Causality and independence

AngleWyrm

Active Member
#1
Player 1 draws a card from a deck, and then player 2 draws a card. This is a causal relationship: Player 1's draw occurred before player 2's draw, and had an effect on what player 2 could draw. The player's hands are not independent events. The claim event A did something to effect the outcome of event B requires that A came before B. Thus a causal relationship requires the use of before/after, a temporal ordering.
We can conclude that if B occurred after A, then the outcome of event B could not have had any influence on the outcome of A.

Whereas with independent events, the roll of die A has no effect on the roll of die B, and neither does the roll of die B have an effect on the outcome of die A. Therefore we can conclude the notion of before/after is meaningless for independent events.
And if we claim the events are simultaneous, that there is no before/after relationship between them, then causality between them cannot be claimed either.

So let's say we use a test for independence on a set of observations, and conclude the events A and B are independent.
Therefore I declare before/after is meaningless and so is causality between these two events.
If you're about to say "that's part of the definition -- that's what independence means!" then congratulations, you agree with me.

What if instead the test for independence on the set of outcomes for A and B showed they're not independent? What can you say about that?
 

hlsmith

Less is more. Stay pure. Stay poor.
#2
I have a strong interest in causality in working in the medical sector. I few comments - temporality is typically a necessary requisite. The second law of thermodynamics is usually quoted in these conversations. In that states typically move towards entropy with time unless intervened on. Example if you wait long enough a melted ice cube wont turn back into an ice cube. Side note, I think it was Kant who postulated that temporality isn't necessary in all causal relationships. For example if you let go of two blocks at the same time, they can collide and balance each other without falling over. So one event did not proceed the other - they were simultaneous.


Now for your post - what are tests for independence - provide an example. And are they just misnomers or dependent on the sample or sampling? If I run a chi-sq test and it 'fails' to show significance, well that could be due to sample size, sampling bias, mis-specification of the terms, etc. Now if I find a difference (independence), this could be related to the same things, explicitly including chance. So is the truth ever known without sufficient random sampling and a define criteria of acceptance (alpha). Non-significance is typically called a failure to find a difference in statistics.

Also, you can find a lack of association between variables due to unfaithfulness in the structure. Meaning a variable may have an effect on a subsequent event through two paths (one negative and one positive) and the net effect cancels out. so there appears to be no overall effect.

P.S., I concur that subsequent card draws would be dependent (even deterministic) and dice rolls independent.
 
Last edited:

AngleWyrm

Active Member
#3
what are tests for independence - provide an example
Pearson's chi squared test for independence

Tests of Independence
If A and B are independent then P(A AND B) = P(A)P(B)
A sample of 300 students is taken. Of the students surveyed, 50 were music students, while 250 were not. Ninety-seven were on the honor roll, while 203 were not. If we assume being a music student and being on the honor roll are independent events, what is the expected number of music students who are also on the honor roll?
 
Last edited:

katxt

Well-Known Member
#4
We can conclude that if B occurred after A, then the outcome of event B could not have had any influence on the outcome of A
Something odd here. What if A didn't look at his card until B had showed hers. Then A's card depends on what B's is, even though she picked second.
 

AngleWyrm

Active Member
#5
Something odd here. What if A didn't look at his card until B had showed hers. Then A's card depends on what B's is, even though she picked second.
You appear to be discussing something that isn't the draw of a card A followed by the draw of a card B.

It looks like you might be talking about a strategy game where two players hold more than one card each, and are choosing to play cards from their hands. Do you agree the cards a player drew for their hand are not the same cards as the ones in the opponent's hand?
 
