Subjective probablity

Scientific American has a brief article about the Sleeping Beauty Paradox. I think that thinking about it may be illuminating for the central importance of the idea of subjective probability:

So my analysis is this:

There’s a 50/50 chance that she’s in the one-time universe. And she knows she’ll be woken up, so waking her up conveys no information. It’s a bit like the show Severance, and so she's entitled to seeing her chances of being in each universe as 50-50.

But from a different perspective: imagine this: there are two staff members in the two-wakings universe, but neither of whom knows which universe they're in. Each one has an independent one-time assignment to wake her up. Because they don't know which universe they're in, they don't know if someone else will have or already has woken her up. (The day of the week thing doesn't matter: ignore it.) There's also one staff member in the one-waking universe who also doesn’t know which universe they’re in assigned to wake her up in. Each of the three -- on a bet -- should bet that they're in the two-wakings universe, since the odds are 2-1 that they are. Thus any particular moment of her waking up is more likely in the two-wakings universe than in the one-waking universe.

That seems obvious. On the other hand Sleeping Beauty gets no more info when she’s woken up than she had before, when it was 50-50 which universe she would be in. So there's no reason for her to believe that she's more likely to be in the two-wakings universe, since she'll be woken up either way, and each experience is completely independent of any others. There is no set of events that she can refer to. Her subjective experience (each time!) is to be woken up once and only once.

So if she sees herself from outside she’s going to bet that the person waking her up is one of the pair who don’t know which universe they’re in but who would rightly bet they’re in the two-wakings universe. But she has to take that circuit, relying on other people’s subjective probability, rather than her own. More simply: subjectively she has one experience of waking up, and there's a 50-50 chance that she's in the one-waking universe when she has that experience. But vicariously she knows that the person waking her up would rightly bet that they're in the two-wakings universe. And they should bet that they're in the two-wakings universe because each would know vicariously that anyone waking her up should bet on being in that universe. The point being that the wakers know that two other wakers also have the task of waking her up, whereas she's the only Sleeping Beauty, and will only have a memory of a single experience of being woken up. (There may be a tense logic to this: her memory is part of a present-tense subjective experience of her own past, whereas the wakers are having a present-tense objective experience of the objective existence in the present of other wakers.)

Anyhow, I think it's like the well-known two-envelopes problem:

There are two envelopes, one of which contains twice the amount of money as the other. You’re given a chance to switch. Is there an advantage to doing so?

The argument for it: It’s just as though you’re flipping a coin, where the stakes are putting half of what you have at risk to double the amount. I have $10; I flip a coin and have a 50% chance of getting $20, and a 50% chance or losing only $5. Of course that’s a good bet, Pascalians!

The argument against: I know one has $10 and one has $20. 50-50 chance I have the $20 envelope and 50-50 that I have the $10 envelope. If I have the $20 and I switch I’ll lose $10. If I have the $10 I’ll gain $10. 50-50 chance either way of gaining or losing $10.

The argument for switching hits a paradox because then why shouldn’t you switch again after you’ve already switched, and do this forever?

The argument against is clear cut and obviously true, even if you use $n and $2n without knowing what n is.

In the argument for switching there seem to be three equally possible amounts of money: n, 2n, n/2. And it’s presented as though all three are possible outcomes. But only two are possible, once the envelopes are sealed. Subjectively 3, but objectively 2. So this is where Bayesian subjective probability hits an infinite loop that frequentist probability wouldn’t.

The infinite loop can be thought of this way as well: every time j you're asked whether you want to switch the expected 1.5n that you got in the envelope last time now becomes a new n_j. We're no longer talking about three possibilities being shoe-horned into 50% chance for each, but 4,5,6..., i.e: an infinite number of possibilities, each of which seems to have a 50% chance of being true at the moment that you consider it. So switching an infinite number of times should yield an infinite amount of money, but that's because you get stuck in a loop that can go on infinitely because the amount of money is indeterminate from the start.

As in a vivid sleep

Here’s an amazing, haunting poem by L.S. Bevington, from 1876:

Twilight

Grey the sky, and growing dimmer,
And the twilight lulls the sea;
Half in vagueness, half in glimmer,
Nature shrouds her mystery.

What have all the hours been spent for?
Why the on and on of things?
Why eternity’s procession
Of the days and evenings?

Hours of sunshine, hours of gleaming,
Wing their unexplaining flight,
With a measured punctuation
Of unconsciousness, at night.

Just at sunset, was translucence,
When the west was all aflame;
So I asked the sea a question,
And an answer nearly came.

Is there nothing but Occurrence?
Though each detail seem an Act,
Is that whole we deem so pregnant
But unemphasizèd Fact?

