Newcomb's paradox is an interesting thought experiment. Some philosophers think it has to do with the problem of free will and causality. Some think it is silly. There are many slightly differing formulations, and some think the details of the formulation matter a lot. I don't take the paradox too seriously and I don't think the details matter, nevertheless it is very interesting to observe people's reaction when they hear this paradox for the first time.
So here is my version of the paradox.
You are presented with two boxes, A and B. The box A is transparent, and inside it is $1,000. The box B is opaque. You are presented with two choices. Either you can take both boxes, or you can take the box B only. And then there is the predictor. With high accuracy the predictor predicts your choice. If he/she/it predicts you will take both boxes, it puts nothing in the box B. If he/she/it predicts you will take the box B only, it puts $1,000,000 in the box B. What is your choice?
Argument 1. The box B has either $1,000,000 or nothing. The predictor already predicted my choice and decided the content of the box B, so my choice doesn't affect the content of the box B in any way. Taking both boxes is always superior to taking the box B only, by the amount of $1,000. Therefore I take both boxes.
Argument 2. Since the predictor is highly accurate, if I choose both boxes it is highly likely that I will find the box B to be empty, and that my income will be $1,000. Since the predictor is highly accurate, if I choose the box B only it is highly likely that I will find the box B to have $1,000,000, and that my income will be $1,000,000. Therefore I take the box B only.
Note that the first argument is independent of the predictor's accuracy. Also note that the mechanism of the predictor's prediction is completely unspecified, and you have no way to assess its accuracy other than that it is "highly accurate".
Let's change that. Suppose that you are an observer, and you observed 10 (100, 1000, whatever) people choosing either both boxes or one box. Suppose that you observed the predictor's prediction to be always correct. Would it change your choice? Does the number of observations matter? Note that if you support the first argument, you have no reason to change your choice.
Also note that unless the mechanism of the prediction is specified, you can argue that the past performance is no proof of the future performance, and to think otherwise succumbs to the gambler's fallacy. The gambler may think that if coin toss turned head ten times in a row, the next toss is likely to turn head (or tail). Of course if coin is fair, each toss is independent and head and tail are equally likely, no matter what.
Suppose that you observed the predictor's prediction to be 90% correct. It is easy to cacluate that the expected value for both boxes is $101,000 and for one box is $900,000. Would the fact that the predictor is not perfect change your choice? Does the exact probability matter? Does the ratio of the amount of money matter? If your choice is based on the expected value, they probably do.
Some argues that the predictor as presented in the paradox can not exist. Suppose that people in the paradox are replaced with AI programs, and the predictor has much faster computer to run programs. Then it is obvious to me that the predictor can indeed predict the behavior. Since I accept the Turing thesis (which states, in short, human-like AI is possible), I also accept that such predictor can exist.
Also note that the predictor in the paradox does not need to be perfect. Public opinion polls routinely make predictions such as that females are more likely to vote for the candidate X, that people who live in Y are more likely to vote for the candidate Z, etc. How can you argue that it is impossible to predict the human behavior?
(P.S. So what is my choice? I choose the box B. My reason shall remain private though.)
2008년 4월 21일 월요일
2008년 4월 5일 토요일
Pinball
There is an increasing danger of practicing tetrapyloctomy, as discussion gets longer. This post is another attempt at discussing learning, research, and related matters. Readers should judge whether I succeeded in avoiding the danger.
Let me first cite my sources. The first part of the argument closely follows the article The Two Cultures of Mathematics by Timothy Gowers. The link contains the complete article. The second part of the argument was influenced by the chapter titled "Pinball" from the book The Soul of a New Machine by Tracy Kidder. I can't link to the full text, but the link contains a good summary of pinball analogy.
Now, here is a short recap of the discussion so far:
1. You are scientists so you should love learning.
2. Scientists learn just enough so that they can do research.
3. This is just a semantic matter; scientists must indeed love learning, else why would they bother to do the research?
4. Consider an analogy to writers. Reading is not what they are paid for; writing is. Similarly, scientists do not learn science; scientists do science.
5. I wrote that scientists must be motivated by a love of learning. What motivates scientists to do research? Certainly not money. I think it is outcome. Perhaps you are imagining a sort of performance-art scientist, for whom the very process of research provides its own reward, regardless of outcome?
Certainly, I am not making ridiculous claims like that scientists love research but not learning, or that writers love writing but not reading. I am talking about the priority. Consider the following two statements:
1. The point of scientific research is to understand the nature better.
2. The point of understanding the nature is to do the better scientific research.
At first, the second statement may seem odd. Some variations should help understand it better: 1. The point of understanding the nature is to apply that understanding; one application is as a basis for an effort to achieve deeper understanding. 2. The point of learning existing knowledge about the nature is to do research (as opposed to intrinsic value of such knowledge).
There is truth in both statements, but one does not necessarily agree with both statements to the same degree. I claim that average scientists are inclined to the second rather than the first. You could argue that I am imagining "performance-art scientist"; actually I am claiming that it is not an imagination, but probably a majority. I will argue two points, that outcome is often not the most important, and that research is often primarily seen as a vehicle for further research.
It seems to me that scientists are often not too conscious about outcome of their research. The case of physicists participated in the Manhattan Project is a classic one.
