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Looking forward to the next part then.
it's pretty unclear from the research whether NIT actually reduces employment (the main side effect of penalizing productivity). Of course theoretically it should, but the data isn't really conclusive in either direction.
I think as a general rule, when something theoretically is supposed to happen and we do not have conclusive practical data, its reasonable to assume that it indeed will happen.
You'd rather have a program that costs $4 trillion with zero maintenance costs, than a similarly impactful program that costs $~650 billion with maintenance costs?
Depends on the maintenance cost, obviously. But again the problem here is that our best theories strongly predict that impact will not be similar. NIT creates perversive incentives that UBI doesn't. And incentives matter - economy isn't a zero sum game.
So a more appropriate question is: Would you rather invest 4 trillions in a company which evaluation will go up or 650 billions in a company which evaluation will go down?
pretty dicey political stance.
I predict that implementation of general NIT will face much more pushback than implementation of UBI. No matter how minor are the required tweaks to EITC they are politically unfeasible in US. Meanwhile, UBI has support all around the political spectrum, from leftists to libertarians.
From your title it seems that you are going to talk about UBI in the context of automation. This context immediately gives us the source of all the wealth that is to be redistributed via UBI - the gains of productivity due to automation . But then suddenly you start talking as if you don't know where to get the money and investigate the option of replacing current welfare programs. This is super confusing. If you wanted to talk about UBI as a replacement for welfare, why mention automation?
When you compare UBI with NIT and existent welfare programs you do not mention the elephant in the room - UBI doesn't penalize productivity. Also it's not clear why would implementing NIT globally would not require dismantling current welfare system or doing a huge tax reform if implementing UBI would.
Another very important andvantage of UBI is that it's easy to maintain. You do not need a huge bureaucracy tracking employment status of multiple people and their income, processing papers and validating that noone is cheating the system. You simply send money to everyone.
Almost any set: only the empty set is excluded. The identities of the elements themselves are irrelevant to the mathematical structure. Any further restrictions are not a part of mathematical definition of probability space, but some particular application you may have in mind.
Any non-empty set can be a sample space for some problem. But we are interested in a sample space for a very specific problem - Sleeping Beauty. This "applications we have in mind" is the whole point of the discussion.
Was it really not clear? I have a whole post dedicated towards exploring different probability spaces that people proposed for the Sleeping Beauty problem, where I explicitly notice tha these probability spaces are valid in principle but not sound when applied to the Sleeping Beauty problem - they describe some other problems instead. Likewise, in this post I point that indexical sample space for one problem can be a sample space for another problem - the one where elements of the set indeed are mutually exclusive outcomes. How did you manage to miss it?
Anyway, when we fixed the problem that we are talking about, there are some specific conditions according to which we can say whether a set indeed is a sample space for this problem. And one of them is that the elements of it has to be mutually exclusive outcomes, such as in every iteration of the probability experiment one and only one of them is realized. Are we on the same page here?
So if we define a set in such a way that it consist of outcomes that happen multiple times during the same trial, it can't be a sample space for this problem anymore. However we can remove this restriction by defining a new entity: indexical sample space for this problem.
In some cases this is reasonable, but in others it is impossible. For example, when defining continuous probability distributions you can't eliminate sample set elements having measure zero or you will be left with the empty set.
Fair point. No disagreement here. I was talking specifically about discrete case.
It is a synonym for probability in the sense that it is a mathematical probability: that is, a measure over a sigma-algebra for which the axioms of a probability space are satisfied. I use a different term here to denote this application of the mathematical concept to a particular real-world purpose.
Our particulr real world purpose is the Sleeping Beuaty problem. Credence that the Beauty has isn't just a value of measure function from some probability space it's the value of probability function from the probability space specifically for this problem. Which sample space has to consist from mutually exclusive outcomes for this problem.
I also don't quite know what you mean by the phrase "stable truth value"
There is this thing called probability experiment. Which I provided you a wiki link to in a previous comment. According to Kolmogorov, it's some complex of conditions that can be repeated and on repetition outputs different results all of which belong to some set. On any iteration of the experiment exactly one result is output. This is how we merge our mathematical model with the phisical universe. We say that these results are elements of the sample space. And so one outcome of the sample space is realized at every iteration of the experiment. And every element of the event space to which the realized outcome belongs to is also considered to be realized. Thus we can say that for every iteration of probability experiment every well-defined event is either realized or not realized - has a stable truth value.
