Not entirely. There's three main parts: liability, comprehensive, and collision.
Liability is the part that's legally required in the US. If the numbers showed that self-driving cars were much less likely to cause an accident, it would be considerably cheaper for them (a much more dramatic version of how anti-lock brakes can make your liability coverage cheaper).
Comprehensive (mainly stuff that happens while you're not in the vehicle) would remain unchanged. But it's also the cheapest part.
Collision is what covers repairs to your own vehicle if you're at fault or if someone at fault isn't insured themselves (you can also skip the first part and just get uninsured motorist coverage). The uninsured at-fault driver case doesn't change until there are enough self-driving cars on the road that accidents as a whole go way down, but the first case (damage to your own vehicle that was your own fault) should be much less likely and thus cheaper to insure.
Liability is the part that's legally required in the US. If the numbers showed that self-driving cars were much less likely to cause an accident, it would be considerably cheaper for them (a much more dramatic version of how anti-lock brakes can make your liability coverage cheaper).
Comprehensive (mainly stuff that happens while you're not in the vehicle) would remain unchanged. But it's also the cheapest part.
Collision is what covers repairs to your own vehicle if you're at fault or if someone at fault isn't insured themselves (you can also skip the first part and just get uninsured motorist coverage). The uninsured at-fault driver case doesn't change until there are enough self-driving cars on the road that accidents as a whole go way down, but the first case (damage to your own vehicle that was your own fault) should be much less likely and thus cheaper to insure.