We’ve talked before about the anthropic principle, which, in physics, is roughly the idea that things are what they are because otherwise we wouldn’t be around to see them. Related are various equilibrium principles which state that things are what they are because, if they weren’t, behavior would change until equilibrium is reached.
An example is the idea that price elasticity of demand should be close to -1. If it’s steeper than -1, then the seller has a motivation to lower the price so as to get more total money; if it’s shallower than -1, then the seller has a motivation to raise the price so as get more total money; equilibrium is only at -1. The price elasticity of demand is not, in general, -1, because there are lots of other costs, benefits, and constraints in any system; the -1 thing is just a baseline. Still, it can be a useful baseline.
Another example is the median voter theorem, which we’ve discussed many times on this blog (see the many links here): to the extent that parties take positions are not close to the median of the voters, the parties should be able to gain votes by moving toward the median. Again, this does not generally happen because of many complicating factors; the median voter theorem can still be helpful as a baseline.
Another example is effect sizes in statistics, a topic that can also be studied empirically.
Today I want to talk about polling error, in particular, this finding from an article with Houshmand Shirani-Mehr, David Rothschild, and Sharad Goel:
Reported margins of error typically only capture sampling variability, and in particular, generally ignore nonsampling errors in defining the target population (e.g., errors due to uncertainty in who will vote). Here, we empirically analyze 4221 polls for 608 state-level presidential, senatorial, and gubernatorial elections between 1998 and 2014, all of which were conducted during the final three weeks of the campaigns. Comparing to the actual election outcomes, we find that average survey error as measured by root mean square error is approximately 3.5 percentage points, about twice as large as that implied by most reported margins of error.
Roughly speaking, nonsampling error is about the same size as sampling error. I want to argue that this fits an anthropic or equilibrium storyline. It goes like this: if you conduct a survey with a huge sampling error, then there will be a clear benefit from increasing your sample size and bringing that sampling error down. From the other direction, it would not make sense to run a state poll with a sample size in the tens of thousands: that would bring down the sampling error but it would not help with nonsampling error.
With independent error components,
sd of total error = sqrt((sd of sampling error)^2 + (sd of nonsampling error)^2).
and the way the math works is that, reducing the smaller of these terms gives diminishing returns.
Again this reasoning is only approximate. For one thing, if the sd of average survey error is twice that of sampling error, then this implies that sampling error is less than nonsampling error (because of that root-mean-square thing), and I guess that kinda makes sense, given that polls are used for information other than the headline number, also polls are analyzed for trends, not just levels. The idea is that you wouldn’t expect nonsampling error to be much less than sampling error or much more than sampling error.
The point of this anthropic reasoning is not to give an exact answer but rather to give some intuition to where we are. It’s related to the general principle that you’d expect variance and squared bias to be comparable to each other, as discussed in Section 4.3 of Regression and Other Stories.
Nice argument. I’m not still sure how it connects to the anthropic principle, which only makes sense for an existing ensemble of parallel possibilities, but it makes economic sense.
Indeed “equilibrium principles” seem related to the “anthropic principle” only in the “principles that can be used to explain things” sense. “This anthropic reasoning” is actually an “equilibrium reasoning” but I guess “anthropic” sounds better. (Tangentially related: there are a lot of comments on Twitter lately about Taleb saying that probability is a kernel and can be negative.)
Carlos:
You can have negative probabilities in problems for which not all probabilities can be measured.
Consider the following example of events A and B. The following probabilities must sum to 1: Pr(A and ~B), Pr(A and B), Pr(~A and B), and Pr(~A and ~B).
Now consider the following assignment of probabilities:
Pr(A and ~B) = 3/4
Pr(A and B) = – 1/2
Pr(~A and B) = 3/4
Pr(~A and ~B) = 0.
These four numbers add to 1, but if you could measure both A and B, they’re unphysical. But if you can only measure A and B, they work: Pr(A) = 1/4 and Pr(B) = 1/4. People have sometimes used such probabilities to model incomplete information, as in the theory of belief functions. I’ve never found such ideas useful, but I do discuss them a bit in this article.
