manipulating betting odds into probabilities and other odds

A colleague and I wanted to enter into a bet on a group-stage game of the World Cup of Football. We wanted our bet to have the following structure: He pays me something if Team A wins, I pay him something if Team B wins, and we push (that is, neither of us pays) if it is a draw between Team A and Team B. We are friends, so we wanted to give reasonable odds on this bet. How to calculate them?

Online, we saw the following odds posted on a reputable sportsbook (yes, such things exist) in European style:

Team A wins1.625
Team B wins6.710
tie 3.910

Is it possible to convert these data into what we want to know? Of course the answer is yes, but we need to make assumptions. Before we continue, I should note that these odds would be written in American style as follows:

Team A wins-160
Team B wins+571
tie +291

Because that may be confusing, let's just check in on what these odds mean. Looking at the European odds, what they mean is that if I bet $100 that Team A wins, I will be paid $162.50 if they do indeed win. That is, I will be paid back my $100 plus $62.50 in winnings. If I bet $100 that there will be a draw, then I will be paid $391.00 (my $100 plus $291 in winnings) if they do indeed draw. Looking at the American odds, what they mean is that if I bet $160 that Team A will win, I will be paid $100 in winnings plus my original bet back, or a total of $260, if they do indeed win. And they mean that if I bet $100 that Team B wins, I will be paid $571 plus my original bet back, or a total of $671, if they do indeed win. The +/- sign at the beginning of those odds makes a big difference! So in this sense, to a scientist, usually the European-style odds writing makes more sense.

Strictly, what's written in European-style odds is not the odds, but the odds-plus-one. If the book was not out to make money—If the book was just trying to break even—then these odds would be the book's approximation to the inverse of the probabilities of the outcomes. If you want to see why, you can think about placing a Dutch bet: Putting money on each outcome in proportion to the probability. If you do that, and the odds are just right, then you will get the same payout no matter what the outcome. That's an exercise to the reader! But if we interpret these European odds as inverse probabilities, then the implied probabilities of the various outcomes would be:

Team A wins0.615
Team B wins0.149
draw 0.256

Which you can get just by inverting the numbers in the first chart. The observant reader will notice that these three numbers don't add up to one! They add up to a bit more than one. Why? Because the book is making money, and so they pay out odds slightly lower than what's fair, which means that they have an edge over the bettors, on average.

Now if we make the assumption that the book is relatively capable (or that the betting public is pushing the book's odds to something sensible), then we can assume that these are close to the correct probabilities for the three outcomes. That's a great start! But remember that we want odds for a win bet, with the draw as a push. The idea is, we want to know: What's the probability that Team A wins given that there isn't a draw. For that we condition on there not being a draw and re-normalize that probabilities. Probabilities for these outcomes are:

Team A wins (conditioned on no draw)0.805
Team B wins (conditioned on no draw)0.195

I got these by re-normalizing the two outcomes my colleague and I care about to get unity. Now we can convert to European-style odds by inverting again, and the odds we get are:

Team A wins (conditioned on no draw)1.242
Team B wins (conditioned on no draw)5.128

That is I should bet $100, and my colleague $24.20, and whoever wins gets it all. If there is a draw, we take our money back. Or at larger stakes, I should bet $412.80, my colleague $100, and so on.


freshwater consumption

I'd like it to be a theme of my writing about teaching that I learn as much as any student in my class when I teach! It certainly is a fact. This semester has been huge for me because I'm teaching a subject I don't know much about (sustainability), and because I am teaching writing, which is a first (yes, in 25 years of teaching, I have never taught writing).

A few weeks ago I gave a (writing) assignment to estimate global freshwater needs by computing your own freshwater needs and extrapolating to the world. Then compare with estimates of global usage. My expectation: That a typical American college kid uses way more water than a typical person on Earth, and this would be a lesson about disparity. I wasn't wrong, but I also wasn't right either:

It turns out that global freshwater use is way larger than any estimate of your own personal water use, extrapolated to the globe! Isn't that odd? It's because global freshwater use is not dominated by household uses like laundry and toilets (which, by the way, don't need freshwater). It is dominated by agriculture. If you add in your own share of agricultural water use, a typical American uses way more than the average human! But it is hard to know or estimate this without significant engineering research.

There was one clue, however, in the essays I got: One student included the amount of water used to water a lemon tree in their dorm room. It was a trivial amount of water, so it was almost a joke. But then if you think about how much food we eat relative to the miniscule annual production of that tiny lemon tree, there is a little window into just how much water we must use for agriculture.


Is the greenhouse effect like a greenhouse?

I am teaching a class at NYU this Spring that I am loving: It is called (with grandiosity) The Art and Science of Approximate Reasoning: Physics, Sustainable Energy, and the Future of Humanity. We are computing what we can about sustainability, from a physics perspective and with order-of-magnitude reasoning. The strangest thing about the class (for me) is that it is a writing class. So I am grading writing assignments, something I am certainly not qualified to do.