As a student in the computer science department, I want to give another example. Computers are often said to be general purpose, which means, unlike calculators, televisons, or MP3 players, it can do many tasks (like calculation, playing video, and playing audio) given appropriate softwares and peripherals. It is also a common knowledge that general purpose computer chips are used heavily in military applications. If one wants to avoid any association with tanks or missiles because one is against wars, there is no way other than ceasing to work on electronic chip design at all. However, in reality, it won't be too much of exaggeration to assert that most electronic chip designers are utterly uninterested in most applications of their work -- applications that range from executing financial strategies in derivative market to producing virtual child pornography.
It also seems that research are often judged in terms of potential for further research, rather than in isolation. One often talks about "profitable line of research". Naive interpretation is that this refers to research that brings profit. It actually means that the research is likely to lead to further research.
So is the goal of the game of pinball. You win one game, you get to play another. For writers, you finish one novel, you get to write another. For mathematicians, you solve one difficult problem, you get to solve another. For engineers, be they civil engineers, computer hardware designers, or software developers, you complete one engineering project, you get to join another. For scientists, you research one subject, you get to research it further.
I am not saying that one particular round of pinball, or one theorem, or one bridge, or one paper is not important; but as or more important is the process of creation, and the hope of better future. Whatever you learn from others, by definition, is a thing of the past. Whatever you learned yourself, becomes a thing of the past when you learned it. Whatever you are yet to learn, especially if nobody has ever learned it, is the most interesting.
Speaking of my experience, I often longed for the beginning of a new software project when one project was almost at the end. The endgame is not as interesting. However, the endgame, or the polish, absolutely matters in commercial software development. I often thought that this is why most research software, as opposed to commercial software, lacks in polish -- because there is less pressure for the polished product.
If Rowling does not want to write another Harry Potter book, I can understand.
Let me first cite my sources. The first part of the argument closely follows the article The Two Cultures of Mathematics by Timothy Gowers. The link contains the complete article. The second part of the argument was influenced by the chapter titled "Pinball" from the book The Soul of a New Machine by Tracy Kidder. I can't link to the full text, but the link contains a good summary of pinball analogy.
Now, here is a short recap of the discussion so far:
1. You are scientists so you should love learning.
2. Scientists learn just enough so that they can do research.
3. This is just a semantic matter; scientists must indeed love learning, else why would they bother to do the research?
4. Consider an analogy to writers. Reading is not what they are paid for; writing is. Similarly, scientists do not learn science; scientists do science.
5. I wrote that scientists must be motivated by a love of learning. What motivates scientists to do research? Certainly not money. I think it is outcome. Perhaps you are imagining a sort of performance-art scientist, for whom the very process of research provides its own reward, regardless of outcome?
Certainly, I am not making ridiculous claims like that scientists love research but not learning, or that writers love writing but not reading. I am talking about the priority. Consider the following two statements:
1. The point of scientific research is to understand the nature better.
2. The point of understanding the nature is to do the better scientific research.
At first, the second statement may seem odd. Some variations should help understand it better: 1. The point of understanding the nature is to apply that understanding; one application is as a basis for an effort to achieve deeper understanding. 2. The point of learning existing knowledge about the nature is to do research (as opposed to intrinsic value of such knowledge).
There is truth in both statements, but one does not necessarily agree with both statements to the same degree. I claim that average scientists are inclined to the second rather than the first. You could argue that I am imagining "performance-art scientist"; actually I am claiming that it is not an imagination, but probably a majority. I will argue two points, that outcome is often not the most important, and that research is often primarily seen as a vehicle for further research.
It seems to me that scientists are often not too conscious about outcome of their research. The case of physicists participated in the Manhattan Project is a classic one.
As a student in the computer science department, I want to give another example. Computers are often said to be general purpose, which means, unlike calculators, televisons, or MP3 players, it can do many tasks (like calculation, playing video, and playing audio) given appropriate softwares and peripherals. It is also a common knowledge that general purpose computer chips are used heavily in military applications. If one wants to avoid any association with tanks or missiles because one is against wars, there is no way other than ceasing to work on electronic chip design at all. However, in reality, it won't be too much of exaggeration to assert that most electronic chip designers are utterly uninterested in most applications of their work -- applications that range from executing financial strategies in derivative market to producing virtual child pornography.
It also seems that research are often judged in terms of potential for further research, rather than in isolation. One often talks about "profitable line of research". Naive interpretation is that this refers to research that brings profit. It actually means that the research is likely to lead to further research.
So is the goal of the game of pinball. You win one game, you get to play another. For writers, you finish one novel, you get to write another. For mathematicians, you solve one difficult problem, you get to solve another. For engineers, be they civil engineers, computer hardware designers, or software developers, you complete one engineering project, you get to join another. For scientists, you research one subject, you get to research it further.
I am not saying that one particular round of pinball, or one theorem, or one bridge, or one paper is not important; but as or more important is the process of creation, and the hope of better future. Whatever you learn from others, by definition, is a thing of the past. Whatever you learned yourself, becomes a thing of the past when you learned it. Whatever you are yet to learn, especially if nobody has ever learned it, is the most interesting.
Speaking of my experience, I often longed for the beginning of a new software project when one project was almost at the end. The endgame is not as interesting. However, the endgame, or the polish, absolutely matters in commercial software development. I often thought that this is why most research software, as opposed to commercial software, lacks in polish -- because there is less pressure for the polished product.
If Rowling does not want to write another Harry Potter book, I can understand.