pEpsilon: p being the referent for the negative base-10 logarithm of whatever follows.
Don’t you mean exponent, not logarithm? I’m more used to seeing this concept as e, like 1e-10.
The primary trouble with this suggestion is this problem; most practical models will not be able to actually distinguish between 99.99th percentile events and 99.9999th percentile events, let alone 1-1e-10 quantile events.
No, I meant “-log10(X)=pX”, in the general sense. That is standard use in chemistry and other fields that regularly deal with very small numbers.
And if your model can’t distinguish between something that happens one in ten thousand times and something that happens one in a million times, you aren’t simulating a probabilistic infinite series, and you can make a deterministic conclusion about your worst-case scenario.
The goal is to be able to calculate things currently relegated to simulation.
No, I meant “-log10(X)=pX”, in the general sense. That is standard use in chemistry and other fields that regularly deal with very small numbers.
Ah, okay, that’s cleared up my linguistic confusion; thanks!
And if your model can’t distinguish between something that happens one in ten thousand times and something that happens one in a million times, you aren’t simulating a probabilistic infinite series, and you can make a deterministic conclusion about your worst-case scenario.
Sort of. These sorts of simulation projects go on in my department, and so I’m familiar with them. For example, a person that used to work down the hall was working on uncertainty quantification for models of nuclear power plants- we have a relatively good idea of how likely any particular part is to break, and how to simulate what happens if those break, and we can get out a number that says “we think there’s a 1.3e-9 chance per year per plant of a catastrophic event,” for example, but we want to get out the number that says “if our estimate of the rate at which valves fail is off by 5%, our final estimate will be off by X%”, and use that to get a distribution on the final catastrophe estimate. (This will give us a better sense of total risk, as well as where we should focus our experiments and improvements.)
So they can say “our model tells us that, with these point inputs, this happens once in a billion times,” but they can’t yet say “with our model and these input distributions, the chance of this happening more than once in a million times is less than one in a thousand,” which must be true for the first statement to be useful as an upper bound of the estimate (rather than an expected value of the estimate).
The goal is to be able to calculate things currently relegated to simulation.
It’s not clear to me what you mean by this, since I see calculation as a subset of simulation. If you mean we’d like to analytically integrate over all unknowns, even if we had the integration power (which we can’t), we don’t have good enough estimates of the uncertainties for that to be all that much more meaningful than our current simulations. With only, say, a thousand samples, it may be very difficult to determine the thickness of the tails of the population distribution, but the thickness of the tails may determine the behavior of the expected value / the chance of catastrophe. These problems show up at much lower quantiles, and it’s not clear to me that just ignoring events more rare than a certain quantile will give us useful results.
To use the bridge example again- for the ‘expected load’ distribution to be faithful above the 99.99th percentile, that would require that we be surprised less than once every thirty years. I don’t think there are traffic engineers with that forecasting ability, and I think their forecasting ability starts to break down more quickly than the rare events we’re interested in, so we have to tackle these problems somehow.
Well, when I was working on a S5W/S3G (MTS 635, SSBN 732 blue) power plant, our baseline “end of the world” scenario started with “a non-isolateable double-ended shear of a main coolant loop”. (half of the power plant falls off). I can’t begin to estimate the likelihood of that failure, but I think quantum mechanics can.
If classical mechanics gives you a failure rate that has uncertainty, you can incorporate that uncertainty into your final uncertainty: “We believe it is four nines or better that this type of valve fails in this manner with this frequency or less.”
And at some point, you don’t trust the traffic engineer to guess the load- you post a load limit on the bridge.
So they can say “our model tells us that, with these point inputs, this happens once in a billion times,” but they can’t yet say “with our model and these input distributions, the chance of this happening more than once in a million times is less than one in a thousand,” which must be true for the first statement to be useful as an upper bound of the estimate (rather than an expected value of the estimate).
Why not? Can’t we integrate over all of the input distributions, and compare the total volume of input distributions with failure chance greater than one in N with the total volume of all input distributions?
Why not? Can’t we integrate over all of the input distributions, and compare the total volume of input distributions with failure chance greater than one in N with the total volume of all input distributions?
The impression I got was that this is the approach that they would take with infinite computing power, but that it took a significant amount of time to determine if any particular combination of input variables would lead to a failure chance greater than one in N, meaning normal integration won’t work. There are a couple of different ways to attack that problem, each making different tradeoffs.
