There was a bunch of discussion over at Andrew Gelman's blog about "bet proof" interpretations of confidence intervals. The relevant paper is here.
Below is what I original wrote in grey. Nope. I was misreading. The actual definition of bet-proofness is also weird. Here it is in essence:
A confidence interval construction procedure is bet proof if for every betting scheme there exists some value of the parameter such that this scheme will lose in the long run.
This is kind of the converse of what I was thinking before. But here's my example of why that makes no sense (quoted here from what I wrote on Andrew's blog after I finally figured out what was going on).
Now, with the above reworded Definition 1, I can see how the game is about revealing X and not revealing Theta. But, I don’t see how it is interesting. Let me give you an example:
Our bet is to sample a random sample of every adult who lives within 2 blocks of me. We will measure their heights X, then you’ll construct a confidence interval CI(X), and I will bet whether after we measure all the rest of them, the population mean Theta will be in the CI.
Now, being a good Bayesian, I use Wald’s theorem and realize that any strategy to decide what my bet winnings will be that doesn’t use a prior and a loss function will be dominated by one that does…. So I’ll go out to Wikipedia and google up info about the population of the US and their heights, and I”ll construct a real world prior and then I’ll place my bet.
Now bet proofness says that because it’s the case that if the actual height of people in a 2 block radius is 1 Million Feet (ie. THERE EXISTS A THETA), my prior will bias my bets in the wrong direction and I will not be able to make money…. that this CI is all good, it’s bet proof.
And how is that relevant to any bet we will actually place?"
The basic principal of bet-proofness was essentially that if a sample of data X comes from a RNG with known distribution that has some parameter , then even if you know exactly, so long as you don't know what the X values will be, you can't make money betting on whether the constructed CI will contain the (the paper writes this in terms of but the principal is the same since f is a deterministic function).
The part that confused me, was that this was then taken to be a property of the individual realized interval... "Because an interval came from a bet-proof procedure it is a bet-proof realized interval" in essence. But, this defines a new term "bet-proof realized interval" which is meaningless when it comes to actual betting. The definition of "bet-proof procedure" explicitly uses averaging over the possible outcomes of the data collection procedure but after you've collected and told everyone what it is, if someone knows and knows they can calculate exactly whether the confidence interval does or does not contain and so they win every bet they make.
So "bet-proof realized confidence interval" is really just a technical term meaning "a realized confidence interval that came from a bet proof procedure" however it doesn't have any content for prediction of bets about that realized interval. The Bayesian with perfect knowledge of and and the confidence construction procedure wins every bet! (there's nothing uncertain about these bets).
Consider things like GDP / capita and how to use them in constructing dimensionless ratios. Now, a person is not an infinitely divisible thing. It's pretty much meaningless to talk about 2.8553 people. Just like molecules, "people" is a count, a dimensionless integer. But, then you can also think about the "mole" in SI units. This has all the qualities that normally make up an arbitrary dimensional unit, you can sub-divide it essentially continuously (at least to 23 decimal places). And that's the key thing about dimensions. The symmetry property of dimensionlessness is essentially that if you define a unit of measure of something, say the Foo and someone else defines the Bar and 1 Bar is equal to x Foos, then you can make all your equations in Bar units into equations in Foo units by multiplying by x Foos/Bar wherever you have something measured in Bars.
This is closely related to the renormalization group. However, it breaks down when the thing you are measuring is not infinitely divisible (or approximately so).
So, in the asymptotic regime where you are describing large aggregates of people (like say countries or states or very large corporations etc) you can calculate statistics in terms of "per capita" and treat capita as if it were an infinitely divisible thing, and therefore as if it represents a "dimension". However in the asymptotic regime where you are discussing a small number of people, you will always use an integer, 1,2,3,4 people etc and so this should be treated as simply a number that is dimensionless and does not enter into the dimensional calculation.
So, when I calculated Total Wages Paid / Total Hours Worked / Market Cap Of Stocks * 1 hr, the dimensions of this are Dollars/Hours / Dollars * Hours = 1 = dimensionless, and this is implicitly because we're discussing what fraction of the total market a single person could buy if they received an average amount of money for their hour of work.
On the other hand, when thinking about the aggregate when I calculated
(GDP/capita * Fraction Of GDP To wages ) / (GDP/Capita * Market Cap as Fraction of GDP * 1 Yr)
We are now explicitly comparing averages across a large pool of people, and we might be interested in how "capita" changed in time in both cases, and "capita" changes in a near-continuous manner because it's an integer, but it's an integer like 325,000,000 so it has 9 significant figures and we can ignore the incremental person's discrete effect.
