4 ms·

I'm not sure that the single number is entirely meaningless, because you would expect reversion to the mean in the absence of a trend, so if you computed the li

by mrow84 11y ago

I'm not sure that the single number is entirely meaningless, because you would expect reversion to the mean in the absence of a trend, so if you computed the likelihood function then I guess it would be weighted toward there being a positive trend, if only slightly. I think :-/.

paulmd 11y ago
The null hypothesis (no warming) would posit independent events with equal probability (the chance of an above-average and below-average year are equivalent). So in that case you're basically asking the question "given that I've flipped a fair coin and gotten heads 9 times in a row, what is the probability of heads on the 10th flip?" It's deceptive because you want to say 0.5^10, but the answer is actually 50% (because it's one throw of a fair coin). The 0.5^10 is the chance of 10 heads in a row occurring, but you can't disregard the prior (you have already thrown 9 heads). Reversion to the mean is a trend, it says nothing about any individual event in a series. In the event that it's not warming, the chance of a warm year is just as likely as a cool year. Now, as a trend, the evidence is highly suggestive of global warming being the case. When you have a sufficient sample of throws with a huge number of heads, the chance of the coin being biased* increases. * Note: you cannot actually bias a flipped coin. But you can bias a die, or a spun coin: https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf
- nonbel 11y ago
  >"The null hypothesis would posit independent events with equal probability (the chance of an above-average and below-average year are equivalent)." Why should that be the null hypothesis? Is that a theory someone subscribes to? That would be a theory of no trends and no auto-correlation in temperature. I doubt you will find someone espousing such a theory, so why should we make something no one believes a null hypothesis? If you can find someone claiming that, I am interested in the reasoning.
- mrow84 11y ago
  I think the difference in this case is that we are looking at a point that exceeds the current series maximum. If we had a single value from a series without a trend, we would expect it to fall below the maximum value observed so far (perhaps after sampling at least 2 points - I will think about that later), and so I reckon that the likelihood of a positive trend will be higher than for a zero or negative trend because of that fact. I may try and actually work it out to see if what I'm saying is correct, but one thing I can tell you that likelihood of me not doing so is greater ;-).
  - paulmd 11y ago
    Yes, looking at the sample minimum/maximum does change the probabilities. You're looking for Extreme Value Theory. https://en.wikipedia.org/wiki/Extreme_value_theory https://en.wikipedia.org/wiki/Extreme_value_theory I kept it simple (above or below average should be equal probability if your average is really average) to try and avoid that. I have no idea what kind of distributions weather follows, and as the other guy noted weather is not fully independent either. But it works for a simple explanation. :) It's been a long time since my stat courses and I don't remember this stuff all that well. Maybe there is something that expresses it better. Either way the fact that we're suddenly having a big run of Nth Warmest Year events is pretty suggestive that something is going on. But you do have to look at it as a trend, particular years aren't all that meaningful.

by mrow84 11y ago

paulmd 11y ago

The null hypothesis (no warming) would posit independent events with equal probability (the chance of an above-average and below-average year are equivalent). So in that case you're basically asking the question "given that I've flipped a fair coin and gotten heads 9 times in a row, what is the probability of heads on the 10th flip?" It's deceptive because you want to say 0.5^10, but the answer is actually 50% (because it's one throw of a fair coin). The 0.5^10 is the chance of 10 heads in a row occurring, but you can't disregard the prior (you have already thrown 9 heads). Reversion to the mean is a trend, it says nothing about any individual event in a series. In the event that it's not warming, the chance of a warm year is just as likely as a cool year. Now, as a trend, the evidence is highly suggestive of global warming being the case. When you have a sufficient sample of throws with a huge number of heads, the chance of the coin being biased* increases. * Note: you cannot actually bias a flipped coin. But you can bias a die, or a spun coin: https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf https://www.stat.berkeley.edu/~nolan/Papers/dice.pdf

nonbel 11y ago
>"The null hypothesis would posit independent events with equal probability (the chance of an above-average and below-average year are equivalent)." Why should that be the null hypothesis? Is that a theory someone subscribes to? That would be a theory of no trends and no auto-correlation in temperature. I doubt you will find someone espousing such a theory, so why should we make something no one believes a null hypothesis? If you can find someone claiming that, I am interested in the reasoning.
mrow84 11y ago
I think the difference in this case is that we are looking at a point that exceeds the current series maximum. If we had a single value from a series without a trend, we would expect it to fall below the maximum value observed so far (perhaps after sampling at least 2 points - I will think about that later), and so I reckon that the likelihood of a positive trend will be higher than for a zero or negative trend because of that fact. I may try and actually work it out to see if what I'm saying is correct, but one thing I can tell you that likelihood of me not doing so is greater ;-).
- paulmd 11y ago
  Yes, looking at the sample minimum/maximum does change the probabilities. You're looking for Extreme Value Theory. https://en.wikipedia.org/wiki/Extreme_value_theory https://en.wikipedia.org/wiki/Extreme_value_theory I kept it simple (above or below average should be equal probability if your average is really average) to try and avoid that. I have no idea what kind of distributions weather follows, and as the other guy noted weather is not fully independent either. But it works for a simple explanation. :) It's been a long time since my stat courses and I don't remember this stuff all that well. Maybe there is something that expresses it better. Either way the fact that we're suddenly having a big run of Nth Warmest Year events is pretty suggestive that something is going on. But you do have to look at it as a trend, particular years aren't all that meaningful.