Use probability without sample space?

[yeet991only]

TL;DR

How to assign values to probabilities when there is no sample space? Is this allowed? Or it will seem like weight when you have a neural network?

Suppose this example:
You said a joke to a girl. She didn't laugh (let this be behaviour/evidence E). Was this because she is mad at you (let it be M) or because she didn't get the joke (let it be NJ , and it be 1 - p(M) = p(NJ)). So we have just 2 hypotheses M and NJ that is all. Now how to assign probabilities to such expression:

Suppose she was mad 2 times before this year.
So p(m) = 2 * average_days_mad/365 ??
I dont have any idea how to assign such probabilities..
How would NJ be assigned then?
Or should they be assigned as a grade out of 10?
Like from 0 to 1 , how likely do I think that M = "Mad" is true.
Like a weight from a neural network?

 

[Dale]

You cannot have probability without a sample space. It would be like trying to have a vector without a vector space. The space that it comes from is part of the definition of the thing itself.

 

[jbergman]

TL;DR Summary: How to assign values to probabilities when there is no sample space? Is this allowed? Or it will seem like weight when you have a neural network?

Suppose this example:
You said a joke to a girl. She didn't laugh (let this be behaviour/evidence E). Was this because she is mad at you (let it be M) or because she didn't get the joke (let it be NJ , and it be 1 - p(M) = p(NJ)). So we have just 2 hypotheses M and NJ that is all. Now how to assign probabilities to such expression:
View attachment 356097
Suppose she was mad 2 times before this year.
So p(m) = 2 * average_days_mad/365 ??
I dont have any idea how to assign such probabilities..
How would NJ be assigned then?
Or should they be assigned as a grade out of 10?
Like from 0 to 1 , how likely do I think that M = "Mad" is true.
Like a weight from a neural network?

You do have a sample space but that is only part of what is required to define a sigma algebra. You also need to declare what sets are measurable and a function that assigns a probability to the measurable sets.

Your question is really about how to assign a probability to different events in your sample space.

That is in some sense a philosophical question and depends on your interpretations of probability.

Subjective Bayesians would say, I believe, that probabilities represent degrees of belief. So you could assign ypur initial probabilities based on that.

 

[Hornbein]

The sample space is the set of events. To not have a sample space you'd have to have no events? I can't see how this is feasible. I think I don't understand the question. If you don't have a sample space, what have you got?

 

[PeroK]

You cannot have probability without a sample space.

I don't want to quote you out of context, but on a recent discussion you said:

Sometimes there is no sample. So a Bayesian can still choose a prior that represents their uncertainty.

The full discussion is here:

https://www.physicsforums.com/threads/what-motivates-bayes-theorem.1066142

 

[Dale]

I don't want to quote you out of context, but on a recent discussion you said:


The full discussion is here:

https://www.physicsforums.com/threads/what-motivates-bayes-theorem.1066142

Yes. An example would be if you wanted to know the probability that the universe will end in the next four years. We have no sample, no data on universes that we observed ending, etc. But the sample space would be all positive real numbers representing the number of years from today and the specific event in question would be the numbers between 0 and 4.

 

[jbergman]

The sample space is the set of events. To not have a sample space you'd have to have no events? I can't see how this is feasible. I think I don't understand the question. If you don't have a sample space, what have you got?

I believe the question was how to assign the probabilities to the events which isn't a question about the sample space. Instead it is a disguised question about how to assign probabilities to something that isn't like a coin toss that you can repeat over and over again.

 

[pbuk]

TL;DR Summary: How to assign values to probabilities when there is no sample space? Is this allowed? Or it will seem like weight when you have a neural network?

As @jbergman suggests, this question does not match the example. Are you interested in an answer to this question or an analysis of your example?

You said a joke to a girl. She didn't laugh (let this be behaviour/evidence E). Was this because she is mad at you (let it be M) or because she didn't get the joke (let it be NJ , and it be 1 - p(M) = p(NJ)).

$1 - p(M) = p(NJ)$ doesn't say what you think it says. For instance this is satisfied by $p(M) = p(NJ) = 0.5$, and here she will laugh 25% of the time. The rest of your model is similarly flawed.

Let's start again:

You said a joke to a girl. She didn't laugh (let this be behaviour/evidence E). Was this because she is mad at you (let it be M) or because she didn't get the joke?
...
Suppose she was mad 2 times before this year.

If by "this year" you mean "in the last 365 days" then the probability that she is mad at you today is $\frac 2 {365}$. As she didn't laugh at your "joke" then the probability that she didn't get it is therefore $\frac{363}{365}$.

If you want to estimate the probability that she doesn't get your jokes then you need more data, for instance what happened when you told her jokes previously.