Whether to treat Capita or People as a dimension is more or less down to a choice of how you are thinking about the problem. When the number of people involved is large and explicitly considered in a ratio, such as in the calculation of GDP/Capita then treating "Capita" as a dimension that needs to cancel makes sense. When the number of people is small and a specific integer: such as "for a family of 3 the cost to purchase food for a month is X Dollars/Month" it would make sense to treat the number of people as a dimensionless count and not calculate something like X/3 Dollars/Month/Person, so if you want to ask what's the relative cost of feeding 3 people per month vs watering a lawn at cost Y dollars/month you can say X/Y is dimensionless ignoring the fact that a family of 3 people is involved.
The most useful case for treating people as a dimension is when there is a natural linear relationship between the number of people and the thing of interest. For example, for adults feeding 3 of them requires about 3 times the mass of food as feeding one. But, because of economies of scale related to cooking and shopping costs, there is no real reason to think that it requires 3 times the dollar cost (for home cooked meals). So, Dollars/Person is not a meaningful statistic at the asymptotic small person count regime. On the other hand, for a whole Battalion of troops, it would be. This is directly related to the "unit conversion symmetry" property I described in the first paragraph. If you define a Battalion of troops as 7000 and a different country they define it as 5300, then you can convert the other countries equations to your units by multiplying their Battalion numbers by 5300 and dividing by 7000, and the assumed linearity of the relationships in the asymptotic limit makes these discrete counts have approximately arbitrary linear-scaling dimensional properties.
Here's a different version of a graph I posted yesterday:
This graph is (GDP/capita) * (Wages As Percentage Of GDP/100) / (GDP/capita * Stock Market Capitalization as Percentage of GDP/100) = Wages as Percent / Stock Market Cap as Percent.
Note also that the per capita cancels on the top and the bottom.
It looks like GDP cancels but note that this is problematic. Conceptually the dimensions of this number are actually 1/[Time] since GDP is not a dollar measure but an INCOME measure, so when they calculate "Stock Market Cap as Percent" they're really calculating "Stock Market Cap as Percent of the GDP Income Rate times 1 year". Whereas wages as a percentage of GDP are in fact in units of Dollars/Time. The overall dimensions are 1/[Time] with units of 1/Year.
So, this graph shows you how if you saved *all* your wages for a year, how big a fraction of the stock market could you buy relative to "the share that everyone would have if the stock market were equally divided among all people".
In 1975 you could buy a little more than "your equal share" (I won't call it "fair" it's just equal). These days you can buy a little less than 30% of an equal share.
Here's another relevant question. How much would an epsilon share of the stock market buy you in consumer goods if you sold it?
This is (stock market capitalization as a fraction of GDP * GDP/capita) / CPI * epsilon where epsilon is a sufficiently small dimensionless number that the results are in dollar amounts you might carry in your wallet.
In 1988 your per capita share of the stock market would buy you consumer goods equivalent to $25 today (3 chicken sandwiches?), whereas by 2000 it would buy you what $80 would buy you today (a sushi dinner for 3?), and it's been on a wild ride ever since. On the other hand, if you look at growth rates prior to 1990, you can see that the extension of those growth rates would put the stock market at the bottom of the 2008 crash about at the "right" level. Hmm... Certainly we do things more efficiently now that we have computers, so I could see an increased growth rate being reasonable, but pretty obviously given the periodic oscillations, there's way too much optimism.
The Federal Reserve Economic Data (FRED) website is a great way to easily be able to play around with measures in the economy and create new dimensionless ratios that are relevant to various interests. They have a system where you can add various datasets to a graph and construct a new measure via a formula. So for example, here is a measure of consumer goods purchasing power of an hour of average labor:
When multiplied by 1 hr marginal extra work, this becomes a quantity in Dollars, so it's a dimensionless ratio multiplied by todays prevailing wages (about $30/hr). You might call this "Real Wages" if you were an economist, but that would be wrong because there's no such thing. This graph shows that the amount of consumer goods you can consume for an hour of wages has after about 20 years stagnating between 1970 and 1990, finally gone up between 1995 and 2010. One suspects this is due to a combination of the Internet improving logistics and Chinese manufacturing reducing costs of production.
But, how about this graph?
These are rescaled to match approximately the $30/hr prevailing wage as of 2015. So they are a dimensionless ratio multiplied by a modern $30/hr wage. They show essentially how big a piece of the money supply (blue) or how big a piece of the stock market (red) your marginal hour of work buys. Because I've rescaled to match the $30/hr today, you can see that a marginal hour of work in 1975 bought you as much stock as if you were paid something like $160/hr in 2015, and as much a fraction of the money supply as if you were paid $60/hr in 2015.
