What about Monte Carlo methods? There are many problems for which Monte Carlo integration is the most efficient method available.
Monte Carlo methods can’t buy you any correctness. They are useful because they allow you to sacrifice an unnecessary bit of correctness in order to give you a result in a much shorter time on otherwise intractable problem. They are also useful to simulate the effects of real world randomness (or at least behavior you have no idea how to systematically predict).
So, for example, I used a Monte Carlo script to determine expected scale economies for print order flow in my business. Why? Because it’s simple and the behavior I am modeling is effectively random to me. I could get enough information to make a simulation that gives me 95% accuracy with a few hours of research and another few hours of programming time. Of course there is somewhere out there a non-randomized algorithm that could do a more accurate job with a faster run time, but the cost of discovering it and coding it would be far more than a day’s work, and 95% accuracy on a few dozen simulations was good enough for me to estimate more accurately than most of my competition, which is all that mattered. But Eliezer’s point stands. Randomness didn’t buy me any accuracy, it was a way of trading accuracy for development time.
What about Monte Carlo methods? There are many problems for which Monte Carlo integration is the most efficient method available.
Monte Carlo methods can’t buy you any correctness. They are useful because they allow you to sacrifice an unnecessary bit of correctness in order to give you a result in a much shorter time on otherwise intractable problem. They are also useful to simulate the effects of real world randomness (or at least behavior you have no idea how to systematically predict).
So, for example, I used a Monte Carlo script to determine expected scale economies for print order flow in my business. Why? Because it’s simple and the behavior I am modeling is effectively random to me. I could get enough information to make a simulation that gives me 95% accuracy with a few hours of research and another few hours of programming time. Of course there is somewhere out there a non-randomized algorithm that could do a more accurate job with a faster run time, but the cost of discovering it and coding it would be far more than a day’s work, and 95% accuracy on a few dozen simulations was good enough for me to estimate more accurately than most of my competition, which is all that mattered. But Eliezer’s point stands. Randomness didn’t buy me any accuracy, it was a way of trading accuracy for development time.