friend asked them about the possibility of Russia invading Poland and the US suspending diplomacy. They think the two parts are relevant, and make sense together. With the first one, they don't see any good explanation offered so they reject the idea. Did it happen for no reason? Bad claim. Did it happen without an invasion of Poland or any other notable event worth mentioning? Bad claim.

People are using valuable real life skills, such as looking for good explanations and trying to figure out what reasonable question people intend to ask, rather than splitting hairs. This is not a horrible bias about X&Y being more likely than Y. It's just common sense. All the conjunction fallacy research shows is that you can miscommunicate with people and then and then blame them for the misunderstanding you caused. If you speak in a way such that you can reasonably expect to be misunderstood, you can then say people are wrong for not giving correct answers to what you meant and failed to communicate to them.

The conjunction fallacy does not exist, as it claims to, for all X and all Y. That it does exist for specially chosen X, Y and context is incapable of reaching the stated conclusion that it exists for all X and Y. The research is wrong and biased. It should become less wrong by recanting.

This insight was created by philosophical thinking of the type explained in _The Beginning of Infinity_ by David Deutsch. It was not created by empirical research, prediction, or Bayesian epistemology. It's one of many examples of how good philosophy leads to better results and helps us spot mistakes instead of making them. It also wasn't discovered by empirical research. As Deutsch explained, bad explanations can be rejected without testing, and testing them is pointless anyway (because they can just make ad hoc retreats to other bad explanations to avoid refutation by the data. Only good explanations can't do that.).

Please correct me if I'm wrong. Show me an unbiased study on this topic and I'll concede.

Human performance, psychometry, and baseball statistics

24 Craig_Heldreth 15 October 2010 01:13PM

I. Performance levels and age

Human ambition for achievement in modest measure gives meaning to our lives, unless one is an existentialist pessimist like Schopenhauer who taught that life with all its suffering and cruelty simply should not be. Psychologists study our achievements under a number of different descriptions--testing for IQ, motivation, creativity, others. As part of my current career transition, I have been examining my own goals closely, and have recently read a fair amount on these topics which are variable in their evidence.

A useful collection of numerical data on the subject of human performance is the collection of Major League Baseball player performance statistics--the batting averages, number home runs, runs batted in, slugging percentage--of the many thousands of participants in the hundred years since detailed statistical records have been kept and studied by the players, journalists, and fans of the sport. The advantage of examining issues like these from the angle of Major League Baseball player performance statistics is the enormous sample size of accurately measured and archived data.

The current senior authority in this field is Bill James, who now works for the Boston Red Sox; for the first twenty-five years of his activity as a baseball statistician James was not employed by any of the teams. It took him a long time to find a hearing for his views on the inside of the industry, although the fans started buying his books as soon as he began writing them.

In one of the early editions of his Baseball Abstract, James discussed the biggest fallacies that managers and executives held regarding the achievements of baseball players. He was adamant about the most obvious misunderstood fact of player performance: it is sharply peaked at age 27 and decreases rapidly, so rapidly that only the very best players were still useful at the age of 35. He was able to observe only one executive that seemed to intuit this--a man whose sole management strategy was to trade everybody over the age of 30 for the best available player under the age of 30 he could acquire.

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Is it rational to be religious? Simulations are required for answer.

-13 Aleksei_Riikonen 11 August 2010 03:20PM

What must a sane person1 think regarding religion? The naive first approximation is "religion is crap". But let's consider the following:

Humans are imperfectly rational creatures. Our faults include not being psychologically able to maximally operate according to our values. We can e.g. suffer from burn-out if we try to push ourselves too hard.

It is thus important for us to consider, what psychological habits and choices contribute to our being able to work as diligently for our values as we want to (while being mentally healthy). It is a theoretical possibility, a hypothesis that could be experimentally studied, that the optimal2 psychological choices include embracing some form of Faith, i.e. beliefs not resting on logical proof or material evidence.

In other words, it could be that our values mean that Occam's Razor should be rejected (in some cases), since embracing Occam's Razor might mean that we miss out on opportunities to manipulate ourselves psychologically into being more what we want to be.

To a person aware of The Simulation Argument, the above suggests interesting corollaries:

  1. Running ancestor simulations is the ultimate tool to find out, what (if any) form of Faith is most conducive to us being able to live according to our values.
  2. If there is a Creator and we are in fact currently in a simulation being run by that Creator, it would have been rather humorous of them to create our world thus that the above method would yield "knowledge" of their existence.

 


1: Actually, what I've written here assumes we are talking about humans. Persons-in-general may be psychologically different, and theoretically capable of perfect rationality.

