Philosophy Mondays: Human-AI Collaboration
Today's Philosophy Monday is an important interlude. I want to reveal that I have not been writing the posts in this series entirely by myself. Instead I have been working with Claude, not just for the graphic illustrations, but also for the text. My method has been to write a rough draft and then ask Claude for improvement suggestions. I will expand this collaboration to other intelligences going forward, including open source models such as Llama and DeepSeek. I will also explore other moda...
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Web3/Crypto: Why Bother?
One thing that keeps surprising me is how quite a few people see absolutely nothing redeeming in web3 (née crypto). Maybe this is their genuine belief. Maybe it is a reaction to the extreme boosterism of some proponents who present web3 as bringing about a libertarian nirvana. From early on I have tried to provide a more rounded perspective, pointing to both the good and the bad that can come from it as in my talks at the Blockstack Summits. Today, however, I want to attempt to provide a coge...
Philosophy Mondays: Human-AI Collaboration
Today's Philosophy Monday is an important interlude. I want to reveal that I have not been writing the posts in this series entirely by myself. Instead I have been working with Claude, not just for the graphic illustrations, but also for the text. My method has been to write a rough draft and then ask Claude for improvement suggestions. I will expand this collaboration to other intelligences going forward, including open source models such as Llama and DeepSeek. I will also explore other moda...
Intent-based Collaboration Environments
AI Native IDEs for Code, Engineering, Science
Web3/Crypto: Why Bother?
One thing that keeps surprising me is how quite a few people see absolutely nothing redeeming in web3 (née crypto). Maybe this is their genuine belief. Maybe it is a reaction to the extreme boosterism of some proponents who present web3 as bringing about a libertarian nirvana. From early on I have tried to provide a more rounded perspective, pointing to both the good and the bad that can come from it as in my talks at the Blockstack Summits. Today, however, I want to attempt to provide a coge...
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Last Uncertainty Wednesday we dug deeper into understanding the distribution of sample means. I ended with asking why the chart for 100,000 samples of size 10 looked smoother than then one for samples of size 100 (just as a refresher, these are all rolls of a fair die). Well, for a sample of size 10, there are 51 possible values of the mean: 1.0, 1.1, 1.2, 1.3 … 5.8, 5.9, 6.0. But with a sample size of 100 there are 501 possible values for the mean. So with the same number of samples (100,000) the distribution will not be approximated as closely. We can fix this by upping the number of samples to say 1 million. Thanks to the amazing speed of a modern laptop even 100 million rolls of a die just take a couple of minutes (this still blows my mind). Here is the resulting chart:
Much smoother than before! We could make that even smoother by taking up the number of runs even further.
OK. So what would happen if we went to sample size 1,000? Well, by now this should be easy to predict. The distribution of sample means will be even tighter around 3.5 (the expected value of the distribution) and in order to get a smooth chart we have to further up the number of runs.
So what is the limit here? Well, this get us to the law of large numbers, which essentially states that the sample average will converge to the expected value as the sample grows larger. There is a strong version and a weak version of the law, a distinction which we may get to later (plus some more versions of the law).
For now though the important thing to keep in mind is that when we have small sample sizes, the sample mean may be far away from the expected value. And as we see above that even for a super simple probability distribution with 6 equally likely outcomes there is considerable variation in the sample mean even for samples of size 100! So it is very easy to make mistakes from jumping to conclusions on small samples.
Next Wednesday we will see that the situation is in fact much worse than that. Here is a hint: every sample has a mean (why?) but does every probability distribution have an expected value?
Last Uncertainty Wednesday we dug deeper into understanding the distribution of sample means. I ended with asking why the chart for 100,000 samples of size 10 looked smoother than then one for samples of size 100 (just as a refresher, these are all rolls of a fair die). Well, for a sample of size 10, there are 51 possible values of the mean: 1.0, 1.1, 1.2, 1.3 … 5.8, 5.9, 6.0. But with a sample size of 100 there are 501 possible values for the mean. So with the same number of samples (100,000) the distribution will not be approximated as closely. We can fix this by upping the number of samples to say 1 million. Thanks to the amazing speed of a modern laptop even 100 million rolls of a die just take a couple of minutes (this still blows my mind). Here is the resulting chart:
Much smoother than before! We could make that even smoother by taking up the number of runs even further.
OK. So what would happen if we went to sample size 1,000? Well, by now this should be easy to predict. The distribution of sample means will be even tighter around 3.5 (the expected value of the distribution) and in order to get a smooth chart we have to further up the number of runs.
So what is the limit here? Well, this get us to the law of large numbers, which essentially states that the sample average will converge to the expected value as the sample grows larger. There is a strong version and a weak version of the law, a distinction which we may get to later (plus some more versions of the law).
For now though the important thing to keep in mind is that when we have small sample sizes, the sample mean may be far away from the expected value. And as we see above that even for a super simple probability distribution with 6 equally likely outcomes there is considerable variation in the sample mean even for samples of size 100! So it is very easy to make mistakes from jumping to conclusions on small samples.
Next Wednesday we will see that the situation is in fact much worse than that. Here is a hint: every sample has a mean (why?) but does every probability distribution have an expected value?
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