Last edited:

katxt

Well-Known Member
#6
You appear to be discussing something that isn't the draw of a card A followed by the draw of a card B.
No. All I'm saying is that the effect of A's card on B's card is exactly the same as the effect of B's card on A's card, irrespective of who choses first.
What you seem to be saying is that if two players cut a pack to decide who deals, it matters whether you cut first or second.
Dependence is not necessarily a case of causality or timing.
 

hlsmith

Less is more. Stay pure. Stay poor.
#7
I will note that once someone draws a card, regardless of whether its face value is known - that then changes the population (available cards) the second person can select. Selection probabilities are different, since things become conditional or the sample changes. I agree that assigning causality intuitively feels weird in this setting.

The go to book on causality is "Causality" by Judea Pearl. Which both him and Rubin state that you can have statistical inference, but in order to have causal inference additional assumptions are needed. Including exchangeability (unconfoundedness), no interference (my treatment doesn't effect your outcome), and a well defined and consistent exposure. Temporality typically helps as well - meaning it is difficult too identify a causal effect if data are collected cross-sectional.
 
Last edited:

AngleWyrm

Active Member
#8
Do you agree that P(A|B) is a mathematical model of dependence?
That is to say P(A) given P(B) states a before/after relationship, and a causality of B effecting A?
 

katxt

Well-Known Member
#9
A coin and a dice are thrown together. P(Head | a six)= 0.5, There is no dependency, before/after relationship, or causality involved in this probability expression.
You have to be wary about using statements in colloquial English to prove or demonstrate rigorously defined probability terms.
 

hlsmith

Less is more. Stay pure. Stay poor.
#10
In my area "P(A|B)" is not a model of dependence, but a model of association. We use them all of the time, say Prob(cancer|smoker), it is the first level causation, second level: prob(cancer| do smoke), meaning the person was assign to smoke or status is manipulatable, last level is Pr(Y^x|X), which is the counterfactual of what would have happened (outcome) if counter to reality, so person did not smoke.

I found the follow quickly online - it may give you an idea:

http://forns.lmu.build/classes/spring-2020/cmsi-432/lecture-2-1.html
 

AngleWyrm

Active Member
#11
If P(cancer | smoker) > P(cancer), cancer is more frequent among smokers, then can we say smoker came before cancer, and shows a causal relationship? Here's the rub I've seen so far, which seems to me Simpson's Paradox: It may also be the case P(smoker | cancer) > P(smoker), smokers are more common among cancer patients -- a possible but not a necessary conclusion.

So I can classify the observed results as either
  1. P(A|B) > P(A) AND P(B|A) > P(B)
  2. #1 is false
Since #1 is a logical AND operation, there are four outcomes which can be presented as a Venn diagram
Untitled.png

Thus P(A|B)>P(A) XOR P(B|A)>P(B) asserts causality, selecting which is the dependent variable.
 
Last edited:

katxt

Well-Known Member
#12
AngleWyrm. I think you need to abandon the idea that conditional probability implies causality or has a before/after slant. It is perfectly acceptable, and quite common, to have the condition occur after the event.
You have a friend X who is reckless driver. One day your hear that X is in hospital. You think that X may have had a motor accident.
We can express this probability as P( X has had a motor accident | X is in hospital). The condition clearly happened after the event.
Most first year probability courses have little stories followed by questions like "Given that it rained on Sunday, what is the probability that it rained on Saturday?"
 

hlsmith

Less is more. Stay pure. Stay poor.
#13
Also, unless data are from a well defined experiment, A|B can be most as identified for many reasons, with a co-founder or collided is involved.
 

AngleWyrm

Active Member
#14
katxt said:
We can express this probability as P( X has had a motor accident | X is in hospital). The condition clearly happened after the event.

If P( accident | hospital) > P(accident) then we have a measured judgement people who've had accidents are more frequently found in hospitals.

The | symbol is pronounced "given," a prerequisite. Which means "given they are hospitalized" is what makes the left probability different from the right probability.

I think Mr. Simpson has brainwashed people.
 
Last edited:

hlsmith

Less is more. Stay pure. Stay poor.
#15
Look at Bayes theorem, which was originally called inverse probability. Since you are working backwards at times, probability of disease given a positive test.