2: At least for some individuals, not necessarily all.

Beauty quips, "I'd shut up and multiply!"

6 neq1 07 May 2010 02:34PM

When it comes to probability, you should trust probability laws over your intuition.  Many people got the Monty Hall problem wrong because their intuition was bad.  You can get the solution to that problem using probability laws that you learned in Stats 101 -- it's not a hard problem.  Similarly, there has been a lot of debate about the Sleeping Beauty problem.  Again, though, that's because people are starting with their intuition instead of letting probability laws lead them to understanding.

The Sleeping Beauty Problem

On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.

Each interview consists of one question, "What is your credence now for the proposition that our coin landed heads?"

Two popular solutions have been proposed: 1/3 and 1/2

The 1/3 solution

From wikipedia:

Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1/3.

Yes, it's true that only in a third of cases would heads precede her awakening.

Radford Neal (a statistician!) argues that 1/3 is the correct solution.

This [the 1/3] view can be reinforced by supposing that on each awakening Beauty is offered a bet in which she wins 2 dollars if the coin lands Tails and loses 3 dollars if it lands Heads. (We suppose that Beauty knows such a bet will always be offered.) Beauty would not accept this bet if she assigns probability 1/2 to Heads. If she assigns a probability of 1/3 to Heads, however, her expected gain is 2 × (2/3) − 3 × (1/3) = 1/3, so she will accept, and if the experiment is repeated many times, she will come out ahead.

Neal is correct (about the gambling problem).

These two arguments for the 1/3 solution appeal to intuition and make no obvious mathematical errors.   So why are they wrong?

Let's first start with probability laws and show why the 1/2 solution is correct. Just like with the Monty Hall problem, once you understand the solution, the wrong answer will no longer appeal to your intuition.

The 1/2 solution

P(Beauty woken up at least once| heads)=P(Beauty woken up at least once | tails)=1.  Because of the amnesia, all Beauty knows when she is woken up is that she has woken up at least once.  That event had the same probability of occurring under either coin outcome.  Thus, P(heads | Beauty woken up at least once)=1/2.  You can use Bayes' rule to see this if it's unclear.

Here's another way to look at it:

If it landed heads then Beauty is woken up on Monday with probability 1.

If it landed tails then Beauty is woken up on Monday and Tuesday.  From her perspective, these days are indistinguishable.  She doesn't know if she was woken up the day before, and she doesn't know if she'll be woken up the next day.  Thus, we can view Monday and Tuesday as exchangeable here.

A probability tree can help with the intuition (this is a probability tree corresponding to an arbitrary wake up day):

If Beauty was told the coin came up heads, then she'd know it was Monday.  If she was told the coin came up tails, then she'd think there is a 50% chance it's Monday and a 50% chance it's Tuesday.  Of course, when Beauty is woken up she is not told the result of the flip, but she can calculate the probability of each.

When she is woken up, she's somewhere on the second set of branches.  We have the following joint probabilities: P(heads, Monday)=1/2; P(heads, not Monday)=0; P(tails, Monday)=1/4; P(tails, Tuesday)=1/4; P(tails, not Monday or Tuesday)=0.  Thus, P(heads)=1/2.

Where the 1/3 arguments fail

The 1/3 argument says with heads there is 1 interview, with tails there are 2 interviews, and therefore the probability of heads is 1/3.  However, the argument would only hold if all 3 interview days were equally likely.  That's not the case here. (on a wake up day, heads&Monday is more likely than tails&Monday, for example).

Neal's argument fails because he changed the problem. "on each awakening Beauty is offered a bet in which she wins 2 dollars if the coin lands Tails and loses 3 dollars if it lands Heads."  In this scenario, she would make the bet twice if tails came up and once if heads came up.  That has nothing to do with probability about the event at a particular awakening.  The fact that she should take the bet doesn't imply that heads is less likely.  Beauty just knows that she'll win the bet twice if tails landed.  We double count for tails.

Imagine I said "if you guess heads and you're wrong nothing will happen, but if you guess tails and you're wrong I'll punch you in the stomach."  In that case, you will probably guess heads.  That doesn't mean your credence for heads is 1 -- it just means I added a greater penalty to the other option.

Consider changing the problem to something more extreme.  Here, we start with heads having probability 0.99 and tails having probability 0.01.  If heads comes up we wake Beauty up once.  If tails, we wake her up 100 times.  Thirder logic would go like this:  if we repeated the experiment 1000 times, we'd expect her woken up 990 after heads on Monday, 10 times after tails on Monday (day 1), 10 times after tails on Tues (day 2),...., 10 times after tails on day 100.  In other words, ~50% of the cases would heads precede her awakening. So the right answer for her to give is 1/2.