If something doesn't have a stable truth value during an iteration of the experiment, it's not a well defined event. This is exactly what happens with statement "Today is Monday" in Sleeping Beauty experiment.
either restrict the universe states in your set to locations or time regions over which all selected propositions have a definite truth value, or restrict the propositions to those that have a definite truth value over the selected universe states. Either way works.
What you are saying is that we can either modify the setting of the experiment so that the statement have a stable truth value or use different statements. This is true. But we want to talk about a specific experimental setting - Sleeping Beauty problem - we can't modify it, otherwise we would be talking about something else, like No-Coin-Toss problem. Therefore, the only way is to acknowledge that some statements are not coherent in the current experimental setting and not use them. This is exactly what I do.
I described the structure under which it is, and you can verify that it does in fact satisfy the axioms of a probability space.
The fact that you can describe such mathematical structure means that there is an experiment where statement "Today is Monday" is a well-defined event. Here we are in agreement. More specifically this is the experiment where awakenings on Monday and Tuesday are mutually exclusive during one trial, such as No-Coin-Toss or Single-Awakening.
But these experiments are not Sleeping Beauty experiment. And this is what I've been talking about the whole time that specifically in Sleeping Beauty experiment, from the perspective of the Beauty "Today is Monday" isn't a well-defined event, it doesn't have a stable truth value.
You seem to be thinking that if something is a well-defined event in some experiment, it has to be a well defined event in Sleeping Beauty experiment as well. Is this our crux?
Otherwise you can't model even a non-coin flip form of the Sleeping Beauty problem in which Beauty is always awoken twice. If the problem asks "what should be Beauty's credence that it is Monday" then you can't even model the question without distinguishing universe states by time.
Yes. This is a completely correct conclusion which I'm planning to talk about in the next post. If a person goes through memory loss and the repetition of the same experience, there is no coherent credence/probability that this experience happens the first time that this person can have.
P(Monday) = 1
P(Tuesday) = 1
The vague inuitive feeling that it has to be 1/2 is once again pointing to weighted probability, which renormalizes the measure function.
Probabilities are measures on a sigma-algebra of subsets of some set, obeying the usual mathematical axioms for measures together with the requirement that the measure for the whole set is 1.
Not just any set. A sample space. And one of its conditions is that its elements are mutually exclusive, so that one and only one happens in any iteration of probability experiment.
That's why I need to define a new mathematical entity indexical sample space, for which I'm explicitly lifting this restriction to formally talk about thirdism.
Applying this structure to credence reasoning, the elements of the sample space correspond to relevant states of the universe, the elements of the sigma-algebra correspond to relevant propositions about those states
A minor point is that outcomes and events can both very well be about map not the territory. If elementary event {A} has P(A) = 0, then we can simply not include outcome A into the sample space for simplicity sake.
and the measure (usually called credence for this application) corresponds to a degree of rational belief in the associated propositions. This is a standard probability space structure.
There is a potential source of confusion in the "credence" category. Either you mean it as a synonym for probability, and then it follows all the properties of probability, including the fact that it can only measure formally defined events from the event space, which have stable truth value during an iteration of probability experiment. Or you mean "intuitive feeling about semantic statements which has some relation to betting", which may or may not be formally definable as probability measure because the statement doesn't have stable truth value.
People tend to implicitly assume that having a vague feeling about a semantic statements has to mean that there is a way to formally define a probability space where this statement is a coherent event, but it doesn't actually has to be true. Sleeping Beauty problem is an example of such situation.
In the Sleeping Beauty problem, the participant is obviously uncertain about both what the coin flip was and which day it is. The questions about the coin flip and day are entangled by design, so a sample space that smears whole timelines into one element is inadequate to represent the structure of the uncertainty
I'm not sure what you mean by "smears whole timelines into one element". We of course should use the appropriate granularity for our outcomes and events. The problem is that we may find ourselves in a situation where we intuitively feel that events has to be even more granular then they formally can.
