What do you mean? Are you saying that everyone with an average IQ is supposed to be able to understand what it means to minimize the within-cluster sum of squared differences, regardless of education? I don’t know what a standard deviation is either. I am able to read Wikipedia, understand what to do and use it. I know what squared means and I know what differences means. I just expected the sentence to mean more than the sum of its parts. Also I do not call the ability to use tools comprehension. What I value is to know when to use a particular tool, how to use it effectively and how it works.
You could teach stone-age people to drive a car. It would still seem like magic to them. Yet if you cloned them and exposed them to the right circumstance they might actually understand the internal combustion engine once grown up. Same IQ. Same as the server WolframAlpha is running on do possess a certain potential. Yet what enables the potential are the five million lines of Mathematica.
I’d be really surprised if one was able to understand the sentence the first time with a self-taught 1-year educational background in mathematics. That doesn’t mean that there are exceptions, I’m not a prodigy.
I think you’re right. “Sum of squared differences” makes sense as a normal thing to do with data points only if you’ve learned that it’s a measure of how spread apart they are, that it’s equivalent to the variance, and that making the variance small is a good way to ensure that a cluster is “well clumped.” There is a certain amount of intuition that’s built up from experience.
I also want to stress the point that I’m a bit biased(?) when it comes to understanding concepts. Surely I could accept any mathematical method or algorithm at face value. After all I’m also able to use WolframAlpha. But I feel that doesn’t count. At least I do not value such understanding. If you taught a prehistoric man to press some buttons he would be able to control a nuclear facility.
Many people are bothered by the counter-intuitive nature of probability. I have never been more confused by probability than by any other branch of mathematics. I believe that people regard probability as more difficult to understand because they learn about it much later than about other mathematical concepts. For me that is very different because it is all new to me. For me P(Y) ≥ P(X∧(X->Y)) is as (actually more) intuitive than a^2 + b^2 = c^2. The first makes sense in and of itself, the second needs context and proof (at least regarding my gut feeling). I just don’t see how 2 + 2 = 4 is more obvious than Bayes’ theorem. You just learnt to accept that 2 + 2 = 4 because 1.) you encounter the problem very often 2.) you can easily verify its solution 3.) you learn about it early on. But it is not self-evident.
I also want to stress the point that I’m a bit biased(?) when it comes to understanding concepts.
This is something people have noticed and it influences their responses. Aggressive “not understanding” is often considered a sign of bad faith, for good reason.
What I noticed is that everyone seems to assume that my problem to understand the sentence ”...within-cluster sum of squared differences...” was regarding “sum of squared differences” and not “within-cluster”. I don’t know the definition of the concept of a mathematical cluster. What might add to the confusion is that I’m not even sure about the meaning of the English word “cluster”. After that I decided to postpone reading the post. I could take the effort to look everything up of course but thought it would be more effective to read it in future.
Your post simply served as an example of how difficult it can be to read Less Wrong without a lot of background knowledge.
What I noticed is that everyone seems to assume that my problem to understand the sentence ”...within-cluster sum of squared differences...” was regarding “sum of squared differences” and not “within-cluster”.
Not really. I actually wrote a basic explanation of the whole sentence concept by concept but trimmed it down to the part that best illustrated dependence on mathematical background. Saying “within cluster is basically a phrase in English that refers to the same thing that’s in the title of the post” wouldn’t have helped convey the point. :P
It does, however, illustrate a different point. There is a trait related not just to intelligence but also to openness to information and flexible thinking that makes some people more suited than others to picking up and following new topics and ideas based on what they already know and filling in the blanks with their best inference. Confidence is part of it but part of it is social competition strategy embodied at the cognitive level.
There isn’t an explicit mathematical concept of a cluster.
Here’s what K-means does. Say, K is 3.
You try all the possible ways to partition your data points into three groups. You pick the partition that minimizes the sum of squared differences within each group. Then you iterate the procedure.
What do you mean? Are you saying that everyone with an average IQ is supposed to be able to understand what it means to minimize the within-cluster sum of squared differences, regardless of education?
No, approximately the opposite of that. Are you sure you didn’t intend this to be a reply to Peter? It seems to be quite an odd reply to me in the context.
You said that you have been polite in what you previously wrote. I parsed that the way that you agree with Peter de Blanc but that you have chosen to communicate this fact in a way that makes it possible to arrive at the conclusion without stating it. In other words, I should have been able to understand the sentence.
I didn’t reply to Peter de Blanc because I don’t know him and he doesn’t know me and so his statement that he would have understood Y without X doesn’t give me much information regarding my own intelligence. But you have actually read a lot of my comments and addressed me directly in the discussion above.
Interestingly I’m having a discussion (see my previous comments) with Roko if one should tell people directly if they are dumb or try to communicate such a truth differently.
