Causal knowledge is $p(z\mid h)$, i.e. the probability that a certain state $h$ causes a certain state of the observable variable $z$. Diagnostic knowledge on the other hand refers to the knowledge we gain from inferring the probability of state $h$ from observation $z$ (like in a medical diagnose).

Diagnostic knowledge is often hard to obtain, while causal knowledge is easy to obtain, so we make use of Bayes' rule to calculate the probability of a state using causal knowledge:

$$p(h\mid z) = \frac{p(z\mid h)p(h)}{\sum\limits_{i=1}^Np(z\mid h_i)p(h_i)}$$

What is the reason that one tends to be harder than the other?

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    $\begingroup$ Who said that? In what context? What exactly are h and z? I'm afraid your question is lacking context that is needed to understand what do you exactly mean. $\endgroup$ – Tim Feb 23 '17 at 16:37
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    $\begingroup$ I disagree that this merits closing. Some gaps might be filled in but let's give @Lenar Hoyt some time to do so. $\endgroup$ – rolando2 Feb 23 '17 at 17:20
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    $\begingroup$ I am unsettled by the liberal use of "causal" here, especially in the phrase "causal knowledge is easy to obtain"... $\endgroup$ – juod Feb 23 '17 at 18:27
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    $\begingroup$ I struggle to understand how we are intended to interpret "harder." It seems like a vague and personal concept. Perhaps this question could be clarified by providing a specific context or example. $\endgroup$ – whuber Feb 24 '17 at 0:41
  • $\begingroup$ This question has nothing to do with causality. Neither conditional probability depends on casual knowledge since they don't involve interventions. $\endgroup$ – Neil G Feb 24 '17 at 2:29

Consider a fire alarm as a diagnostic tool in determining whether or not there is a fire. The alarm can be in one of two states: $R$ meaning Ringing or $R^c$ meaning Not Ringing. Let $F$ denote the event that there is a fire. Now, the manufacturer of the alarm can test his product by repeatedly setting fires in his testing area and seeing how often the alarm starts ringing (correct detection), and also by not having a fire and seeing how often the alarm goes off anyway (false alarm). That is, $P(R\mid F)$ and $P(R\mid F^c)$ are "easy" for the manufacturer to determine, and can even be used in advertising: "Be safe with Super-Duper Fire Alarm. It detects fires with probability $0.99999999$ and only goes off at random with probability $0.00000001$". But your task, as you are wakened by the alarm going off at 3 a.m. on a snowy winter morning, is to decide whether there really is a fire and so you should rush out of the frat house in your jammies to wait for the fire engines to arrive, or is it a false alarm that can be safely ignored as you stay in the warm comfort of your bed. That is, you need what you call the diagnostic probabilities $P(F\mid R)$ and $P(F^c\mid R)$, and these depend on $P(F)$ (and $P(F^c) = 1 - P(F)$) which only you can know or estimate (how much of a firetrap is your frat house? is it a party night when the level of alcohol consumption can be assumed to be higher than usual? were your frat brothers playing with matches when you went to bed? etc).

Thus, as you say, causal knowledge is easy to obtain (and the manufacturer of the fire alarm can provide it to you) but diagnostic knowledge is much harder to obtain since it requires knowledge or estimation of the current situation.

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