I have learned a huge amount in this class. Here's one tiny, tiny example:

How does a greenhouse work, and is it anything like how the greenhouse effect on Earth works? The answer is extremely cool. In both cases (greenhouse and greenhouse effect) the system permits visible radiation to enter, but deters infrared radiation from escaping. In this sense, the two effects are very similar.

However, there are critical differences: In the case of the greenhouse, much of the effect is that the glass walls cut off convective cooling to the upper atmosphere. That is, they trap the heat near the heated ground, not letting it mix with the rest of the atmosphere. This effect probably dominates over the infrared-reflectance effect of normal glasses and plastics. The fundamental point is that (to first order) the Sun heats the Earth, which then heats the atmosphere. If part of the Earth is put inside a glass box, that part gets to hold its heat without sharing it entirely with the surrounding atmosphere. And indeed, greenhouses control their temperatures through ventilation management (management of convection to the outside).

In the case of the greenhouse effect, increasing the CO2 (and water and methane and so on) in the atmosphere increases the opacity at infrared wavelengths, so the Earth can't cool radiatively as efficiently as when the atmosphere is more infrared-transparent. That certainly is an effect in greenhouses, but not the only effect, and probably not the dominant effect.

The Earth's atmosphere can only cool (to space) radiatively, so the greenhouse effect is purely an adjustment of that radiative cooling. The human-built greenhouse can cool radiatively or convectively, and it uses both of those mechanisms for heat management.


the total eclipse

[This is an excerpt from a longer description I wrote for friends. The eclipse took place (for us) at 10:21 local time on 2017-08-21.]

We viewed the eclipse from high ground in central Oregon near Walton Lake in the Ochoco National Forest. We chose a viewing location with clear views to the WNW, and set up tripods, camera equipment, and a small refracting telescope with a solar filter. As the eclipse began, forest service and fire crews showed up at our location to watch. We gave them some of our excess eclipse glasses. Late in the partial phase, our shadows became really strange, with corners and pinpricks in our shadows turning into thin crescents. The light became strange—dark and direct, like the world was under tinted glass—and it started to get cold.

The eclipse glasses were magical; the full disk of the Sun could be viewed comfortably and inspected, and the cut from the Moon was dramatic. Through our small telescope, sunspots were visible. Another crew at our site had a larger reflecting telescope set up, also with a Solar filter. They had an astounding view of the sunspots.

But the transition from even a tiny sliver of visible Sun to totality is astonishingly stark! At totality, the sky became dark enough that we could see a few stars and planets. The corona of the Sun could be viewed directly without the glasses, and it was visible by eye out to a full Solar Diameter away from the eclipse limb. There was lighter sky at the horizon, and it was pink like a sunset in every direction! That, along with the enormity of the corona, was the most surprising thing about the eclipse for me. It was dark everywhere like late dusk, with late sunset on every horizon. It got cold enough that those of us fortunate enough to have brought sweaters to the viewing point put them on. The corona of the Sun was almost triangular in shape, with a dark black hole where the Moon lay.

Towards the end of totality (that is, about 2 minutes after the beginning of totality), we could see the light racing towards us from the WNW: The more distant hills lit up first. At the very last moment of totality a tiny pinprick of sunlight appeared on the limb of the sun making, with the ring-like corona a "diamond ring". Instantly it was too light to look at directly and we put our glasses back on.


does the Earth really go around the Sun?

tl;dr: Executive summary: It is not fundamentally true that the Earth goes around the Sun; it is just easier to calculate things that way.

We like to say that the critical event that started the scientific revolution is the discovery that the Earth goes around the Sun, and not the other way around. This was incredibly important; the hypothesis by Copernicus led to the immensely important data-taking by Tycho Brahe and the quantitative, theoretical explanation of it by Kepler. Galileo's discovery of moons of Jupiter bolstered the case in important ways, and Newton's quantitative description of it all in terms of the inverse-square law solidified it all into an edifice of great importance, that is just as important and valuable today as it was then. It is also a great example of how a scientific discovery requires both observational and theoretical backing to become confidently adopted by the community.

In the 20th Century, Einstein brought us General Relativity, with the eponymous generality granting us immense coordinate freedom. That is, there are (infinitely) many ways we can make decisions about what is stationary and what is moving, and what we choose as reference points. In some choices, calculations are harder. In other choices, calculations are easier. In yet others, certain symmetries become more obvious or more valuable for making predictions. That is, GR delivers to us lots of choices about how to think about what's moving and how.

So the crazy insane thing is this: In GR, there is no answer to the question of whether the Earth goes around the Sun or whether the Sun goes around the Earth. There is literally no observational answer to the question, and no theoretical answer. All observations can be incorporated to an analysis from either perspective. The question of which goes around which is not a question you can ask in the theory.

