Somewhere else, I called *Ghosts*, by Argentinian writer César Aira, “a charming, meandering book of philosophical musings disguised as a novel.” That’s meant as a compliment. The story is nominally about the laborers, mostly Chilean immigrants, who are working on the construction of a luxury condominium high-rise in Buenos Aires. The construction site is haunted by ghosts, bawdy naked men covered in construction dust. The ghosts are invisible to the architects, designers, and project developers, as well as to the building’s wealthy future residents. They can only be seen by the workmen, who are about as visible to the project developers and future residents as the ghosts are. The Chilean foreman and his family live on the construction site as caretakers, and the ghosts take a special interest in the foreman’s thoughtful, dreamy, teenage stepdaughter Patri.

It’s a slim novel at 139 pages, and the plot itself is about enough for a short story. The rest of the book is filled with digressions, some by the omniscient narrator, some put in the thoughts of the characters. I picked up the novel expecting a regular ghost story, so at first the book seemed slow — it is slow — but after I adjusted to its reality, I enjoyed it. Aira rambles on about art, literature, architecture, how the arrangement of buildings in an African village reflect the expected modes of interaction among the villagers, homeopathy, marriage, the difference between the complacent Argentinian upper-middle-class and the immigrant Chileans…

This is a long-winded way of saying that arguing with the book while reading it seems like a perfectly natural thing to do.

What happened? Late in the book, the ghosts invite Patri to their New Year’s eve party. She has to think about it.

Invitations to a magic party with ghosts were obviously going to be very rare. There might be another chance, but for Patri that was beside the point. She was wondering how many such invitations there could be in eternity. That was a different question. Repetition in eternity was not a matter of probabilities, no matter how large the numbers. In eternity, as distinct from “in life” or “outside life,” this party was an absolutely unique occasion.

No, no, NO! I wanted to shout at the book (I was in a restaurant, so I didn’t). In all eternity, this invitation is *bound* to happen again.

I’ve worked for many years at the intersection of statistics, computer science, and engineering that people currently call “data science.” The trendy mantra among the tech gurus these days is “Big Data.” As in gigabytes, terabytes of data, collected from the web, from sensor systems, from the cellular network, from wherever.

It’s not the panacea everyone would imply, but the thing about Big Data, good or bad is up to you, is that rare events *will* happen. Imagine a “one in a million event”: one that you have to sift through a million events (however event is defined) to observe. “One in million” is casual English for “incredibly rare.” But if you have a terabyte of data — that’s a million million bytes — an event that happens once in a megabyte can happen around a million times. Still rare, proportionally speaking, but not “never.” Suppose that “incredibly rare” event is a certain type of transcription error while writing down your data. Back in the good old days, you probably would never have worried about it, but now, you might have to. And if we’re not just talking about a million million, but about eternity….

There’s another way of looking at it. In casual English, if you say something has “zero probability” of happening, you mean that it’s impossible. It can’t happen. That’s not what zero probability means in statistics. Think of probability as a physical space — the space of potential events. Then the probability of an event is its “measure”: length, area, volume, depending on the dimensionality of the space.

The surface of a sphere in three dimensions has measure zero because it has no volume. The boundary of a circle in two dimensions has measure zero because it has no area. A point on the (continuous) line has measure zero because it has no length. Being at a specific point on the line, that point and no other, is a probability zero event.

But points exist (in the mathematical sense), boundaries exist, surfaces too. Clearly, they aren’t “impossible” in the everyday sense. Imagine that time is an infinite line going away into the past and into the future. One of the points on that line is the moment of my birth (let’s say that we can pick an exact time, before which I didn’t exist in the world, and after which, I did). The probability that I was born at that exact moment, and no other, is zero. My birth was “impossible.”

And yet here I am, and you, too. The events of our births were “impossible,” miraculous. We’re all miracles, and not in some happy-clappy, fuzzy spiritual way. It’s cold, hard statistics — if you want to think about it this way. And at this moment, I do, because it’s cool.

So from that point of view, then the invitation from those ghosts, to that girl, at that moment of time, in all eternity — yes, it is a unique event.

Maybe that’s what Mr. Aira meant by what he wrote; I don’t know. But I am sure that he’d be amused that his ramblings have inspired this silly little essay.

The image above is a remix of a snapshot of the relevant quote in *Ghosts*, taken by me, with an image created by John Mount’s Genetic Art project. Click on the “Gallery” tab from the link to see thumbnails of more images.

Pingback: A Moment’s Digression | Nina B. Zumel

Nice ! Here are some good ghost stories you may like:

http://www.amazon.com/INDIAN-GHOST-STORIES-Saptarshi-Bhattacharyya-ebook/dp/B00ON3L6JO/ref=sr_1_1?ie=UTF8&qid=1414304387&sr=8-1&keywords=indian+ghost+stories

Thanks for the recommendation! I will check it out.

I hope you will help me with this, prompted by your interesting post. With respect to the probability of your birth at a particular moment, aren’t you talking about predicting that moment in advance? Isn’t such a prediction (with no other information to suggest when you might be born) just about as close to zero as possible? Obviously, you exist, but I wonder if some may get the wrong idea about how statistics are commonly used. You are the statistician, so please feel free to correct me.

I am being a bit loose with the statistics, but, no it’s not really got to do with prediction in the sense of the word that you mean. If I came up to you at a cocktail party and asked you “Guess when I was born?” and supplied you with the necessary probability distributions (about the ages of the guests at the party, or something), then all the reasoning I talked about applies.

You could estimate the (nonzero) probability that the moment of my birth was sometime between, oh, 1960 and 1980, or even the probability of it being sometime on the day January 27, 1975. But the probability of it being

exactlyJanuary 27, 1975, at 1:45.0937 AM is zero (or very very near zero, if we call a millisecond very very near a “point” of time). The key idea is that you can only estimate the probability ofintervalsof time, not exact points of time — unless time is measured discretely, say only in years. If you only had to guess what year I was born, then the probability that I was born in 1975 is (or at least, could be) nonzero.Hope that helps. And January 27, 1975 is not my birthday.