> You can have negative probabilities
When “probability” has been used for a century to refer to something well-defined mathematically (if not philosophically) it may be better to use another name to refer to other things. The term “quasiprobability” is often used in physics, for example. Different people may have different opinions on the subject so you can have negative probabilities if you want to have them – and you can’t if you don’t.
> I’ve never found such ideas useful, but I do discuss them a bit in this article.
I started to read that article a few weeks ago but I couldn’t make it that far. There also words seemed to mean something unexpected rendering the exposition extremely difficult to understand.
https://statmodeling.stat.columbia.edu/2024/08/07/a-welcome-rant-on-betting-knowledge-belief-and-the-foundations-of-probability/#comment-2377101
Negative probabilities are no different from irrational probabilities.
Stick to the original combinatoric definition and all that “philosophical” confusion goes away.
How do pollsters measure their own nonsampling error? How do they then design polls to attain their preferred level of nonsampling error?
Anon:
I think they just do their best to bring their nonsampling error as close to zero as they can.
I would like some reassurance that this isn’t totally unscientific. Like how do we know that polling based forecasts are better than astrology or vibes?
Anon:
It’s basically impossible to compare a forecast to “vibes,” because “vibes” will make use of all currently available information, including polls and public forecasts.
Can we tell if there’s predictive information in them that isn’t in RDI + battle deaths (or some such model)? Is there a standard formalized theory according to which polls do a good job at predicting election outcomes?
Anon:
For the fundamentals-based forecast, there’s no way based on data alone to choose or combine predictors; there just aren’t enough national elections, even when predicting vote share (which makes much more sense than predicting win/loss).
Regarding polls, yes, it’s clear that they improve upon a pure fundamentals-based forecast, at least at the state level. At the national level, again, it’s hard to say.
This is related to the principle I used to use to select sample sizes for forensic surveys of construction defects. In essence, it made sense to set the sample size so that the cost of an additional investigation was about the same size as the expected reduction in the cost of settling the lawsuit.
In settling lawsuits in general you don’t pay the best estimate of the amount owed to the plaintiff, because there’s uncertainty in that estimate, and this means there’s a good probability that the plaintiff sees things differently and thinks they’re still losing out. Instead you pay something like mean + 1 sd of the posterior for the estimate, because this bias in the payout ensures the plaintiff is willing to take the offer.
By investigating how much damages there are at the site (say of 500 buildings) you can’t really strongly control the mean amount, but you can reduce the posterior standard deviation, creating less uncertainty. So you should spend money to reduce this uncertainty, until reducing the uncertainty costs about as much as collecting another data point.
This principle produces sample size estimates that are VASTLY lower than typical “power” calculations in this context, because one data point might cost you $35,000 to obtain. (say, tearing open the roof of a building to quantify the degree of water intrusion and then replacing the roof after).
If you’ve got 500 buildings, you might be selecting say 10 or 12 buildings for investigation, via random sampling, because those first few data points dramatically lower the uncertainty, but not doing substantially more, because they each cost $35k and don’t reduce the cost of settlement by an accompanying amount to make it worthwhile.
I’ve never liked the anthropic principle in physics (I don’t think it actually explains anything) but this makes sense.
I like it but, same as you, I don’t think it _explains_ anything. How I see the anthropic principle is not as an explanation, but as a… uh, what would you call it… agh, can’t think of the right word! But it’s related to what some religious people think — let me give you an example.
I was talking with mormons, and I was interested to know why they had adopted that faith. One of them gave the rather common explanation that there have been strange coincidences in her life that _can not_ be other than God’s leading. As an example she talked about all the small things that had to happen for her to meet her husband. For all those things to happen just that way and at the right time — she exlaimed — is so improbable that it MUST be God’s doing!
But of course we understand that to be misleading, and I won’t elaborate further on that point, since I think that’s so obvious to readers of this blog. But in my mind that’s kind of a “statistical anthropic principle”: it obviously doesn’t EXPLAIN why she met her hubby etc. but it makes us see the supposed “improbabilities” in a different light; more realistically mayhaps.