If each data point is prohibitively expensive, then the only thing I can suggest is limiting the permissible input distributions. If that’s not possible, I think the historical path is to continue to store the waste in pools at each power plant while future research and politics is done on the problem.
“Logarithm” and “exponent” can name the same thing as viewed from different
perspectives. Example: 3 is the base-10 logarithm of 1000, which is another
way of saying that 10^3 = 1000 (an expression in which 3 is the exponent).
Don’t you mean exponent, not logarithm? I’m more used to seeing this concept as e, like 1e-10.
The primary trouble with this suggestion is this problem; most practical models will not be able to actually distinguish between 99.99th percentile events and 99.9999th percentile events, let alone 1-1e-10 quantile events.
No, I meant “-log10(X)=pX”, in the general sense. That is standard use in chemistry and other fields that regularly deal with very small numbers.
And if your model can’t distinguish between something that happens one in ten thousand times and something that happens one in a million times, you aren’t simulating a probabilistic infinite series, and you can make a deterministic conclusion about your worst-case scenario.
The goal is to be able to calculate things currently relegated to simulation.
Ah, okay, that’s cleared up my linguistic confusion; thanks!
Sort of. These sorts of simulation projects go on in my department, and so I’m familiar with them. For example, a person that used to work down the hall was working on uncertainty quantification for models of nuclear power plants- we have a relatively good idea of how likely any particular part is to break, and how to simulate what happens if those break, and we can get out a number that says “we think there’s a 1.3e-9 chance per year per plant of a catastrophic event,” for example, but we want to get out the number that says “if our estimate of the rate at which valves fail is off by 5%, our final estimate will be off by X%”, and use that to get a distribution on the final catastrophe estimate. (This will give us a better sense of total risk, as well as where we should focus our experiments and improvements.)
So they can say “our model tells us that, with these point inputs, this happens once in a billion times,” but they can’t yet say “with our model and these input distributions, the chance of this happening more than once in a million times is less than one in a thousand,” which must be true for the first statement to be useful as an upper bound of the estimate (rather than an expected value of the estimate).
It’s not clear to me what you mean by this, since I see calculation as a subset of simulation. If you mean we’d like to analytically integrate over all unknowns, even if we had the integration power (which we can’t), we don’t have good enough estimates of the uncertainties for that to be all that much more meaningful than our current simulations. With only, say, a thousand samples, it may be very difficult to determine the thickness of the tails of the population distribution, but the thickness of the tails may determine the behavior of the expected value / the chance of catastrophe. These problems show up at much lower quantiles, and it’s not clear to me that just ignoring events more rare than a certain quantile will give us useful results.
To use the bridge example again- for the ‘expected load’ distribution to be faithful above the 99.99th percentile, that would require that we be surprised less than once every thirty years. I don’t think there are traffic engineers with that forecasting ability, and I think their forecasting ability starts to break down more quickly than the rare events we’re interested in, so we have to tackle these problems somehow.
Well, when I was working on a S5W/S3G (MTS 635, SSBN 732 blue) power plant, our baseline “end of the world” scenario started with “a non-isolateable double-ended shear of a main coolant loop”. (half of the power plant falls off). I can’t begin to estimate the likelihood of that failure, but I think quantum mechanics can.
If classical mechanics gives you a failure rate that has uncertainty, you can incorporate that uncertainty into your final uncertainty: “We believe it is four nines or better that this type of valve fails in this manner with this frequency or less.”
And at some point, you don’t trust the traffic engineer to guess the load- you post a load limit on the bridge.
Why not? Can’t we integrate over all of the input distributions, and compare the total volume of input distributions with failure chance greater than one in N with the total volume of all input distributions?
The impression I got was that this is the approach that they would take with infinite computing power, but that it took a significant amount of time to determine if any particular combination of input variables would lead to a failure chance greater than one in N, meaning normal integration won’t work. There are a couple of different ways to attack that problem, each making different tradeoffs.
If each data point is prohibitively expensive, then the only thing I can suggest is limiting the permissible input distributions. If that’s not possible, I think the historical path is to continue to store the waste in pools at each power plant while future research and politics is done on the problem.
“Logarithm” and “exponent” can name the same thing as viewed from different perspectives. Example: 3 is the base-10 logarithm of 1000, which is another way of saying that 10^3 = 1000 (an expression in which 3 is the exponent).