With all the cold weather of late, I have been thinking of how we should update our beliefs about changes in climate parameters as extreme weather events occur. This analysis applies, of course, to the warm weather events we had in early winter, and the earlier cold snaps back in September around here. My son is also studying conditional probability in high school right now, so this is good for him as well.
There are at least two “takeaways” from this little excursion. The first is that, as we should have suspected, extreme weather events should indeed lead to some updating of our beliefs about climate change, but not a lot. More important, and my second takeaway, is that this excursion is a great way to see some of the basic ideas of Bayesian statistics.
So I am considering how my beliefs about the probability of climate change taking place change when I observe a (local) extreme cold weather event. Intuitively, a cold event should make us wonder if climate change – which should result in warmer temperatures – is actually occurring. But also intuitively, we should be really surprised if we learn too much from one single weather event.
Our starting point is Baye’s Rule:
Pr(climate change| event) =
{Pr(event | climate change) * Pr(climate change)} / Pr(event)
In words: The “posterior” probability of climate change conditional upon observing a weather event equals: The “prior” probability of climate change, Pr(climate change), times the probability of the event happening conditional upon climate change occurring, all divided by the unconditional probability of the weather event occurring.
If you multiply both sides by Pr(event), you see that both sides of the above equation give the probability of both climate change and the weather event occurring. The equality obviously holds therefore.
We normally use the term “posterior probability” for the left hand side of Baye’s Rule. The Rule tells us how our posterior probability is different from our “prior” probability after observing an event.
It is neat to see how Baye’s rule plays out in our climate context.
Let’s suppose that we have “diffuse” prior beliefs on climate change, that is, that Pr(climate change) from the right hand side of the equation equals ½. The question is how that prior belief changes after we observe an extreme weather event.
To do the analysis, we need to know Pr(event | climate change). Again in words, this is the probability of the weather event conditional upon climate change having happened. This is a key calculation. I take “climate change having happened” to mean that the distribution of daily temperatures in my local location having shifted to the right. Specifically, I assume that climate change has increased the average daily temperature by one degree. Let me also define my extreme weather event as a new record low temperature for the day April 16 in Hanover, NH. What is the probability of this event, conditional upon climate change having occurred? Well, it turns out that the long term average daily temperature for April 16 in Hanover is 33 degrees. And the standard deviation, I will assume, is about 7.5 degrees. The record low for this date is 15 degrees, having occurred in 1940.
If we assume that daily temperatures are normally distributed, then the probability of breaking that 15 degree record given the historical mean of 33 degrees is .008198. With climate change, the average shifts up to 34 degrees and the probability of breaking the record goes down: to .005649.
We now have everything that we need to complete Baye’s rule. The denominator of the equation is a little tricky; it is the unconditional probability of the new low temperature record. To find this, we use this equation:
Pr(event) = Pr(event | climate change) * Pr (climate change)
+ Pr(event |no climate change) * Pr(no climate change)
This is just adding up the different ways that we could observe the extreme weather event: there are two ways, either through climate change or not through climate change.
We have all of these probabilities mentioned in the above discussion. If we calculate it out, we get Pr(event) = (.005649*.5) + (.008198*.5) = .0069235.
Now take the first term of Baye’s Rule and divide by the denominator; this is called the likelihood ratio. In our case, this ratio is .005649/.0069235 = .8159168.
Our posterior belief about climate change after observing the extreme cold event is about 82% of our prior belief before we observed the extreme event.
This is more than I would have expected going into this exercise. What determines how much our prior beliefs are affected?
First, note that the stronger our prior beliefs, the less we would change our priors after observing the cold event: The denominator of Baye’s Rule is a weighted average of the probability of the cold event under two different scenarios, climate change or no climate change. The weight on climate change increases as our prior belief on climate change increases, and this makes the weighted average move closer to the probability of the extreme event conditional on climate change.
As the denominator moves closer to the probability of the extreme event conditional on climate change, the likelihood ratio moves closer to one, and our prior belief is affected very little by the extreme cold event. Intuition: With strong prior beliefs on climate change, a cold event does little to affect them. With weak prior beliefs on climate change, a cold event makes us even more skeptical. Wonder why people are affected differently by events? Bayes would not be surprised!
The other thing that affects our much our prior beliefs change is the term
Pr(event | climate change) relative to Pr(event | no climate change). This gets to the heart of how informative the cold event is. Think about it: If climate change really doesn’t do anything to the probability of cold events happening, then these two probabilities would be equal, and the likelihood ratio in Baye’s Rule would equal one. (Why might climate change not affect the probability of extreme cold events? Well, if climate change did nto affect the distribution of daily temperatures very much, that would work.) And in that case, our posterior probability equals our prior probability. Makes sense, no?