For example, one of the relevant states of the universe may be "the Sleeping Beauty experiment is going on in which the coin flip was Heads and it is Monday morning and Sleeping Beauty is awake and has just been asked her credence for Heads and not answered yet".
The fact that something is a semantic statement about the universe doesn't necessary mean that it's well-defined event in a probability space.
One of the measurable propositions (i.e. proposition for which Sleeping Beauty may have some rational credence) may be "it is Monday" which includes multiple states of the universe including the previous example.
No it can't. Semantic statement "Today is Monday" is not a well-defined event in the Sleeping Beauty problem. People can have credence about it in the sense of "vague feeling", but not in the sense of actual probability value.
You can easily observe yourself that there is no formal way to define "Today" in Sleeping Beauty if you actually engage with the mathematical formalism.
Consider No-Coin-Toss or Single-Awakening problems. If Monday means "Monday awakening happens during this iteration of probability experiment" and, likewise, for Tuesday, we can formally define Today as:
Today = Monday xor Tuesday
On every iteration of probability experiment either Monday or Tuesday awakenings happen. So we can say that the participant knows that "she is awakened Today", meaning that she knows to be awakened either on Monday or on Tuesday.
P(Today) = P(Monday xor Tuesday) = 1
We can express credence in being awakened on Monday, conditionally on being awakened "Today" as:
P(Monday|Today) = P(Monday|Monday xor Tuesday) = P(Monday)
This is a clear case where Beauty's uncertainty about which day it is can be expressed via probability theory. Statement "Today is Monday" has stable truth values throughout any iteration of probability experiment
Now consider No-Memory-Loss problem where Sleeping Beauty is completely aware which day it is.
Now statement "Today is Monday" doesn't have a stable truth value throughout the whole experiment. It's actually two different statement: Monday is Monday and Tuesday is Tuesday. The first one is always True, the second one is always False. So Beauty's uncertainty about the question which day it is can't be expressed via probability theory. Thankfully, she doesn't have any uncertainty about the day of the week.
So we can do a trick. We can describe No-Memory-Loss problem as two different non-independent probability experiments in a sequential order. First one is Monday-No-Memory-Loss, where the Beauty is sure that it's is Monday and uncertain about the coin. The second is Tuesday-Tails-No-Memory-Loss where the Beauty is sure that it's Tuesday and the coin is Tails. The second happens only if the coin was Tails in the first.
In Monday-No-Memory-Loss, Today simply means Monday:
Today = Monday
And statement "Today is Monday" is a well defined event with trivial probability measure:
P(Monday|Today) = P(Monday|Monday) = 1
Similarly with Tuesday-Tails-No-Memory-Loss:
Today = Tuesday
P(Monday|Today) = P(Monday|Tuesday) = 0
And now when we consider regular Sleeping Beauty problem the issue should be clear. If we define Today = Monday xor Tuesday, the Beauty can't be sure that this event happens, because on Tails both Monday and Tuesday are realized.
And we can't take advantage of Beauty's lack of uncertainty about the day as before, because now she has no idea what day it is. And so the statement "Today is Monday" is not a well-defined event of the probability space. It doesn't have a coherent truth value during the experiment - it's True xor ( True and False).
We can still talk about events "Monday/Tuesday awakening happens during this iteration of probability experiment".
P(Monday) = 1
P(Tuesday) = 1/2
And we can use them in betting schemes. If the Beauty is proposed to bet on the statement "Today is Monday" she can calculate her optimal odds the standard way:
E(Monday) = P(Monday)U(Monday) - P(Tuesday)U(Tuesday)
Solving E(Monday) = 0:
U(Monday) = U(Tuesday)/2
So 1:2 odds.
And the last question is: What was then this intuitive feeling about the semantic statement "Today is Monday"? For which the answer is - it was about weighted probability that Monday happens in the experiment.
No, introducing the concept of "indexical sample space" does not capture the thirder position, nor language.
And what does it not capture in thirder position, in your opinion?
You do not need to introduce a new type of space, with new definitions and axioms. The notion of credence (as defined in the Sleeping Beauty problem) already uses standard mathematical probability space definitions and axioms.
So thirder think. But they are mistaken, as I show in the previous posts.
