Please specify in what sense the first line was correct, or declare it an error. Pronouncing assertions known to be incorrect and then just shrugging that off shouldn’t be acceptable on this forum.
One wants an universal probability space where one can find the probability of any event. This is possible:
One way of making such a space is to take all recursive functions of some universal computer, run them, and storing the output, resulting in an universal probability space because every possible set of events will be there, as the results of infinitely many recursive functions, or programs as they are called. The probabilities corresponds to the density of these outputs, these events.
A counterargument is that it is too dependent on the actual universal computer chosen. However, theorems in algorithmic information theory shows that this dependence converges asymptotically as information increases, because the difference of densities of different outputs from different universal computers can at most be 2 to the power of the shortest program simulating the universal computer in another universal computer.
One way of making such a space is to take all recursive functions of some universal computer, run them, and storing the output,
OK....
resulting in an universal probability space because every possible set of events will be there
what!? You haven’t yet described a probability space. The aforementioned set is infinite, so the uniform distribution is unavailable. What probability distribution will you have on this set of recursive-function-runs. And in what way is the resulting probability space universal?
All that will be answered if you study “algorithmic information theory”, or “Kolmogorov komplexity” as it is also called. You can find some of it on the net of course, or you can read the book “An Introduction to Kolmogorov Complexity and its Applications” by Ming Li & Paul Vitányi. Thats the book I read, some years after I invented it myself.
You are wrong because I did specify a probability space.
The probability space I specified was one where the sample space was the set of all outputs of all programs for some universal computer, and the measure was one from the book I mentioned. One could for instance choose the Solomonoff measure, from 4.4.3.
From your writings I conclude that is it quite likely that you are neither quite aware of the concept, nor understanding what I write, while believing you do.
You are wrong because I did specify a probability space.
No, you specified the points and not the measure.
The probability space I specified was one where the sample space was the set of all outputs of all programs for some universal computer, and the measure was one from the book I mentioned. One could for instance choose the Solomonoff measure, from 4.4.3.
OK! Now we’ve got a space. Of course if you wanted to talk about solomonoff measure, why didn’t you just sayso 5 comments ago. Pretty much everyone reading less wrong would have immediately known what you were talking about. You still haven’t justified calling the solomonoff space “universal”.
From your writings I conclude that is it quite likely that you are neither quite aware of the concept, nor understanding what I write, while believing you do.
Now you’re just being rude. You don’t know me, you certainly don’t know what I do or don’t know.
It is universal, because every possible sequence is generated.
It is universal, because it is based on universally recursive functions.
It is universal, because it uses an universal computer.
People knowing algorithmic complexity know that it is about probability measures, spaces, universality, etc. You apparently did not, while nitpicking instead.
I’m not nitpicking, you’re wrong. “Universal” in this context means, to quote the original poster
i.e. the probability space of all events that could occur at some time during the existence of the universe
What the heck does that have to do with every possible sequence being generated? For that matter what does it have to do with sequences at all? The solomonoff measure is a measure over sequences in a finite alphabet, or to put it simpler, Integers. How do I express an event like “it will rain next tuesday” as a subset of the integers?
Whatever you are using the word “universal” to mean, it is not anything like what the OP had in mind. The Solomonoff measure is an interesting mathematical object for sure, and it may be quite relevant to the topic of real-world Bayesian reasoning, but it’s obviously not universal in that sense.
also: what the heck does “universally recursive” mean? Did you just make up that term right now? Because I’ve never heard it before, it only has 10 google hits, and none of them are relevant to this discussion.
Events, and the universe itself, are encodable as sequences.
This means that events, and possible universes, are a subset of the sequences generated from the universal computer.
Algorithmic information theory can then be used to find probabilities for events and universes.
This is one of the CENTRAL POINTS of Algorithmic Information Theory.
What I am doing now, is teaching you A.I.T., while you wrongly claim you understand it, and wrongly claim I do not, despite an amount of evidence to the contrary. I therefore conclude that you are not very rational.
kim0, you are trolling now. You are not communicating clearly, and then claim that the objections to your unclear communication are invalid, because you can retroactively amend the bad connotations and ambiguities, but in the process of doing so, you introduce further false-sounding and ambiguous statements. You should choose your words more carefully.