On the other hand, what would make the extreme cold event very informative? This would happen if Pr(event | climate change) relative to Pr(event | no climate change) was large. This would be the case if climate change made extreme cold events very unlikely, so that the ratio of these two terms would be very small. Then the likelihood ratio would be small, and our posterior probability of climate change would be small relative to our prior probability.
Well, in conclusion, this got a little more complicated than I thought, but I learned a bit from it.
One lingering concern I have is over a “data mining” sort of issue. Let me put it this way: Seeing a record cold event in SOME locality in the country is much less informative than seeing a record cold event in one particular locality such as Hanover. Somewhere in the country is going to see a record cold event pretty often; one locality will see one only rarely. The above analysis does not apply to the event of just seeing a record cold event SOMEWHERE (well, it does apply, but the probabilities need serious adjusting -- especially the probabilities of observing extreme weather events).
All of this was stimulated because I wanted to spend the night at my camp. When I got there, this late Nor’Easter had blown out the electricity and the downdraft in my woodstove was so extreme I couldn’t get a fire started. Real pain. I was tempted to stay and cook my dinner on the Coleman stove, but came back home instead. Nasty weather.
Sunday, April 15, 2007
The Fall of Imus
I have never been a huge fan of Imus, finding him underwhelming with his reasoning.
But boy, did he fall fast. Now once I understood what the word "hos" means (showing my age and ignorance of popular culture there!) I did have to agree that what he said was really bad.
But was an apology not acceptable? Imus did way more than apologize: he grovelled. He grovelled to everyone, even to Al Sharpton. And everyone was after him, especially some of the Presidential candidates. Is the real story here that they all were not looking forward to having to appear on Imus' show several times?
But boy, did he fall fast. Now once I understood what the word "hos" means (showing my age and ignorance of popular culture there!) I did have to agree that what he said was really bad.
But was an apology not acceptable? Imus did way more than apologize: he grovelled. He grovelled to everyone, even to Al Sharpton. And everyone was after him, especially some of the Presidential candidates. Is the real story here that they all were not looking forward to having to appear on Imus' show several times?
One Sad Tax Code
April 15th is upon us, and my taxes have been filed. What was my marginal tax rate this year? Well, that is a really good question. It looks to me like I was right at the cusp of getting hit with the Alternative Minimum Tax. Last year, I did get caught by that, but this year, for some odd reason unknown to me, I did not. It was close; the calculation required computes your tax under the regular code and under AMT, and if AMT is greater, then you pay according to that.
In an earlier post this year, I noted my uncertainty then as to my marginal tax rate, saying it would be either 26% or 42%. It turned out to be more like the 42%, the regular code rate plus Medicare and the effects of phasing out exemptions and deductions.
What kind of tax code do we have, where during the year when I am making income-earning decisions, I cannot determine what my effective tax rate will be? I mean, let's just totally screw up our incentives. If I had made an additional $10,000, I would have netted $7200 (AMT, with the rate this year of 28%) or $5800 (regular code). Anyone who says that a 14% difference in your rate of pay will not make a difference in what you do, is talking nonsense.
As I said before, the Democrats have a chance to show the country that they can respect individual privacy (easy for them); construct a reasonable tax code and manage government spending (not too hard; Bill Clinton did a fair job); and maintain a strong national defense posture (seemingly impossible for the current Democrats). Probably if they got two out of three, they would have a chance to govern for a while. One out of three should mean the wilderness for them.
In an earlier post this year, I noted my uncertainty then as to my marginal tax rate, saying it would be either 26% or 42%. It turned out to be more like the 42%, the regular code rate plus Medicare and the effects of phasing out exemptions and deductions.
What kind of tax code do we have, where during the year when I am making income-earning decisions, I cannot determine what my effective tax rate will be? I mean, let's just totally screw up our incentives. If I had made an additional $10,000, I would have netted $7200 (AMT, with the rate this year of 28%) or $5800 (regular code). Anyone who says that a 14% difference in your rate of pay will not make a difference in what you do, is talking nonsense.
As I said before, the Democrats have a chance to show the country that they can respect individual privacy (easy for them); construct a reasonable tax code and manage government spending (not too hard; Bill Clinton did a fair job); and maintain a strong national defense posture (seemingly impossible for the current Democrats). Probably if they got two out of three, they would have a chance to govern for a while. One out of three should mean the wilderness for them.