Thirder credence fits the No-Coin-Toss problem where Monday and Tuesday don't happen during the same iteration of the experiment and on awakening the person indeed learns that "they are awaken Today", which can be formally expressed as an event .
Not so with Sleeping Beauty, where the participant completely aware that Monday awakening on Tails is followed by Tuesday awakening, therefore, event doesn't happen in 50% cases, so instead of learning that the Beauty is awakened today she can only learn that she is awakened at least once.
In Sleeping Beauty problem being awakened Today isn't a thing you can express via probability space. It's something that can happen twice in the same iteration of the experiment, just like getting a ball in the example from the post. And so we need a new mathematical model to formally talk about this sort of things, therefore weighted probability space.
I suppose you've read all my posts on the topic. What is the crux of our disagreement here?
If this was true, then we could not observe the event „any other coin sequence“ as well since this event is by definition not being tracked.
When you are tracking event A you are automatically tracking its complement.
In fact, in order to detect a correspondence between a coin sequence that we have in mind and the actual sequence, our brain has to compare them to decide if there is a match. I can hardly imagine how this comparison could work without observing the specific actual sequence in the first place. That we classify and perceive a specific sequence as „any other sequence“ can be the result of the comparison, but is not its starting point.
Oh sure, you are of course completely correct here. But this doesn't contradict what I'm saying.
The thing is, we observe a particular outcome and then we see which event(s) it corresponds to. Let's take an example: a series of 3 coin tosses.
So, in the beginning you have sample space which consist of all the elementary outcomes:
And an event space, some sigma-algebra of the sample space, which depends on your precommitments. Normally, it would look something like this:
Because you are intuitively paying attention to whether there all Heads/Tails in a row. So your event space groups individual outcomes in this particular way, separating the event you are tracking and it's complement.
When a particular combination, say is realized in a iteration of the experiment, your mind works like this:
- Outcome is realized
- Therefore every event from the event space which includes is realized.
- Events and are realized.
- This isn't a rare event and so you are not particularly surprised
So, as you see, you do indeed observe an actual sequence, it's just that observing this sequence isn't necessary an event in itself.
Sure, I don‘t deny that. What I am saying is, that your probability model don‘t tell you which probability you have to base on a certain decision
It says which probability you have, based on what you've observed. If you observed that it's Monday - you are supposed to use probability conditionally on the fact that it's Monday, if you didn't observe that it's Monday you can't lawfully use the probability conditionally on the fact that it's Monday. Simple as that.
There is a possible confusion where people may think that they have observed "this specific thing happened" while actually they observed "any thing from some group of things happened", which is the technicolor and rare event cases are about.
Suppose a simple experiment where the experimenter flips a fair coin and you have to guess if Tails or Heads, but you are only rewarded for the correct decision if the coin comes up Tails. Then, of course, you should still entertain unconditional probabilities P(Heads)=P(Tails)=1/2. But this uncertainty is completely irrelevant to your decision.
Here you are confusing probability and utility. The fact that P(Heads)=P(Tails)=1/2 is very much relevant to our decision making! The correct reasoning goes like this:
P(Heads) = 1/2
P(Tails) = 1/2
U(Heads) = 0
U(Tails) = X,
E(Tails) = P(Tails)U(Tails) - P(Heads)U(Heads) = 1/2X - 0
Solving E(Tails) = 0 for X:
X = 0
Which means that you shouldn't bet on Heads at any odds
What is relevant, however, is P(Tails/Tails)=1 and P(Heads/Tails)=0, concluding you should follow the strategy always guessing Tails.
And why did you happen to decide that it's P(Tails|Tails) = 1 and P(Heads|Tails) = 0 instead of
P(Heads|Heads) = 1 and P(Tails|Heads) = 0 which are "relevant" for you decision making?
You seem to just decide the "relevance" of probabilities post hoc, after you've already calculated the correct answer the proper way. I don't think you can formalize this line of thinking, so that you had a way to systematically correctly solve decision theory problems, which you do not yet know the answer to. Otherwise, we wouldn't need utilities as a concept.
Another way to arrive at this strategy is to calculate expected utilities setting U(Heads)=0 as you would propose. But this is not the only reasonable solution. It’s just a different route of reasoning to take into account the experimental condition that your decision counts only if the coin lands Tails.
