monte carlo question

[Kuciwalker]

Let's say you have a Monte Carlo simulation of a system with a free parameter (that is, the value of this parameter is chosen before running the simulation). You have a number of candidate values for this parameter, representing what you consider plausible scenarios. You do not know or have an assumption about the probabilities of these different scenarios.

Is it appropriate to use the same set of random numbers for each scenario? I believe yes.

Later, you develop an assumption about the probabilities of each scenario. Is it appropriate to use those probabilities to weight the output of each and then combine them (e.g. by averaging), even though this means you are reusing random numbers? I believe yes, but if I'm not please explain why.

If I'm right, would it therefore be inappropriate to use different random numbers for the different scenarios? I believe this would strictly degrade your result, though it wouldn't affect the asymptotic behavior.

[Kuciwalker]

Hm. Thinking about the system with 1 coin flip, where the coin is unfair with an unknown bias, suggests that I should use different random numbers if I combine the results of multiple scenarios.

[KrazyHorse]

Neither is "inappropriate"

They simply give different error estimators.

I will expound further tonight if you wish

[KrazyHorse]

Kuci: in more detail,

1) If, for example, I was attempting to compare the difference (one minus the other, or one divided by the other) between the results of two MCs given two sets of input parameters I would reuse random numbers. This would reduce the MC error on the difference or the ratio (it is a naive form of antithetic sampling)

2) If the problem is that I have some random variable P (input parameter) with known distribution, I have a MC process to go from P to V(P) and I want to know, e.g. E(V(P)) then for each parameter value I would use new random numbers, all else being equal. There is more to be said on this second point, however. Getting V(P) for some values of P and then inegrating numerically is unlikely to be the most efficient algorithm.

[Kuciwalker]

Thanks. I had thought about (1) in my opening post, where I was considering P in a Bayesian sense (say P represents the strength of a building and I'm simulating random stresses applied to that building).

I'm not sure yet what I actually want to do with each of my individual simulations |P. I agree that running each |P simulation and then aggregating results is suboptimal to including P as one of the simulated random variables for the specific case of (2).

[KrazyHorse]

If your distribution of P is significantly different from uniform you should consider using it as your weighting for importance sampling (running more MC iterations where P is more likely to be)

[Agent Sox]

I haven't dealt with this stuff in awhile but KrazyHorse is on the right track.

[Wezil]

He likes it when you suck up to him.

I'd do a bit more of that.

[giblets]

I haven't dealt with this stuff in awhile but KrazyHorse is on the right track.

Wow, you sure look smart!

[Agent Sox]

I am not trying to suck up or anything. This isn't my area of expertise so I don't really have anything to add.

[loinburger]

The probability P that any of Agent Sox's weird conspiracy theories hold any water is 0.

[Wezil]

I am not trying to suck up or anything. This isn't my area of expertise so I don't really have anything to add.

You don't need expertise. Just tell him he's right.

You will be rewarded later.

[Agent Sox]

The probability P that any of Agent Sox's weird conspiracy theories hold any water is 0.

Well you really cannot properly figure out the probability because you don't have all the variables.

[loinburger]

You are the only variable needed to determine the probability

[Agent Sox]

That doesn't make a lot of sense. You might think I'm crazy or something but that doesn't mean I have my facts wrong.