Conditional probability does not describe causality at all, since description is a layer of detail about some state, action, scenario, or phenomenon. Causality is a level of abstraction above description. Conditional probability does not imply causality either, because implication is also an abstraction, above causality and description.
In a boxing match, a punch and a person knocked out may occur in succession indicating a high probability that the punch caused the knock out. It may be reasonable to say that the repetition of the trend of knock outs following punches supports this probability of causality, supporting the correlation between conditional probability and causality, but the correlation coefficient is certainly < 1.0 for the general case.
Most that would be studying conditional probability already realize that the two observations (punch, knock out) represent a minuscule subset of all observations that could be made with the right instrumentation. The person may have fainted from dehydration immediately after being punched and the temporal alignment of the two events leads to an incorrect conclusion about cause among the observers. On a higher level of abstraction, the person may only appear knocked out but is acting in a boxing sequence on a realistically choreographed movie shoot.
If the appearance of impact precedes the appearance of unconsciousness, we can correctly make probabilistic statements about causality within an abstract system we constructed about the class of phenomena about which the observations are a part. The announcer might say, "And a round house from Thompson brings Pauly swirling to the floor." The fight is judged on the basis of conclusions about causality that stem from observing multiple fights. Media channels are filled with such conclusions.
Human beings are swimming in a pool of unproven statements presented as facts, so most humans are unaware of the basis upon which they draw conclusions. Diligent and well educated scientists are more likely to remember these concepts:
- Only in virtual scenarios brought to life in human constructions such as chess games and digital circuits can observation be exhaustive.
- Purely unidirectional causality is rare, such as the moonlight on a camera lens. The sun is unaffected by the reflection of light off of the camera lens but the exposure of the image is the detection of solar nuclear reaction. Within the biosphere, physical phenomena more commonly exhibit bidirectional information flow. A acts on B while B acts on A.
- Most descriptions of an event are of a trend found in a cacophony of tiny interactions, such as a punch and knock out being a large number of electrical events in a set of neurons, the elastic deformation of a large number of molecules within cells and bone material, and a large number of electrical events in a second set of neurons.
Human cognitive abilities embrace all of these abstractions of composition and causality, and many thoughtful humans attach some level of doubt about causality when observations appear to indicate it. We can hear reports like, "I think the shot came from behind me and I saw the President's head jerk back." The causality can nonetheless be debated for decades.
We see a panic and crash in a market, but there is no singular panic and singular crash. No person's trade, public message, personal financial concern, or verbal expression causes the entire trend we examine in a graph part way through or after the sequence of singular events marking such a scenario.
In electronics, we take great pains to permit circuit A's control of circuit B without allowing the back flow of causality from B to A. We like digital electronics because we get a purified universe where causality is unidirectional except where we specify feedback, as in the case of digital learning that depends on successive approximation.
All of these things are part of the framework of concepts that decouple conditional probability from conclusive causality.
A statement concluding that we don't have causality is not particularly compelling as a scientific statement. Causality is not something we can have or not have universally, and what one thinks they have another may think they don't, so the first person plural is presumptuous. A statement concluding that physical equations are symmetric can be correct if the symmetry to which the author is referring is regarding bidirectional impact, not reversibility, and equations are restricted to equalities. However, such a statement is not generally true either.
It may be reasonable to say any of these statements.
- Any consistent temporal pattern of asymmetry gives us information about causality but cannot be conclusive.
- Causal interaction may be bidirectional without being balanced in magnitude such that one side may seem to cause the other when what appears as cause and effect both occur as part of a single unified phenomenon.
- Alleged causality written into historic accounts can be unreliable.
- At a quantum level, causality is not strictly applicable.
- The concept of causality is an abstraction that can be misleading if applied in scenarios that have bidirectional imposition of forces or fields or bidirectional information exchange.
The concluding question is interesting, but the approaches of educators and media figures in answering such questions are usually context dependent.
How would Pearl respond to someone saying that conditional probability already captures all we need to show causal relationships?
It is easy to imagine a defensive retort, a response indicating thoughtful consideration of the proposal, or (in the middle ground) a pedantic one. A private conversation over a coffee or some sushi may result in the more thoughtful conversation.