Note polite enough to lie but polite enough to leave off all the caveats and exceptions. Some here could, understand the sentence even with no education in mathematics. Even so, the essentials of what I said was sincere. Piecing together that kind of jargon from the scraps of information available in the context is a far harder task than just understanding the article itself.
I was being polite.
What do you mean? Are you saying that everyone with an average IQ is supposed to be able to understand what it means to minimize the within-cluster sum of squared differences, regardless of education? I don’t know what a standard deviation is either. I am able to read Wikipedia, understand what to do and use it. I know what squared means and I know what differences means. I just expected the sentence to mean more than the sum of its parts. Also I do not call the ability to use tools comprehension. What I value is to know when to use a particular tool, how to use it effectively and how it works.
You could teach stone-age people to drive a car. It would still seem like magic to them. Yet if you cloned them and exposed them to the right circumstance they might actually understand the internal combustion engine once grown up. Same IQ. Same as the server WolframAlpha is running on do possess a certain potential. Yet what enables the potential are the five million lines of Mathematica.
I’d be really surprised if one was able to understand the sentence the first time with a self-taught 1-year educational background in mathematics. That doesn’t mean that there are exceptions, I’m not a prodigy.
I think you’re right. “Sum of squared differences” makes sense as a normal thing to do with data points only if you’ve learned that it’s a measure of how spread apart they are, that it’s equivalent to the variance, and that making the variance small is a good way to ensure that a cluster is “well clumped.” There is a certain amount of intuition that’s built up from experience.
I also want to stress the point that I’m a bit biased(?) when it comes to understanding concepts. Surely I could accept any mathematical method or algorithm at face value. After all I’m also able to use WolframAlpha. But I feel that doesn’t count. At least I do not value such understanding. If you taught a prehistoric man to press some buttons he would be able to control a nuclear facility.
Many people are bothered by the counter-intuitive nature of probability. I have never been more confused by probability than by any other branch of mathematics. I believe that people regard probability as more difficult to understand because they learn about it much later than about other mathematical concepts. For me that is very different because it is all new to me. For me P(Y) ≥ P(X∧(X->Y)) is as (actually more) intuitive than a^2 + b^2 = c^2. The first makes sense in and of itself, the second needs context and proof (at least regarding my gut feeling). I just don’t see how 2 + 2 = 4 is more obvious than Bayes’ theorem. You just learnt to accept that 2 + 2 = 4 because 1.) you encounter the problem very often 2.) you can easily verify its solution 3.) you learn about it early on. But it is not self-evident.
This is something people have noticed and it influences their responses. Aggressive “not understanding” is often considered a sign of bad faith, for good reason.
What I noticed is that everyone seems to assume that my problem to understand the sentence ”...within-cluster sum of squared differences...” was regarding “sum of squared differences” and not “within-cluster”. I don’t know the definition of the concept of a mathematical cluster. What might add to the confusion is that I’m not even sure about the meaning of the English word “cluster”. After that I decided to postpone reading the post. I could take the effort to look everything up of course but thought it would be more effective to read it in future.
Your post simply served as an example of how difficult it can be to read Less Wrong without a lot of background knowledge.
Not really. I actually wrote a basic explanation of the whole sentence concept by concept but trimmed it down to the part that best illustrated dependence on mathematical background. Saying “within cluster is basically a phrase in English that refers to the same thing that’s in the title of the post” wouldn’t have helped convey the point. :P
It does, however, illustrate a different point. There is a trait related not just to intelligence but also to openness to information and flexible thinking that makes some people more suited than others to picking up and following new topics and ideas based on what they already know and filling in the blanks with their best inference. Confidence is part of it but part of it is social competition strategy embodied at the cognitive level.
There isn’t an explicit mathematical concept of a cluster.
Here’s what K-means does. Say, K is 3.
You try all the possible ways to partition your data points into three groups. You pick the partition that minimizes the sum of squared differences within each group.
Then you iterate the procedure.
No, approximately the opposite of that. Are you sure you didn’t intend this to be a reply to Peter? It seems to be quite an odd reply to me in the context.
You said that you have been polite in what you previously wrote. I parsed that the way that you agree with Peter de Blanc but that you have chosen to communicate this fact in a way that makes it possible to arrive at the conclusion without stating it. In other words, I should have been able to understand the sentence.
I didn’t reply to Peter de Blanc because I don’t know him and he doesn’t know me and so his statement that he would have understood Y without X doesn’t give me much information regarding my own intelligence. But you have actually read a lot of my comments and addressed me directly in the discussion above.
Interestingly I’m having a discussion (see my previous comments) with Roko if one should tell people directly if they are dumb or try to communicate such a truth differently.
Note polite enough to lie but polite enough to leave off all the caveats and exceptions. Some here could, understand the sentence even with no education in mathematics. Even so, the essentials of what I said was sincere. Piecing together that kind of jargon from the scraps of information available in the context is a far harder task than just understanding the article itself.