So which is it? Has an average wage gone up from $25/hr in 1995 to $30/hr in 2015 (first graph)? Or has it gone down from $160/hr in 1975 to $30/hr in 2015? The answer is BOTH. Relative to the consumer goods people choose to buy, they are able to buy more today, but the goods they buy change through time (the CPI is not a fixed basket). So perhaps people have funneled their money towards less expensive items. Relative to investments, wages have gone WAY down (red 2nd graph). And, you'll see that as you might expect, when you can't get much for your dollar in the stock market, savings declined dramatically:
Here's savings rate as a percent of income, which is already dimensionless. Sure enough in 1975 when a marginal hour of work bought you what $160 would by you today in the stock market, savings rates were around 12.5% of income. Today, they're around 5%.
How big is todays savings as a fraction of what it was in the 70's?
The investment part of a 1975 marginal hour of work bought you 13 times as much investment as the investment part of todays marginal hour of work. YIKES!
There are many dimensionless ratios in the economy that are important to consumers, and others that are important to producers, or investors. It's a mistake to think in terms of things like "Real Dollars" as if it were possible to simply rescale things "for inflation" with one measure like the CPI, and recapture the past in modern money quantities.
You know what would be a good idea to improve on the ACA? Require any company selling health insurance to offer a plan that costs exactly 1% of GDP/capita per year in premiums for either a 40 year old Male or Female. Allow them to use a 3rd order polynomial in age to adjust the premiums by Age between 0 and 110 years old requiring that the coverage level be the same for all ages, but the premium adjusts so that income-expenses has minimal variation across age subject to the 40 year old anchor point. Let them compete on what coverage they can offer for that level, with no specific coverage requirement whatsoever except that there must be some annual out of pocket maximum.
What would this do? It'd make available a real insurance policy, one where you pay an affordable level of premium, and get protection from extreme events, and participation in pre-negotiated pricing. You'd wind up paying more out of pocket for services than on more plush policies, but if enough people actually did that, there would be downward pressure on health care prices, so that would be a good thing to some extent. Furthermore it would make it so that the individual mandate made sense for everyone. 1% of GDP/capita is today around $560/yr or $47/mo, which is more or less the cost of 5 meals per month.
Finally, what kind of out of pocket max would it be possible to offer. I did a little searching on hospitalization costs a while back and found order of magnitude estimates that a typical hospital stay costs something like $15,000 and that something like 10% of people have a hospitalization... if pay in equals pay out on average (just as an order of magnitude estimate), then and X = 4700. So you'd expect that insurance companies could offer you somewhere between $5k and $10k ceilings on annual expenses for 1% of GDP/capita.
EDIT: Error in the equation for ceiling, but it's fixed, basic conclusion didn't change.
Look, there's a crapload of shitty information on the web about what the Federal Reserve Bank does, including a crapload FROM the Fed. Do they "print money" or not for example?
The problem seems to be that people are generally stuck on a very literal view of what it means to "print". So, this punning between two concepts, kind of like "statistical significance" and "practical significance" causes a HUGE amount of confusion.
"Since central bank money currently exists mainly in the form of electronic records (electronic money) rather than in the form of paper or coins (physical money), open market operations can be conducted by simply increasing or decreasing (crediting or debiting) the amount of electronic money that a bank has in its reserve account at the central bank. This does not require the creation of new physical currency, unless a direct payment bank demands to exchange a part of its electronic money against banknotes or coins."
So, the Federal Reserve in its open market operations buys bonds from say Bank Of America. How does it do this? It simply increments a counter in the electronic records that describes how much reserve BofA has at the Fed. This then alters the regulatory status of BofA because it now has more reserves to meet regulatory reserve requirements. Therefore, it can lend more money without getting in legal trouble. Where did the money that went into the BofA reserve account come from? Nowhere. The FED takes in a promise to be paid money in the future (a bond) and gives out money it creates now (a reserve balance increment). The FED creates money. Traditionally, creating money is called "printing" but if you use "printing" as a technical term of the paper industry... you can claim "the FED doesn't print".
