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Are two heads better than one?

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Re: Are two heads better than one?

#61

Post by dxdm »

dsmurrell wrote: post #57
thelinesnloops wrote: post #19

Sailors in the past had a similar adage: “Never go to sea with two chronometers; take one or three.” They relied on precise clocks to calculate longitude.

I'd much rather take three than one... you might step on the one and crush it.

You seem to agree that two is not a good number. Better bring four then, so that you're not left with only two after your mishap.

Or bring only two, but step on one immediately, to get rid of the cursed pair situation, and also to get the clumsiness out of the way early. Old sailor's trick.

Re: Are two heads better than one?

#62

Post by yccs27 »

maxbond wrote: post #27
thelinesnloops wrote: post #19

Sailors in the past had a similar adage: “Never go to sea with two chronometers; take one or three.” They relied on precise clocks to calculate longitude.

I think it is different in the continuous case though, because you can average two (reasonably accurate) chronometers and get a better measurement. But we can't average true and false, at least not in the context of this problem definition. But the chronometers are will sync with each other if you don't store them apart, which would result correlated noise that an average won't fix.

The saying probably assumes that each chronometer has a certain small probability of malfunctioning, resulting in a significant error (basically a fat-tailed error distribution). With three chronometers, you can use a robust estimator of the true value (consensus value or median). With two, there's no robust estimator and if you use the mean, you have twice the probability of being significantly wrong (though only by half as much).

Re: Are two heads better than one?

#63

Post by BurningFrog »

maxbond wrote: post #45
BurningFrog wrote: post #41

Earlier quoted context omitted.

To spell out the point: If the chronometer error rate is 1%, averaging two will give you a 2% error rate.

You will have an error rate of less than or equal to 1%. You can't average two measurements and get a result with a higher error rate than the worst of the original measurements had. You wouldn't be well served by averaging a measurement with a 1% error and a measurement with a 90% error, but you will have still have less than or equal to 90% error in the result. If the errors are correlated, you could end up with a…

My reasoning is that a clock is either right or wrong.

The average of a right and a wrong clock is wrong. Half as wrong as the wrong one, but still wrong.

If this is a good mental model for dealing with clock malfunctions depends on the failure modes of the clocks.

Re: Are two heads better than one?

#64

Post by kevinrineer »

fisian wrote: post #59

This principle is also highly relevant in safety critical systems for using redundant sensors. Just adding a second sensor is often not enough. Because if they disagree, which one do you trust. One example of this is in airplanes.

I'm having trouble with this one due to a lack of experience, but if there is no consensus between the two parties, my assumption would be that you trust neither and ask again. Why is that not the case in a split-brain scenario here? Do you /have/ to make an immediate decision?

Re: Are two heads better than one?

#66

Post by MarkusWandel »

d1ss0nanz wrote: post #47

As expected by Shannon's information theory (error correction, entropy, channel capacity).

Shannon's information theory asssumes optimal coding, but doesn't say what the optimal coding is.

Anyway if a single observer who lies 20% of the time gives you 4 out of 5 bits correct, but you don't know which ones...

And N such observers, where N>2, gives you a very good way of getting more information (best-of-3 voting etc), to the limit, at infinite observers, of a perfect channel...

then interpolating for N=2, there is more information here than for N=1. It just needs more advanced coding to exploit.

Re: Are two heads better than one?

#68

Post by MarkusWandel »

MarkusWandel wrote: post #66
d1ss0nanz wrote: post #47

As expected by Shannon's information theory (error correction, entropy, channel capacity).

Shannon's information theory asssumes optimal coding, but doesn't say what the optimal coding is. Anyway if a single observer who lies 20% of the time gives you 4 out of 5 bits correct, but you don't know which ones... And N such observers, where N>2, gives you a very good way of getting more information (best-of-3 voting etc), to the limit, at infinite observers, of a perfect channel... then interpolating for N=2, t…

I don't have the math for this, but a colleague does, and after spending a few minutes on Matlab came up with about 0.278 bits/flip of channel capacity (Shannon) for the single observer, and I think around 0.451 bits/flip of channel capacity for the dual observers. That's the theoretical capacity with optimal coding. Whatever coding schemes need to be employed to get there, i.e. what redundancy to add to the bit stream... that's the hard part.

Re: Are two heads better than one?

#69

Post by donnerdave »

No one commenting on the inaccuracy (or at least imprecision) of the Python output cited in the article??

"A:T, B:T - chances - H 6.0% | T 94.0% | occurs 34.0% of the time"

By the simplest of math for unrelated events, the chance of both A & B lying about the coin is 20% of 20%, or .2 * .2 = 0.04, or 4.0% ...

The "Let's prove it" section contains the correct analysis, including that our chance of being correct is 80% with two friends.

The code output for three players is similarly flawed, and the analysis slight misstates our chance of being correct as 90.0% (correctly: 89.6%).

Or am I missing something about the intent or output of the Python simulation?

Re: Are two heads better than one?

#70

Post by fc417fc802 »

kevinrineer wrote: post #64
fisian wrote: post #59

This principle is also highly relevant in safety critical systems for using redundant sensors. Just adding a second sensor is often not enough. Because if they disagree, which one do you trust. One example of this is in airplanes.

I'm having trouble with this one due to a lack of experience, but if there is no consensus between the two parties, my assumption would be that you trust neither and ask again. Why is that not the case in a split-brain scenario here? Do you /have/ to make an immediate decision?

You could see that as a lack of detail of the problem as posed. Alternatively it's a breakdown when applying it as an analogy.

Time critical scenarios are one possibility.

In a safety critical scenario intermittent sensor failure might be possible but keep in mind that consistent failure is too.

A jury scenario is presumably one of consistent failure. There's no reason to expect that an intentional liar would change his answer upon being asked again.

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