You can do it with enough causal assumptions (e.g. not "from nothing"). There is a series of magical papers, e.g. this:

http://www.cs.helsinki.fi/u/phoyer/papers/pdf/hoyer2008nips.pdf

which show you can use additive noise assumptions to orient edges.

I have a series of papers:

http://www.auai.org/uai2012/papers/248.pdf

http://arxiv.org/abs/1207.5058

which show you don't even need conditional independences to orient edges. For example if the true dag is this:

1 -> 2 -> 3 -> 4, 1 <- u1 -> 3, 1 <- u2 -> 4,

and we observe p(1, 2, 3, 4) (no conditional independences in this marginal), I can recover the graph exactly with enough data. (The graph would be causal if we assume the underlying true graph is, otherwise it's just a statistical model).

People's intuitions about what's possible in causal discovery aren't very good.

It would be good if statisticians and machine learning / comp. sci. people came together to hash out their differences regarding causal inference.