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What does it mean to say that A is better evidence of a causal process than B?

There are cases where this seems very intuitive. For example, if one observes a person's fingerprints on a dead body's throat, it may serve as a form of preliminary evidence that the person may have murdered. If instead it was found that the person's DNA is on the person's body, it may be considered better evidence.

If fingerprints is O1 and DNA is O2, one can argue, quite reasonably, that the P(person being a murderer|O2) is higher than P(person being a murderer|O1). This is partially justifiable because of prior data. DNA has served as better evidence in the past than fingerprints in these kinds of matters.

But what about cases where there seems to be no priors? For example, suppose you are evaluating whether a psychic exists. This psychic makes a prediction. Let's say his prediction is that a person will be murdered tomorrow in the city. Let's call this O1. Now let's say the psychic instead made a prediction that a person with a name starting with the letter J will get murdered tomorrow. Let's call this O2. Assume that both of those predictions materialized.

O2 is a more specific prediction than O1. However, most would say that it seems completely unreasonable to think that the person is a psychic based on either O1 or O2. However, should O2 be considered as better evidence than O1? Should P(psychic powers|O2) be higher than the P(psychic powers|O1)? Many would say yes but I'm having trouble understanding this. We don't know if psychics actually exist. Sure, if we knew in advance that one of those predictions came true because of psychic powers, it might be wise to bet on the second. But we don't have any evidence of this.

If psychic powers truly are a matter of fiction, then by definition O2 should not serve as better evidence of psychic powers than O1, given that it is impossible for a psychic to predict anything. If psychics don't exist, P(psychic powers|O1) = P (psychic powers|O2) =. 0. On the other hand, I can imagine a scenario where if a psychic made enough predictions, it would be considered better evidence.

So the question is: when should an observation be considered better evidence than another, especially in cases of theories where we have zero prior evidence of? In my example, should one increase their confidence in psychics more after O2 compared to O1, or should their confidence remain the same after observing either piece of data.

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  • (Assuming we are applying the scientific method) This sounds like you would be preferring some observations, and that's an error. If one observation contradicts another, you cannot discard it, or the other. It is imperative to strictly keep all results and formulate conclusions considering the contradictions. Now, if you have multiple theories, that's a different history, evidently, you can prefer the theory which observations fit the predictions better and discard the others (discard the theories and the observations). Or else, if you use the best predictions to formulate a new theory, ok.
    – RodolfoAP
    Jan 22, 2023 at 15:41
  • If testing a hypothesis (eg H being the case) one has to take account of data coinciding by mere chance (this can be the Null Hypothesis, mere chance). If that is the case (result close to mere chance) hypothesis is not validated, else there is some evidence for the hypothesis.
    – Nikos M.
    Feb 2, 2023 at 17:50

4 Answers 4

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You seem to think that coincidence per se should affect one's view of probabilities. Suppose I hypothesised that a malignant magic kangaroo on Mars was going to kill someone in the city tomorrow, and someone is indeed murdered in the city the next day. Does that increase the probability that my hypothesis is correct? Suppose I had hypothesised that the malignant magic kangaroo on Mars was going to kill someone whose name began with a J, and the person was called Joe, would that increase the probability of my hypothesis even more? If you believe the answer to either of those questions is yes then I suggest you book yourself into the nearest insane asylum.

When you evaluate probabilities you must consider more than just the coincidence between the proposed cause and the effect. You have to have some credible basis for there being a link between the two. The case of the psychic is no more credible than the case of the magic kangaroo unless you have a suggested mechanism for how the psychic power works.

When in your thought experiment you assess the probability of there being psychic powers given O2, you are overlooking a vast amount of contrary evidence. You should be asking what is the probability of there being psychic powers given O2 and everything else we know that is relevant, the answer to which is negligibly close to zero.

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  • Can you show me where I seem to think that? I do not think that. I am asking IF we should think that. Anyways, of course it seems ridiculous to believe in psychic powers based on the examples. But this is evading the question. The question is whether P(psychics|O2) should be higher than P(psychics|O1). The question was not asking how negligible these probabilities are.
    – user62907
    Jan 22, 2023 at 23:04
  • I suggest you read about signal to noise ratios and statistical significance. You seem to think it is possible to quantify the relative probabilities of an unproven phenomenon with no known cause or mechanism being true based on two observations, which is just nonsense. It is like hypothesising that the average weight of a pebble on Chisel Beach is 0.9833462981349827765 kg, and then picking two pebbles at random, one of which weighs 0.5kg the other 1.2kg, and asking which pebble makes your hypothesis more probable... Jan 23, 2023 at 7:53
  • ...the answer is that neither does. Based on such a small sample size you have absolutely no idea how representative the two pebbles are. Jan 23, 2023 at 7:55
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Suppose there are 2 tests (A and B) that can be conducted to connect a person to a crime.

Test A is such that the probability of the corresponding evidence E1 matching a person other than the defendant is 10%. That means, in a small town of 30,000 people, there'll be roughly 3000 people who could've also commited the crime.

Test B, on the other hand, has a probability of 0.01% that the corresponding evidence E2 matches a person other than the defendant. In the same town, there'll only be 300 other possible suspects.

Test B is a better test than test A because it's less likely that the defendant didn't commit the crime [compare 1/3000 to 1/300]. To cut to the chase an observation O2 (positive test result) in test B is better (evidentially) than an observation O1 (positive test result) in test A for the hypothesis that the person we're talking about is our guy.

The same rationale for psychics as well. In very simple terms, the less likely experts are able to make a similar prediction, the more likely is a person psychic. ESP is about bypassing normal channels of acquiring knowledge and the easiest way to do that is to use experts as a yardstick.

Being specific is to simply narrow down what in math is known as possibility space i.e. you're identifying a unique event/person/place/time which means the probability that you could simply guess that event/person/place/time is 1/x where x will obviously be a astronomically huge number. In other words you have a snowball's chance in hell of getting it right by fluke. We already know that even the experts can't make such a prediction. Rule out the impossible and whatever remains, no matter how improbable, is the truth - ESP!

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A single observation — one measured event — is effectively meaningless, scientifically speaking. It's a single data point, with no context or orientation. Literally, if we saw the sun rise once and only once, that would tell us nothing about the universe: we'd have no idea whether the sun would set, or in what direction it would fall, or whether (you know) someone is psychically lifting the sun with their mind. One data point is not enough to develop a pattern, and any pattern we tried to create would be little more than guesswork. So if we are merely looking at two singular data points, neither is better, or worse, or indicative of anything at all. It boils down to: "I saw that, and I saw that," with maybe an obligatory "Weird, huh?".

Meaning comes from the way we conceive relationships between observed events. This is true even in your prosaic 'crime' example. Fingerprints and DNA can be transferred to a crime scene in numerous different ways, and only gain importance through relationships to other observations: e.g., fingerprints taken from the throat of the victim are far stronger evidence than touch DNA taken from a sleeve: the former suggests violence whereas the latter could be the result of casual contact. As we place more and more singular pieces of evidence into relationship with each other, we begin building a pattern of events that precludes many theoretical possibilities and makes us more certain that some given (theorized) relationship between these events is correct, but that certainty is not vested in any single observation.

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So the question is: when should an observation be considered better evidence than another, especially in cases of theories where we have zero prior evidence of?

Never. This type of thinking leads to cherry-picking of data.

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