Google gives 0 hits on “recursive probability space”. Blanket assertions like this need to be technically precise.
I refer interested readers to the Algorithmic probability article on Scholarpedia.
The technically precise reference was this part:
“This is algorithmic information theory,..”
But if you claim my first line was too obfuscated, I can agree.
Kim Øyhus
Please specify in what sense the first line was correct, or declare it an error. Pronouncing assertions known to be incorrect and then just shrugging that off shouldn’t be acceptable on this forum.
O.K.
One wants an universal probability space where one can find the probability of any event. This is possible:
One way of making such a space is to take all recursive functions of some universal computer, run them, and storing the output, resulting in an universal probability space because every possible set of events will be there, as the results of infinitely many recursive functions, or programs as they are called. The probabilities corresponds to the density of these outputs, these events.
A counterargument is that it is too dependent on the actual universal computer chosen. However, theorems in algorithmic information theory shows that this dependence converges asymptotically as information increases, because the difference of densities of different outputs from different universal computers can at most be 2 to the power of the shortest program simulating the universal computer in another universal computer.
Kim Øyhus
OK....
what!? You haven’t yet described a probability space. The aforementioned set is infinite, so the uniform distribution is unavailable. What probability distribution will you have on this set of recursive-function-runs. And in what way is the resulting probability space universal?
All that will be answered if you study “algorithmic information theory”, or “Kolmogorov komplexity” as it is also called. You can find some of it on the net of course, or you can read the book “An Introduction to Kolmogorov Complexity and its Applications” by Ming Li & Paul Vitányi. Thats the book I read, some years after I invented it myself.
I have. I’m not an expert in it, but I’m quite aware of the concept.
You have not specified a probability space, and you have not made any attempt to justify calling the space you didn’t specify “universal”
uh huh.
You are wrong because I did specify a probability space.
The probability space I specified was one where the sample space was the set of all outputs of all programs for some universal computer, and the measure was one from the book I mentioned. One could for instance choose the Solomonoff measure, from 4.4.3.
From your writings I conclude that is it quite likely that you are neither quite aware of the concept, nor understanding what I write, while believing you do.
No, you specified the points and not the measure.
OK! Now we’ve got a space. Of course if you wanted to talk about solomonoff measure, why didn’t you just say so 5 comments ago. Pretty much everyone reading less wrong would have immediately known what you were talking about. You still haven’t justified calling the solomonoff space “universal”.
Now you’re just being rude. You don’t know me, you certainly don’t know what I do or don’t know.
It is universal, because every possible sequence is generated.
It is universal, because it is based on universally recursive functions.
It is universal, because it uses an universal computer.
People knowing algorithmic complexity know that it is about probability measures, spaces, universality, etc. You apparently did not, while nitpicking instead.
I’m not nitpicking, you’re wrong. “Universal” in this context means, to quote the original poster
What the heck does that have to do with every possible sequence being generated? For that matter what does it have to do with sequences at all? The solomonoff measure is a measure over sequences in a finite alphabet, or to put it simpler, Integers. How do I express an event like “it will rain next tuesday” as a subset of the integers?
Whatever you are using the word “universal” to mean, it is not anything like what the OP had in mind. The Solomonoff measure is an interesting mathematical object for sure, and it may be quite relevant to the topic of real-world Bayesian reasoning, but it’s obviously not universal in that sense.
also: what the heck does “universally recursive” mean? Did you just make up that term right now? Because I’ve never heard it before, it only has 10 google hits, and none of them are relevant to this discussion.
Events, and the universe itself, are encodable as sequences.
This means that events, and possible universes, are a subset of the sequences generated from the universal computer.
Algorithmic information theory can then be used to find probabilities for events and universes.
This is one of the CENTRAL POINTS of Algorithmic Information Theory.
What I am doing now, is teaching you A.I.T., while you wrongly claim you understand it, and wrongly claim I do not, despite an amount of evidence to the contrary. I therefore conclude that you are not very rational.
kim0, you are trolling now. You are not communicating clearly, and then claim that the objections to your unclear communication are invalid, because you can retroactively amend the bad connotations and ambiguities, but in the process of doing so, you introduce further false-sounding and ambiguous statements. You should choose your words more carefully.