If I know the Pearson correlation between A and B and also know the Pearson correlation between A and C, can I infer anything about the correlation between B and C?
Assume that I no longer have access to the raw values for A, B, and C.
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On the whole, the answer to your question is "no". Structurally, it's easy to generate data so that A->B is significant and B->C is significant but A is not related to C.
There are two circumstances where this question crops up:
B may be considered a "mediator" in that A causes the B which causes the C. So if you are interested in A->B->C there are specific tests for that (like in SEM) BUT it's rare I believe such results (when significant) if we don't at least find A->C in the sample. It's important to be sure that B is not a confounder in this circumstance. Take smoking (A), dietary intake (B) and cardiovascular risk. A->C (positive) and B->C (positive) but A->B negative. if you looked just at the directional effects here, you would conclude smoking is beneficial for heart disease since it reduces appetite.
C was not actually measured. From a strictly practical perspective, in general scientists will greatly shoot down any attempt to make inference of that nature. Suppose I am interested in treating lowstage breast cancer with a mild chemotherapeutic agent. Tumors have to shrink before they disappear, so tumor shrinkage might be my B whereas survival is my C. Chemotherapy is very toxic, so I wouldn't be surprised to find I achieve B but not having measured C I cannot conclude that the A->C effects are "negligible".