Evidential Decision Theory – EDT – is a branch of decision theory which advises an agent to take actions which, conditional on it happening, maximizes the chances of the desired outcome. As any branch of decision theory, it prescribes taking the action that maximizes utility, that which utility equals or exceeds the utility of every other option. The utility of each action is measured by the expected utility, the averaged by probabilities sum of the utility of each of its possible results.
One way to see the difference between evidential utility and causal utility is to contemplate the contrasting sentences:
Causal Decision Theory – CDT – says only through causal process one can influence the chances of the desired outcome 1. Lesswrong's favored "logicial decision theory / functional decision theory" is usually though not always seen by LWers as a special case of CDT in this sense. Eg, FDT evaluates, "If Lee Harvey Oswald's algorithm hadn't output 'Shoot John F. Kennedy', nobody else would've." This is still a causal counterfactual, only evaluated on a logical proposition instead of a physical event.
EDT, on the other hand, requires no causal connection. The action only has to be Bayesian evidence for the desired outcome. So EDT is widely regarded critically as favoring auspiciousness over causal efficacy2; "an irrational policy of managing the news".
A standard example of where EDT and CDT/FDT diverge is said to be Smoking lesion, but as this is needlessly confusing one may wish to consider the Toxoplasmosis dilemma instead:
Mice infected with Toxoplasmosis gondii become less scared of cats, and infected mice being eaten by cats is favorable to the lifecycle of toxoplasmosis. Suppose that early experiments suggesting that humans infected with Toxoplasmosis gondii are likewise more fond of cats had replicated. Suppose furthermore (going into the realms of thought experiment) that of people who choose to pet cats given a chance, 20% are found to have latent toxoplasmosis, vs 10% of those not choosing to pet cats.
You are offered a cute cat, guaranteed to itself be free of toxoplasmosis or other diseases you could contract by petting it; there is no way that petting the cat can cause you to contract toxoplasmosis. However, if you pet the cat it will be evidence that you have already contracted toxoplasmosis; and a 10% higher absolute chance of having toxoplasmosis is much worse to you than the goodness of petting a cat a few times.
Would you feel free to pet that cute clean cat if you wanted?
CDT/FDT say there is no harm in petting the cat; any harm here has already been done or not done.
EDT, considering the action-conditional world in which it has pet the cat, updates evidentially within this world on the fact of it having observing itself to pet the cat, concludes in that visualized world that it has a higher chance of toxoplasmosis, and recommends refraining.
A more sophisticated agent following the recommendations of EDT would recognize that if they observe that they have the desire to pet the cat, then actually petting or not would provide no more evidence for having the paraside; that is, the "tickle" of introspectively visible desire screens off petting from parasite. This is known as the "tickle defence".
(FDT would likewise observe that if EDT agents end up universally espousing a policy of never petting cats -- if this is in general the output of EDT -- then refraining from petting the cat cannot be evidence about parasites conditioned on the agent knowing itself to be an EDT agent.)