Clarification about conditional probability and causal relationship

Question

I'm a reading a beginner book about bayesian statistics (Think Bayes by Allen Downey). At the very beginning it reads:

Epidemiologist have identified many factors that affect the risk of heart attacks; depending on those factors, my risk might be higher or lower than average. I am male, 45 old, and I have borderline high cholesterol. Those factors increase my chances. However, I have low blood pressure and I don't smoke, and those factors decrease my chances.

I guess that "Epidemiologist have identified many factors that affect the risk of heart attacks" means that epidemiologist have found positive correlations between e.g. high cholesterol and heart attacks.

However, such a correlation does not necessarily mean that if I decrease somehow my cholesterol my probabilities heart attack will decrease. Am I right?

Answer 1

You are right.

The meaning of your quote is: if you look at the probability $P(H)$ of people being hit by a heart attack occuring in a given population, you are looking at the "average" probability in that population.

Differently, if you look at the probability of people being hit by a heart attack given their cholesterol level is higher than some threshold, say event $C$, you are considering the conditional probability $P(H | C)$, i.e. the probability of being hit by heart attach in that population given a high cholesterol level.

If events $H$ and $C$ are not independent, then $P(H) \neq P(H|C)$. That's it. I think that conditional probability is used to introduce Bayes' theorem, and for the moment you can stop right there.

Further interpretations require more analyses. For example, let say that $P(L|S)$ is the probability of getting a lung cancer when you smoke, while $P(L|NS)$ is the probability of getting a lung cancer when you do not smoke. We can safely assume that $P(L|S) > P(L|NS)$. However, if a man smoked cigarettes for 50 years, does stopping his habit reduce his probabilities of dying?

Moreover, the event you are conditioning on (e.g. some blood value) could be just a confounding factor, and changing its level might not have any effect on your target variable.

More generally, when you want to draw cause-and-effect relationships, you have to assume (and possibly validate) a causal model.

Answer 2

I'm trying to imagine the context of the paragraph.

I think the author is just saying 'imagine a complicated situation in which many variables affect an outcome variable and maybe each other too'.

Don't worry about correlation just yet, read on! There should be material later in the book about how to interpret more specific case studies.