This is not "another way". This is the right way. It has the proper formalization and actually allows us to arrive to the correct answer even if we do not yet know it.
If the optimal betting sheme requires you to rely on P(Heads/Red or Blue)=1/2 when receiving evidence Blue, then the betting sheme demands you to ignore your total evidence.
You do not "ignore your total evidence" - you are never supposed to do that. It's just that you didn't actually receive the evidence in the first place. You can observe the fact that the room is blue in the experiment only if you put your mind in a state where you distinguish blue in particular. Until then your event space doesn't even include "Blue" only "Blue or Red".
But I suppose it's better to go to the comment section Another Non-Anthropic Paradox for this particular crux.
God is good*
*for a very specific definition of "goodness", which doesn't actually capture the intuition of most people about ethics and is mostly about iteraction of sub-atomic particles.
3First of all, it‘s certainly important to distinguish between a probability model and a strategy. The job of a probability model is simply to suggest the probability of certain events and to describe how probabilities are affected by the realization of other events. A strategy on the other hand is to guide decision making to arrive at certain predefined goals.
Of course. As soon as we are talking about goals and strategies we are not talking about just probabilities anymore. We are also talking about utilities and expected utilities. However, probabilities do not suddenly change because of it. Probabilistic model is the same, there are simply additional considerations as well.
My point is, that the probabilities a model suggests you to have based on the currently available evidence do NOT neccessarily have to match the probabilities that are relevant to your strategy and decisions.
Whether or not your probability model leads to optimal descision making is the test allowing to falsify it. There are no separate "theoretical probabilities" and "decision making probabilities". Only the ones that guide your behaviour can be correct. What's the point of a theory that is not applicable to practice, anyway?
If your model claims that the probability based on your evidence is 1/3 but the optimal decision making happens when you act as if it's 1/2, then your model is wrong and you switch to a model that claims that the probability is 1/2. That's the whole reason why betting arguments are popular.
If Beauty is awake and doesn‘t know if it is the day her bet counts, it is in fact a rational strategy to behave and decide as if her bet counts today.
Questions of what "counts" or "matters" are not the realm of probability. However, the Beauty is free to adjust her utilities based on the specifics of the betting scheme.
All your model suggests are probabilities conditional on the realization of certain events.
The model says that
P(Heads|Red) = 1/3
P(Heads|Blue) = 1/3
but
P(Heads|Red or Blue) = 1/2
Which obviosly translates in a betting scheme: someone who bets on Tails only when the room is Red wins 2/3 of times and someone who bets on Tails only when the room is Blue wins 2/3 of times, while someone who always bet on Tails wins only 1/2 of time.
This leads to a conclusion that observing event "Red" instead of "Red or Blue" is possible only for someone who has been expecting to observe event "Red" in particular. Likewise, observing HTHHTTHT is possible for a person who was expecting this particular sequence of coin tosses, instead of any combination with length 8. See Another Non-Anthropic Paradox: The Unsurprising Rareness of Rare Events
Meta:
I find this situation quite ironic. From my perspective, I painstakingly cited and answered your comments piece by piece even though you didn't engage much neither with the arguments in my posts nor with any of my replies.
I'm not sure how I could have missed "parts where you try to communicate the structure of the question". The only things which I haven't directly cited in your previous comment are:
and
Which I neither disagree nor have any interesting to add.
Which does not have any argument in it. You just make statements without explaining why they are supposed to be true. I obviously do not agree that in Sleeping Beauty case our model should be treating time states as individual outcomes. But simply proclaming this disagreement doesn't serve any purpose after I've already presented comprehensive explanation why we can't lawfully reason about time states in this particular case as if they are mutually exclusive outcomes, which you, once again, failed to engage with.
I would appreciate if you addressed this meta point because I'm honestly confused about your perspective on this failure of our communication.
Meta ended.
If you were specifically referring to Sleeping Beauty problem then your previous comment doesn't make sense.
Either you are logically pinpointing any sample space for at least some problem, and then you can say that any non-empty set fits this description.
Or you are logically pinpointing sample space specifically for the Sleeping Beauty problem, then you can't dismiss the condition of mutually exclusivity of outcomes which are relevant to the particular application of the sample space.