Of course, this would be absurd reasoning.  Beauty knows heads has a 99% chance initially.  But when she wakes up (which she was guaranteed to do regardless of whether heads or tails came up), she suddenly thinks they're equally likely?  What if we made it even more extreme and woke her up even more times on tails?

Implausible consequence of 1/2 solution?

Nick Bostrom presents the Extreme Sleeping Beauty problem:

This is like the original problem, except that here, if the coin falls tails, Beauty will be awakened on a million subsequent days. As before, she will be given an amnesia drug each time she is put to sleep that makes her forget any previous awakenings. When she awakes on Monday, what should be her credence in HEADS?

He argues:

The adherent of the 1/2 view will maintain that Beauty, upon awakening, should retain her credence of 1/2 in HEADS, but also that, upon being informed that it is Monday, she should become extremely confident in HEADS:
P+(HEADS) = 1,000,001/1,000,002

This consequence is itself quite implausible. It is, after all, rather gutsy to have credence 0.999999% in the proposition that an unobserved fair coin will fall heads.

It's correct that, upon awakening on Monday (and not knowing it's Monday), she should retain her credence of 1/2 in heads.

However, if she is informed it's Monday, it's unclear what she conclude.  Why was she informed it was Monday?  Consider two alternatives.

Disclosure process 1:  regardless of the result of the coin toss she will be informed it's Monday on Monday with probability 1

Under disclosure process 1, her credence of heads on Monday is still 1/2.

Disclosure process 2: if heads she'll be woken up and informed that it's Monday.  If tails, she'll be woken up on Monday and one million subsequent days, and only be told the specific day on one randomly selected day.

Under disclosure process 2, if she's informed it's Monday, her credence of heads is 1,000,001/1,000,002.  However, this is not implausible at all.  It's correct.  This statement is misleading: "It is, after all, rather gutsy to have credence 0.999999% in the proposition that an unobserved fair coin will fall heads."  Beauty isn't predicting what will happen on the flip of a coin, she's predicting what did happen after receiving strong evidence that it's heads.

ETA (5/9/2010 5:38AM)

If we want to replicate the situation 1000 times, we shouldn't end up with 1500 observations.  The correct way to replicate the awakening decision is to use the probability tree I included above. You'd end up with expected cell counts of 500, 250, 250, instead of 500, 500, 500.

Suppose at each awakening, we offer Beauty the following wager:  she'd lose $1.50 if heads but win $1 if tails.  She is asked for a decision on that wager at every awakening, but we only accept her last decision. Thus, if tails we'll accept her Tuesday decision (but won't tell her it's Tuesday). If her credence of heads is 1/3 at each awakening, then she should take the bet. If her credence of heads is 1/2 at each awakening, she shouldn't take the bet.  If we repeat the experiment many times, she'd be expected to lose money if she accepts the bet every time.

The problem with the logic that leads to the 1/3 solution is it counts twice under tails, but the question was about her credence at an awakening (interview).

ETA (5/10/2010 10:18PM ET)


Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1/3.

Another way to look at it:  the denominator is not a sum of mutually exclusive events.  Typically we use counts to estimate probabilities as follows:  the numerator is the number of times the event of interest occurred, and the denominator is the number of times that event could have occurred. 

For example, suppose Y can take values 1, 2 or 3 and follows a multinomial distribution with probabilities p1, p2 and p3=1-p1-p2, respectively.   If we generate n values of Y, we could estimate p1 by taking the ratio of #{Y=1}/(#{Y=1}+#{Y=2}+#{Y=3}). As n goes to infinity, the ratio will converge to p1.   Notice the events in the denominator are mutually exclusive and exhaustive.  The denominator is determined by n.

The thirder solution to the Sleeping Beauty problem has as its denominator sums of events that are not mutually exclusive.  The denominator is not determined by n.  For example, if we repeat it 1000 times, and we get 400 heads, our denominator would be 400+600+600=1600 (even though it was not possible to get 1600 heads!).  If we instead got 550 heads, our denominator would be 550+450+450=1450.  Our denominator is outcome dependent, where here the outcome is the occurrence of heads.  What does this ratio converge to as n goes to infinity?  I surely don't know.  But I do know it's not the posterior probability of heads.

The Cameron Todd Willingham test

3 Kevin 05 May 2010 12:11AM

In 2004, The United States government executed Cameron Todd Willingham via lethal injection for the crime of murdering his young children by setting fire to his house. 