Slabo:
Yes, I think your Mormon friend’s marriage is a great example of the anthropic principle. My argument is not that a supernatural superbeing brought them together (I think Douglas Adams called that the “infinite improbability drive,” but that, conditional on these two people having found each other, that implies that certain conditions, which might be considered a prior unlikely, must have happened.
I mean, yeah, an anthropic principle type concept can be used that way to say “given that we know X is true, Y and Z which seem a priori unlikely must also be true because they are linked to X”.
But it isn’t a *causal* explanation (and even non-causally is only as strong as our confidence that X is actually linked to Y and Z).
Saying the universe has to be more or less the way it is because otherwise we wouldn’t be there to observe it is kind of reversing cause and effect.
Hello:
In general elasticity =1 is not an equilibrium or optimum. If costs were zero, then elasticity of 1 would be optimal (i.e. profit maximizing). But when costs are positive, equilibrium elasticities are generally bigger than 1 in absolute value.
The price elasticity of demand can’t be *less* than 1 in absolute value because that means profit increases when you raise price. (That’s because revenue (=p*q) increases when you raise price if elasticity < 1, and cost falls.)
But elasticity of demand can certainly be greater than 1.
Dan:
Price elasticity of demand can definitely be flatter than -1 in real life. Take a look at the wikipedia article on the topic; it gives lots of examples. Price elasticity of demand is equal to 1 under equilibrium under some very strong and typically unrealistic assumptions, just as the median voter theorem holds some very strong and typically unrealistic assumptions. I think these equilibrium ideas still provide useful insight.
> The price elasticity of demand can’t be *less* than 1 in absolute value because that means profit increases when you raise price.
Assuming that there is no competition. (Competition is also ignored in the original post – unless it’s somehow included in those “constraints”.)
Carlos:
Yes, lack of competition is assumed with that price elasticity of demand = 1 thing, in the same way that primary election competition is ignored in the median voter theorem.
If we’re being precise, we can say economic theory says the price elasticity of demand should be in the region [-1, 0]. All the major deviations from -1, such as production costs or competition, push the elasticity towards zero.
Joseph:
Yes, and this is consistent with the wikipedia page on the topic, which gives many examples of estimated elasticities, almost all of which are between 0 and -1. Your point has an analogy to the median voter theorem, where the major deviations pull the parties apart in equilibrium.
Setting economics aside entirely, I often tell students that we should expect elasticity to be between 0 and 1, for reasons related to regression to the mean. If you increase some relevant input by 10%, you can expect the output to increase by something, but not the full 10%. Not always, but this is a reasonable starting point, and it helps us interpret regression coefficients when the input and output variables are on the log scale.
Regarding the median voter therom:
the reason it doesn’t make sense for people to vote for a politician that claims to be more centrist than its party is that the politician is almost always lying. In the end it will vote with its party regardless of whatever position it claimed on an issue during the previous election. Voters know this and take it into account when voting. This is why pro-choice Republicans and anti-abortion Democrats have had such a difficult time turning their positions into political capital. The obvious outcome of this situation is selection of candidates toward the party extremes, since there’s no point in running for politicians toward the center, because voters won’t trust them to act on their campaign claims.
I don’t think this is totally true, or at least it has only become true in like the past 4 years. Sen. Manchin in WV holding on long after WV became a true red state, and Sen. Tester in MT winning reelection in 2018, and Sen. Murkowski winning reelection in 2010 *against the actual Republican candidate as well as the Democrat*, and arguably Sen. Collins in Maine winning reelection in 2020 though Maine went fairly strongly blue presidentially, I think show that there can be a real benefit.
But I think this only works for *incumbents* who have an actual record to show that they are more centrist than the rest of their party. A new politician showing up running as a moderate D/R, not so much … which is why we are losing the center in Congress.
(Also, even the incumbent effect may be weakening. Manchin didn’t even try for reelection this time and Tester may well lose. Another caveat is that ME, MT, and AK are not as well aligned with national politics as most states outside the northernmost tier.)