Or, when dusk is in the hollows
Of the hill-side and the wave,
Are things just so much in earnest
That they cannot but be grave?

Nay, the lesson of the Twilight
Is as simple as ’tis deep;
Acquiescence, acquiescence,
And the coming on of sleep.

It’s one in a set of four – “Morning,” “Afternoon,” “Twilight,” and “Midnight” – and I think the best of those though they are all good. Louisa Bevington was an anarchist (Kropotkin came to her funeral, as did the Rossettis) and atheist poet, well versed (so to speak) in Darwin’s theory of evolution. For Tennyson, for the Tractarians, for many others, evolution by natural selection was an unspeakably depressing idea, but Bevington could think about it in a way looking forward to Wallace Stevens’s “Waving Adieu, Adieu, Adieu”: “In a world without heaven to follow, the stops / Would be endings, more poignant than partings, profounder, / And that would be saying farewell, repeating farewell, / Just to be there and just to behold.”

A world without heaven to follow is one in which the present moment, if it signifies anything at all, can only signify repetition, can only mean its own meaning as a present moment. The possibility of temporary repetition is all it offers. Stevens’s repetition is like Bevington’s “Occurrence.” Things happen and the world is a series (hence the idea of repetition) of things happening. What we, empirical, limited, biological people do is acquiesce, as the penultimate line repeats.

What I find so hauntingly perfect about this poem is what that acquiescence is to. On a first reading it looks allegorical: the coming on of sleep is a harbinger of death, which is always coming on (as in Ashbery’s tonally similar poem “Fear of Death,” with its allusion to Hamlet's very last words). But I think what makes this poem so good is that it doesn’t make sleep a figure for death. It’s not about how we Darwinians have to to acquiesce to our own mortality. Rather our experience as biological beings is the experience of acquiescing to our own experience as natural beings. There is a sort of unmeditated kindness in nature, as simple as it’s deep: the kindness that means we fall asleep every night, as mammals do.

We acquiesce, without any sort of decision, to the part of our nature that is the everyday part of nature. I think what’s most amazing about this is the way that the moment in reading the poem that you realize that it’s about sleep, not about death, is the moment you repeat the experience that it describes. Sleep is real, death is notional. There’s no decision there but simply the fact that we do fall asleep, no working through to a resolution of some ontological anxiety but just a little sleep, a little slumber, a little folding of the hands to sleep. We're fundamentally like all other mammals: we acquiesce as they do to living in the natural world -- the world whose existence we contribute to by being natural beings. The poem doesn’t solve the problem it sets up: it is (as Wittgenstein will say (Tractatus 6.251) about the vanishing of the problem.

Another Abe

Joke title, referring to Kafka's idea of another Abraham.  I am thinking of Kōbō Abe in fact as another Kafka. 

Because I was teaching Woman in the Dunes yesterday -- mainly the movie but also Abe's novel.  He  wrote the very faithful screenplay.  Among the writers Abe knew well was Kafka -- Abe had visited Prague a few years earlier (after the Hungarian revolution, whose suppression disgusted him), so roughly the time the movie and novel start, and when there he did the Kafka tour.

One of the important things I think he saw in Kafka was just how realistic Kafka is.  You're thrown into the world not your own and not yourself but the only world there will ever be for you now, and you live in it.  We all do.

So the explicit allusions to Kafka are at least these: the man in the movie (and novel) is an amateur entomologist, looking for a new kind of beetle (the tiger beetle), which is to say that Gregor Samsa might be there in the sand somewhere.  Well he is -- the easiest irony in the movie is that the man is just like the beetles he's collecting, trapped in the dunes as they in their jars. Eventually he becomes focused on the crows around the pit where he lives, trying to catch one (which he can't), to treat as an unimperial messenger, putting one in mind of Kafka's parable: "The crows maintain that a single crow could destroy the heavens. There is no doubt of that, but it proves nothing against the heavens, for heaven simply means: the impossibility of crows."

As has long been pointed out, the word for crow in Czech is a pun on Kafka's name (the aphorism is in German, but his name does mean crow.  Murakami will do something similar with the character Crow in Kafka on the Shore).  And there's the complaint that the man makes that he is living "like a dog," Josef K's last words in The Trial.

But I think the most crucial connection may be in the moral of the story, which is never quite specific, although the man tells the woman that he has no desire to be in Tokyo, the place she imagines is so wonderful. If he'd liked Tokyo he wouldn't be doing entomology in the dunes.  The moral seems to be from Kafka, from the land-surveyor K's sublime rhetorical question: "Was hätte mich denn in dieses öde Land locken können, als das Verlangen hierzubleiben?“ - "What could have drawn me to this desolate land, if not the desire to stay here?"