That said, it really is far, far easier to do calculations in the Copernican frame. Indeed, absolutely all calculations of Solar System dynamics are done in this frame with post-Newtonian code. The way I see it (with modern eyes) is that Copernicus's hypothesis was based on parsimony or simplicity and was adopted for that reason. Brahe and Kepler confirmed that the data are consistent with Copernicus's simple model (though with the eccentricities added). After Brahe and Kepler it was still possible to understand the observations in an Earth-centered (or even stranger) coordinate system, but was far, far easier to do calculations in the heliocentric frame.

Even today, now that GR is our model of gravity, we still calculate the Solar System with Newtonian codes (with adjustments to approximate GR corrections). And even today, now that we have this amazingly accurate model of the Solar System, we still often calculate the positions of celestial bodies by looking at paths on the celestial sphere, as did Ptolemy. How we calculate something is incredibly context-dependent, and doesn't always respect our most fundamental ideas. And the truth of Copernicus's hypothesis really just represents the pragmatism of the present-day mathematical tools. All these thoughts bolster my rejection of scientific realism and play into questions of social construction and so on. It also bolsters my view that Ockham's Razor should be thought of as a statement about calculation, not truth.

Sure the Earth goes around the Sun! But let's remember that this is a statement about calculation and pragmatism, not the fact of the matter.


where's the information?

It came up frequently in discussions this summer (and last): Where is the information (in, say, a spectrum of a star) about some parameter of interest (say, the potassium abundance of the star, or the radial velocity), and how much information is there? The answer is very simple! But the issues can be subtle, because there is only calculable information within the context of some kind of model. And by “model” here, I mean a probability density function for the data, parameterized by the parameters of interest. That is, a likelihood function.

The fast answer is this: The information about parameter θ is related to the (inverse squared) amount you can move parameter θ and still get reasonable probability for the data. The nice thing is that you can compute this, often, without doing a full inference. It is easiest in linear (or linearized) models with Gaussian noise! That's the question we will answer here.

When you have a linear or linearized model with Gaussian noise, there are derivatives of the expectation Y for the data with respect to the parameter of interest, dY/dθ. Here (for now) Y is an N-vector the size N of your data, and θ is a scalar parameter (let's call it the velocity!). So the derivative dY/dθ is an N-vector. The information about θ in the data is related to the dot product of this vector with respect to itself: The accuracy with which you can measure θ given data with Gaussian noise with N×N covariance matrix C (possibly diagonal if the N data points are independent) is:

σθ-2 = [dY/dθ]T C-1 [dY/dθ]

where σθ is the uncertainty on θ. That is, the inverse variance on the θ parameter is the inner product of the derivative vectors, where that inner product uses the inverse variance tensor of the noise in the data as its metric! Here we have implicitly assumed that the vectors are column vectors. When the N data points are independent, the C matrix is diagonal, as is its inverse. Note the units too: The inverse variance tensor has inverse Y-squared units, the inner product uses the derivatives to change this to inverse θ-squared units.

(When there are multiple parameters in θ—say K parameters—the inner product generalizes to making a K×K inverse covariance matrix for the parameter vector, and the expected variance on each parameter is obtained by inverting that inverse variance matrix and looking at the diagonals.)

But we started with the question: Where is the information in the data? In this case, it means: Where in the spectrum is the information about the velocity? The answer is simple: It is where the data—or really the inverse variance tensor for the noise the data—makes large contributions to the inverse variance computed above for θ. You can think of splitting the data into fine chunks, and asking this question about every chunk; the chunks or pixels or data subsets that contribute most to the scalar inverse variance are the subsets that contain the most information about θ.


walk or take the elevator?

I'm just generally excited about getting back into the classroom after a long sabbatical. I'm thinking about problem-set problems for the Physics Majors. Here's what's in my head right now:

NYC has had a hot summer, with most buildings running air conditioning on a thermostat continuously. To save energy, NYU (and other large entities in NYC) asked their employees to conserve energy in various ways, some of which we might take issue with. Here's an uncontroversial one: You should take the stairs, not the elevator.

But is that uncontroversial? What considerations are required to figure out whether this policy would reduce or increase energy consumption? Obviously—if you take the stairs—you use less elevator energy, but then you drop a metabolic load on the building air-conditioning. Which uses more power in the end? Use a combination of web research and simple physical arguments to make cases, and identify weaknesses in your argument as you change assumptions. Things that matter include: Neither humans nor elevators are 100-percent efficient delivery vehicles for potential energy (in fact, can you see a fundamental argument that elevators must spend more than 50 percent of their energy generating heat?). Elevators are heavy but counter-weighted. Some buildings have very busy elevators, so your contribution to the elevator load is only the marginal contribution; in other buildings you are typically the only person in the elevator. Air conditioning systems have efficiencies limited by fundamental ideas in thermodynamics, but are probably much less efficient than the limits. And so on!

Thanks to Andrei Gruzinov (NYU) for starting me thinking about this one.