Foundations of probability theory
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Foundations of probability theory
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Re: Foundations of probability theory
#2I have a question that might be related to the topic. I have found that often, when solving problems related to probability theory, it is more convenient to think in terms of combinations than to think in terms of probabilities (perhaps because I'm a programmer). Would it be possible to devise a programming language that allows me to program a filter that selects the desired outcomes out of the complete set of combinations, and to have the compiler automatically deduce a closed-form formula for the probability (without enumeration)? If this is not possible in general, what would the restrictions on this programming language be, to make it work in practice?
So for example, given the question what the probability is that, when throwing 4 coins, 2 of which will be heads; I could write a function that generates all possible outcomes "TTTT", "TTTH", etc. And I could write a filter function that returns true for "TTHH", "THTH", "THHT", etc.
Re: Foundations of probability theory
#3Looked at the text to be used,
Durrett. There's more and with
higher quality in any of
M. Loeve, J. Neveu, K. Chung,
and L. Breiman.
For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that.
And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
Re: Foundations of probability theory
#4Looked at the text to be used, Durrett. There's more and with higher quality in any of M. Loeve, J. Neveu, K. Chung, and L. Breiman. For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that. And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
Could you expand on this? I'm reading his post now, but from the outside, it seems unlikely that Terry got the concepts completely wrong, and more likely that he's using slightly different definitions for the concepts than you are expecting.
Re: Foundations of probability theory
#5I have a question that might be related to the topic. I have found that often, when solving problems related to probability theory, it is more convenient to think in terms of combinations than to think in terms of probabilities (perhaps because I'm a programmer). Would it be possible to devise a programming language that allows me to program a filter that selects the desired outcomes out of the complete set of combin…
You might find this interesting: http://blog.plover.com/prog/haskell/probmonad.html
Doesn't produce "closed-form formula", but, well, those tend to scale to programming-sized tasks poorly anyhow.
Re: Foundations of probability theory
#6Looked at the text to be used, Durrett. There's more and with higher quality in any of M. Loeve, J. Neveu, K. Chung, and L. Breiman. For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that. And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
Out of curiosity, what's the background that lets you be so dismissive of a Fields Medal winner?
At this point I see a blog post written by a well respected mathematician whom I feel comfortable trusting and it's being brushed aside by I don't know who.
Re: Foundations of probability theory
#7Looked at the text to be used, Durrett. There's more and with higher quality in any of M. Loeve, J. Neveu, K. Chung, and L. Breiman. For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that. And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
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Re: Foundations of probability theory
#8I have a question that might be related to the topic. I have found that often, when solving problems related to probability theory, it is more convenient to think in terms of combinations than to think in terms of probabilities (perhaps because I'm a programmer). Would it be possible to devise a programming language that allows me to program a filter that selects the desired outcomes out of the complete set of combin…
This is possible, and in fact probably implemented in some probabilistic programming languages, but I think you are looking at the wrong direction.
The point is that even for fairly simple real use cases, the computation complexity is so huge, that all computers in the world couldn't compute it in your lifetime if you don't employ some approximation or optimization and stick to naive algorithms.
So, that is what the whole field of machine learning is about: finding some clever ways to deal with random variables in a computationally feasible way...
Re: Foundations of probability theory
#9Looked at the text to be used, Durrett. There's more and with higher quality in any of M. Loeve, J. Neveu, K. Chung, and L. Breiman. For what Tao wrote on his page about determinism or whatever on that page -- just f'get about that. And what he wrote, confusing a sample space and a probability space, just say that he had a bad day that day.
> In the standard foundations of probability theory, as laid out by Kolmogorov, we can then model these events and random variables by introducing a sample space (which will be a probability space) to capture all the ambient sources of randomness; events are then modeled as measurable subsets of this sample space, and random variables are modeled as measurable functions on this sample space.
This matches the definitions of Wikipedia pretty closely: "A probability space consists of three parts: A sample space [...], A set of events [...], The assignment of probabilities to the events".
So either you misunderstood that sentence or both Wikipedia and Terrence Tao are wrong.
Re: Foundations of probability theory
#10I have a question that might be related to the topic. I have found that often, when solving problems related to probability theory, it is more convenient to think in terms of combinations than to think in terms of probabilities (perhaps because I'm a programmer). Would it be possible to devise a programming language that allows me to program a filter that selects the desired outcomes out of the complete set of combin…
Most introductory books on probability I have seen open with the combinatorical approach you describe. When it is applicable, this approach is very powerfull. However there is essentially nothing novel in combinatorical probability relative to combinatorics, so research and advanced topics in this field are done under the title of combinatorics, instead of probability. Probability becomes a field in its own right once we move beyond the cases where a combinatorical approach will work.