That's what the entomologist realizes at the end, his desire to stay there.  The story is about his understanding that he's not the main character.  The woman is.  It's her sorrow, her need, her mourning, her experience that he must learn to take seriously.  So the absolute realism of the movie is this: it's a realistic portrait of a marriage -- of the best that a marriage can be, perhaps, or that human relations can be over time -- which is learning to commit yourself to what you've already been committed to, what circumstances, fate, life, being in the world, have committed you to.  A commitment to commitment in spite of everything.  To others in the same boat, wrecked (as in an early shot in the movie) on the dunes.

William James in 1884: "Our mental life, like a bird's life, seems to be made of an alternation of flights and perchings."

Henry James perhaps remembering this in 1898, has the Governess begin her narrative:  "I remember the whole beginning as a succession of flights and drops, a little seesaw of the right throbs and the wrong.

Unexpected Reunion

It just occurred to me that Johann Peter Hebel's amazing story "Unexpected Reunion" -- which Kafka famously called "the most wonderful story in the world -- is a version of Orpheus and Eurydice.  Or perhaps it might be called Eurydice and Orpheus, with all such a converse might apply.  At any rate it's the Eurydice figure who turns back, Eurydice who's been exiled in this world for all those years.

Ophuls' Black Orpheus, problematic as its real world construction is (for short: not the fact that it depicts an exotic celebration per se, but the exoticization of the actors), is still a brilliant and beautiful movie, and its most brilliant part is what Orpheus sees when he turns back: Eurydice as a very old woman.  What he sees is the truth of marriage, time, aging, death.  A truth, anyhow: the other truth is that these things are okay if one doesn't turn back, doesn't seek to turn back.  

In Hebel's story, too, the woman becomes very old, in her vast separation from her "young husband" ("θαλερὸς παρακοίτης," as Andromake calls Hektor).  But it is she who turns back to see his youth, and to mourn their lives and their parting, she who is more Eurydice than ever.

"With ships the sea was sprinkled far and nigh"

WITH ships the sea was sprinkled far and nigh,

Like stars in heaven, and joyously it showed;

Some lying fast at anchor in the road,

Some veering up and down, one knew not why.

A goodly vessel did I then espy

Come like a giant from a haven broad;

And lustily along the bay she strode,

Her tackling rich, and of apparel high.

The ship was nought to me, nor I to her,

Yet I pursued her with a lover's look;

This ship to all the rest did I prefer:

When will she turn, and whither? She will brook

No tarrying; where she comes the winds must stir:

On went she, and due north her journey took.

This sonnet has always haunted me, without my thinking much about why.  But today I realized it's the amazing twelfth line.  All the other lines are end-stopped (or could be) with no sentences ending midline.  But then we get that amazing caesura, just in the question about when the ship would turn: "When will she turn, and wither?"  And then the only enjambment, as she does turn, and another clausal ending after "tarrying" in line 13.  You read the last three lines as a kind of three line poem-within-the-poem, and they're pure blank verse in this Petrarchan sonnet.  The sense of enjambed, even Miltonic, blank verse -- except that it's purely Wordsworthian -- overlays and displaces the sonnet form that contains it, and that's what the ship is doing too -- brooking no tarrying, commanding the winds, sailing due north.

The Sacred James

 I'm reading Henry James's bizarre last unfinished novel, The Sense of the Past, which I guess he started around the time of The Sacred Fount and just after Turn of the Screw.  Anyhow, the way it treats its central character's relation to a cousin whose intelligence waxes and wanes during a single conversation reminds me of The Sacred Fount, which treats intelligence and insight as a kind of fluid quantity that flows back and forth between characters.  That's what literally happens in each chapter, as characters go from complete imbecility to supersubtle analysts, back to imbecility: with the observing narrator also needing to worry about his own susceptibility to this coming and going of accurate insight in himself.

And it occurred to me today (maybe this is a brief waxing of insight) that James is explicitly parodying what happens in all his novels, parodying the way Isabel or Strether or Milly Theale or Maggie Verver go from being less intelligent and insightful than those around them to being far more so.  It's as though James thought to give this another, how shall I say?... turn of the screw, in order to see what would happen if the dynamic would shift back and forth.

Why would he do this?  Well, partly for fun, mainly for fun, but partly as an experiment in style, since it's style alone that can suggest insight that then becomes so fine that (as Eliot says) no idea can violate it, at which point it becomes obtuseness, an obtuseness so intense that it can't help becoming self-aware and turning into insight again, in an incessant dialectic that can go nowhere except into the subtlety of its own endlessly elaborating, endlessly self-modifying sentences.