If you add up all the balances in BofA, plus the balance it has with the Fed, the total has increased. There is more money available to the world. Unlike Energy, money is not conserved, money is created out of thin-air (computing) by just increasing the balance on the accounting sheets at the Fed. Only the Fed is allowed to do this, everyone else has to *exchange* money (that is, to buy a sandwich you use a debit card to take money out of your checking account and put it into the checking account of the sandwich maker... the total quantity of money is unaffected because your balance decreased by the same amount that the sandwich maker's increased). If you use a credit card, the situation is the same, except with a greater time-delay and potentially if you're not paying off the credit card at the end of the month, with interest payments through time to the credit card issuer and soforth.
Is the Fed "printing money" in the sense of actually telling the Treasury to roll special paper through printing presses and putting special ink on the paper, and then shipping physical bills to banks? No. But, a balance in the Reserve Account at the Fed is as good as a promise to print physical bills whenever the bank wants them. That is, more or less, what the FDIC is, a big fat promise to run the presses when a bank can't hand out all the currency it's supposed to be able to.
In extreme situations, money might be destroyed. With the FDIC for example if you have more than some amount of money in your checking account, and there is a run on your bank, and the bank goes bankrupt, the FDIC will if necessary physically print you an amount of money equal to whatever your deposit amount was at the bankrupt bank... but only up to some limit, like $100k. So people who had $1M in their checking account might lose money in a puff of smoke... But this is an irrelevancy in most cases. It's also true that you can have $5000 in bills in your desk drawer when your house burns down... Again, not relevant to the overall picture in the world... which is that the Fed is able to create money when "money" is defined as all the balances in all the checking accounts and similar accounts, plus all the currency in vaults, plus all the reserve balances at the Fed. (The M1 money supply).
Practically speaking, whenever someone says that "the FED doesn't print money" (which as you see linked above, is something that even the FED says!) they are weasel-word lying. The FED doesn't print money in the same way that Artists don't make pen-and-ink drawings..... they just make the *ideas* that lead to the pen and ink drawings, it's an ink company that makes the ink, and a paper company that makes the paper... A stupid and meaningless distinction.
Look, it can't possibly matter what the number on a price tag is. The reason is, because I haven't told you what the currency is! The only thing that matters is the ratio of two prices, because this ratio is independent of what currency the two prices are listed in.
Lots of people don't understand dimensionless ratios, even lots of scientists. But, dimensionless ratios are key to understanding mathematical models. Creating a new unit, called the "Danner" which is exactly 33 USD at all times would let me list all my prices as 1/33 of the ones in dollars, but doesn't actually change anything about how the world works (except it probably makes you really irritated if you shop in my store).
Also, it can't possibly be the case that the economy relies entirely on a single dimensionless ratio. So, there is no such thing as "real dollars".
"Real Dollars" are another way of saying "Previous prices multiplied by one of the dimensionless ratios at work in the economy" and the dimensionless ratio involved is the one that tells you a certain weighted average of the ratio of a fairly randomly chosen good today to the price you'd have paid at some specified previous time in the past.
But in any particular economic problem (say decision making about where to live for example) the dimensionless ratios at work are things like AfterTaxMonthlyIncome / CostToPurchaseBasicMonthlyRequirements, and (AfterTaxIncome-CostToPurchaseBasicMonthlyRequirements)/CostToPurchaseMyPreferredExtras and InterestRateOnMortgages * DurationOfMortgage, and AnnualTaxesOnProperty/(CostToBuyHouse/30yrs)
In each real-world scenario, there are a variety of important dimensionless ratios which vary from time to time and place to place and person to person. The fact that Economists haven't actually figured this out and started publishing important ratios in data releases from the Bureau of Labor Statistics etc is just further evidence that Economics is still fairly pre-scientific. (I'm making a dig at Economists here, but only because I really want them to get with the program because Economics is really important!)
So, in understanding how well people are doing economically, it is important to figure out where they stand with respect to dimensionless ratios that are actually important to their everyday decisions.
So, here are some that I propose:
- After tax monthly income of a single earner household / Cost to live in a typical one bedroom apartment with electricity and heating/cooling and eat enough food for one person for a month. We can label this ratio because at is the Threshold for Poverty. Note, if you calculated this using real everyday data you would find a VERY different number than the "Federal Poverty Guidelines" which just goes to show that the federal poverty guidelines are a politicized crock of shit.
- After Tax Cost To Purchase and Maintain a House / present value of lifetime cost to rent a similar house. This describes the relative cost to own. If the number is less than 1 it costs less to own a house than to rent one. If it's greater than 1 you are paying extra for the privilege of ownership. Housing is something we simply can't go without for very long. Sure some people are homeless, but then when it gets bitterly cold they go to homeless shelters or they freeze to death. So while they may not own or rent shelter, they do need to consume it. We should really make this a function of size. So let's use the ratio S = (Size of house) / (minimum reasonably functional size of one kitchen one bedroom one bathroom). We can call this (for ownership advantage of house as function of size).