Friday, April 06, 2007
An Excellent Blog for Climate Change
I continue to be impressed by Robert Pielke Sr.'s blog on climate science.
Here is a quote from him on a likely underlying goal of the climate change alarmists: As discussed on Climate Science and Scitizen (e.g. see and see), the underlying reason for this aggressive campaign to focus on the human emissions of carbon dioxide from fossil fuels as the main culprit is to promote energy policy changes, not to develop an appropriate comprehensive climate policy.
The most recent guest post by Professor Ben Herman of the University of Arizona is also worth reading for anyone who is willing to question some of the predictions and policy prescriptions being bandied about.
Here is a quote from him on a likely underlying goal of the climate change alarmists: As discussed on Climate Science and Scitizen (e.g. see and see), the underlying reason for this aggressive campaign to focus on the human emissions of carbon dioxide from fossil fuels as the main culprit is to promote energy policy changes, not to develop an appropriate comprehensive climate policy.
The most recent guest post by Professor Ben Herman of the University of Arizona is also worth reading for anyone who is willing to question some of the predictions and policy prescriptions being bandied about.
Apple and DRM
Lot's going on with Apple and DRM. The EU is threatening antitrust action against iTunes, and Apple and EMI announced a new deal on DRM-free music.
Let's think about the EMI deal. EMI, with Apple's partnership, is going to make its library (ex-Beatles!) available on iTunes without DRM, and in close-to-CD audio quality.
The catch? Instead of the usual $.99 per song, EMI's songs will go for $1.29.
What accounts for the higher price? We have a few choices, not mutually exclusive:
1. Songs unencumbered by DRM are worth more, so the price is higher.
2. Songs of higher audio quality are worth more, so the price is higher.
3. Songs without DRM are going to spread more, reducing purchases by other users of the same song, so they have a higher marginal cost and therefore should sell for more.
4. As I pointed out in this post, Apple should be concerned that if iTunes songs are priced below their "stand-alone" profit-maximizing price, then entry by competitors into the portable musice device market is aided. From the perspective of pricing iTunes songs and iPods, Apple should be pricing songs lower than their stand-alone price, but above marginal cost. To eliminate the aid to entry, Apple has incentive to introduce DRM, preventing iTunes songs from playing on other devices.
If Apple drops DRM for some songs, then to maintain a deterrent to entry, it should increase the price of songs.
5. Last argument, it may be that EMI is simply charging a higher royalty to Apple for its songs (why?) and Apple is simply charging the relevantly higher price.
My choice of answers? A combination of all of these. Look at it from EMI and Apple's perspectives. Everything points to a higher price for the DRM-free songs (and by the way, the DRM-encumbered songs are still available at a lower price). "Let's try it and see what happens."
Based on my own recent experience in hitting the 5-machine limit, I would probably opt for the higher priced version. If there were no choice available, I would likely buy almost the same number of songs at $1.29 instead of $.99, when the higher price was for a higher quality and DRm-free version.
Let's think about the EMI deal. EMI, with Apple's partnership, is going to make its library (ex-Beatles!) available on iTunes without DRM, and in close-to-CD audio quality.
The catch? Instead of the usual $.99 per song, EMI's songs will go for $1.29.
What accounts for the higher price? We have a few choices, not mutually exclusive:
1. Songs unencumbered by DRM are worth more, so the price is higher.
2. Songs of higher audio quality are worth more, so the price is higher.
3. Songs without DRM are going to spread more, reducing purchases by other users of the same song, so they have a higher marginal cost and therefore should sell for more.
4. As I pointed out in this post, Apple should be concerned that if iTunes songs are priced below their "stand-alone" profit-maximizing price, then entry by competitors into the portable musice device market is aided. From the perspective of pricing iTunes songs and iPods, Apple should be pricing songs lower than their stand-alone price, but above marginal cost. To eliminate the aid to entry, Apple has incentive to introduce DRM, preventing iTunes songs from playing on other devices.
If Apple drops DRM for some songs, then to maintain a deterrent to entry, it should increase the price of songs.
5. Last argument, it may be that EMI is simply charging a higher royalty to Apple for its songs (why?) and Apple is simply charging the relevantly higher price.
My choice of answers? A combination of all of these. Look at it from EMI and Apple's perspectives. Everything points to a higher price for the DRM-free songs (and by the way, the DRM-encumbered songs are still available at a lower price). "Let's try it and see what happens."
Based on my own recent experience in hitting the 5-machine limit, I would probably opt for the higher priced version. If there were no choice available, I would likely buy almost the same number of songs at $1.29 instead of $.99, when the higher price was for a higher quality and DRm-free version.
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