In 2009, David Grann wrote an extended examination of the evidence in the Willingham case for The New Yorker, which has called into question Willingham's guilt. One of the prosecutors in the Willingham case, John Jackson, wrote a response summarizing the evidence from his current perspective. I am not summarizing the evidence here so as to not give the impression of selectively choosing the evidence.

A prior probability estimate for Willingham's guilt (certainly not a close to optimal prior probability) is the probability that a fire resulting in the fatalities of children was intentionally set. The US Fire Administration puts this probability at 13%. The prior probability could be made more accurate by breaking down that 13% of intentionally set fires into different demographic sets, or looking at correlations with other things such as life insurance data.

My question for Less Wrong: Just how innocent is Cameron Todd Willingham? Intuitively, it seems to me that the evidence for Willingham's innocence is of higher magnitude than the evidence for Amanda Knox's innocence. But the prior probability of Willingham being guilty given his children died in a fire in his home is higher than the probability that Amanda Knox committed murder given that a murder occurred in Knox's house.

Challenge question: What does an idealized form of Bayesian Justice look like? I suspect as a start that it would result in a smaller percentage of defendants being found guilty at trial. This article has some examples of the failures to apply Bayesian statistics in existing justice systems.

The Last Number

4 Stuart_Armstrong 10 April 2010 12:09PM

"...90116633393348054920083..."

He paused for a moment, and licked his recently reconstructed lips. He was nearly there. After seventeen thousand subjective years of effort, he was, finally, just seconds away from the end. He slowed down as the finish line drew into sight, savouring and lengthening the moment where he stood there, just on the edge of enlightenment.

"...4...7...7...0...9...3..."

Those years had been long; longer, perhaps, in the effects they had upon him, than they could ever be in any objective or subjective reality. He had been human; he had been frozen, uploaded, simulated, gifted with robotic bodies inside three different levels of realities, been a conscript god, been split into seven pieces (six of which were subsequently reunited). He had been briefly a battle droid for the army of Orion, and had chanted his numbers even as he sent C-beams to glitter in the dark to scorch Formic worlds.

He had started his quest at the foot of a true Enlightened One, who had guided him and countless other disciples on the first step of the path. Quasi-enlightened ones had guided him further, as the other disciples fell to the wayside all around him, unable to keep their purpose focused. And now, he was on the edge of total Enlightenment. Apparently, there were some who went this far, and deliberately chose not to take the last step. But these were always friends of a friend of an acquaintance of a rumour. He hadn't believed they existed. And now that he had come this far, he knew these folk didn't exist. No-one could come this far, this long, and not finish it.

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Late Great Filter Is Not Bad News

14 Wei_Dai 04 April 2010 04:17AM

But I hope that our Mars probes will discover nothing. It would be good news if we find Mars to be completely sterile. Dead rocks and lifeless sands would lift my spirit.

Conversely, if we discovered traces of some simple extinct life form—some bacteria, some algae—it would be bad news. If we found fossils of something more advanced, perhaps something looking like the remnants of a trilobite or even the skeleton of a small mammal, it would be very bad news. The more complex the life we found, the more depressing the news of its existence would be. Scientifically interesting, certainly, but a bad omen for the future of the human race.

— Nick Bostrom, in Where Are They? Why I hope that the search for extraterrestrial life finds nothing

This post is a reply to Robin Hanson's recent OB post Very Bad News, as well as Nick Bostrom's 2008 paper quoted above, and assumes familiarity with Robin's Great Filter idea. (Robin's server for the Great Filter paper seems to be experiencing some kind of error. See here for a mirror.)

Suppose Omega appears and says to you:

(Scenario 1) I'm going to apply a great filter to humanity. You get to choose whether the filter is applied one minute from now, or in five years. When the designated time arrives, I'll throw a fair coin, and wipe out humanity if it lands heads. And oh, it's not the current you that gets to decide, but the version of you 4 years and 364 days from now. I'll predict his or her decision and act accordingly.

I hope it's not controversial that the current you should prefer a late filter, since (with probability .5) that gives you and everyone else five more years of life. What about the future version of you? Well, if he or she decides on the early filter, that would constitutes a time inconsistency. And for those who believe in multiverse/many-worlds theories, choosing the early filter shortens the lives of everyone in half of all universes/branches where a copy of you is making this decision, which doesn't seem like a good thing. It seems clear that, ignoring human deviations from ideal rationality, the right decision of the future you is to choose the late filter.

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