I could go on and on, but I think I'll stop here, because honestly.... I could go ON AND ON.
In fact, I have got the American Community Survey microdata and plan to try to infer some of these things, you know, in between doing all the other stuff.
Note to Economists: I'm sure some of this stuff is kind of known and actually calculated by some Economists in some places, and there are a variety of places where you can get some of this stuff, off various data sites... But let's face it, a lot of people still talk about "Real GDP" and "Federal Poverty Guidelines" and "Seasonally Adjusted Heating Oil Prices in 2013 dollars" and practically speaking these are the kinds of things that actually are available out there and discussed widely regarding policy, and they're just muddled ways of thinking about how to measure what's important to actual decisions people make every day.
The next obsession that Economists have is with causal identification. Like if you want to measure the "Return on Education" which you could say is something like (Present value of future earnings - Present value of earnings if you had a high school education)/CostToGetHigherEducation. And the issue is that you don't observe the same person in both conditions. Fine. Just please go out and calculate "observed difference in earnings between the two populations"/CostToGetHigherEducation and publish this regularly, and let's at least start from there and agree that the return to education is probably less than this number by a dimensionless ratio between 0 and 1... The fact is, we're worse off if we have no measurements of a thing than if we have measurements of something that is at least on a similar order of magnitude to that thing. It's a lot easier for people to imagine that they're going to go to school and double their income if they don't have any comprehensive data on the ratio of incomes of actual people out there in the world. And, if the cost of servicing their student loan debt is crushing all the advantages (ie. the denominator increases too!), we need to see that right away in a data series not 15 years after the fact when people think to put together a new measurement.
I know, I know, I'm all about the UBI these days. Well, I'm not going to apologize. The world really needs better solutions to poverty and even just to working and middle class. For example, 2011 data from the Census says that in the US the median black family has a net worth of $6300, which is just about enough to buy a used Toyota Corolla. This means a single $200 a week summer camp for a black child costs about 3% of the median black families whole net worth!
So, here is my intuition about the information theoretic properties of UBI. In a high dimensional space, a frequency distribution over anything always has a typical set that is far from the highest density region (translation into English: most people have many ways in which they are not like the average). Why is this? Even if your "histogram" has a peak density at some location, call it 0, the volume of a spherical shell at radius r away from that point grows like where is the dimension, whereas because the density is at a peak, it falls slowly at first, so as you move away from the "most probable" (highest density) location you find that the volume of the shell grows much faster than the density of the function declines. So, it's far more probable to be at some radius away from the "most probable" location, and the volume of this space is itself very large (there are LOTS of attributes so many many ways to be different from "average"). Eventually when you get far enough away, the requirement that the probability distribution integrate to 1 requires the density to fall extremely fast, and then the probability to be at very large radius also falls (very few people who are different from the average in EVERY way).
Now, suppose the space we are considering is the space where we talk about the quantity of each good purchasable in the market. Suppose we even just consider markets like in Rural Kenya. There must still be something like tens of thousands of goods you could purchase, fishing nets, tin roofing panels, plywood, goats, chickens, eggs, fish, rope-making labor, cart wheel repair labor... so if you were going to choose something to give to someone, you'd need to choose a vector of 10000 or more numbers to say how much of each good you'd give them. Of course, a charity typically isn't going to give people a massive basket of goods (one bean, one square inch of plywood, two fish scales)... they'll give 1 or 3 or 10 goods, a water pump, a cow, some fishing tackle, fencing to keep predators away from their livestock, whatever.... a few things.
Now, the mathematics of high dimensional spaces says that EVEN IF you choose a thing which is near the maximal frequency of desirability for an individual thing (ie. almost everyone wants a cook stove).... it will be pretty far from the region where you need to be. So for example, giving people a cow, or an apartment in a section 8 housing authority (why limit ourselves to Rural Kenya?) might be a pretty good thing to have, but it's nowhere near as good as the best set of things for a given individual that costs the same.
So, from an information theory perspective, we expect giving people Money and letting them choose a basket of goods that meets their individual needs, would be potentially VASTLY improved over even giving people some average basket of goods or some most-desirable among all people individual good etc.
So, from the perspective of why we should favor UBI over all other forms of welfare, it's ultimately that *real wealth* will be vastly increased, at no dollar cost.
The same argument applies, whether it's in Rural Kenya or Brooklyn. The best most efficient way to eliminate the bad effects of poverty, is to give some basic quantity of money.