Statistics with Zhuangzi

Was I a dependent variable dreaming that I was an independent variable, or an independent variable dreaming that I was a dependent variable?

In response to an absolute bare minimum of popular demand on Twitter, here’s a thread of statistical principles as derived from the “What Comes From Without” chapter of the Zhuangzi.

Let us begin.

What comes from without cannot be determined beforehand. So it was that Long-feng was killed; Bi-gan immolated; and the count of Ji (made to feign himself) mad, (while) Wu-lai died, and Jie and Zhou both perished. Rulers all wish their ministers to be faithful, but that faithfulness may not secure their confidence; hence Wu Yuan became a wanderer along the Jiang, and Chang Hong died in Shu, where (the people) preserved his blood for three years, when it became changed into green jade. Parents all wish their sons to be filial, but that filial duty may not secure their love; hence Xiao-ji had to endure his sorrow, and Zeng Shen his grief.

This chapter feels kind of herky-jerky if you don’t know that the overarching theme is related to statistical errors, mostly involving over/under-fitting or over/under-specification. We begin with a helpful announcement of the key lesson of the chapter: cause and effect are not always linear. One additional unit of loyalty will not necessarily result in a corresponding unit of rewards. However, just because relationships are not linear does not mean that they are not predictable. They just need a little more work.

Less helpfully, he then chucks us right into the deep end with a really difficult multivariable optimisation problem + hidden traps:

When wood is rubbed against wood, it begins to burn; when metal is subjected to fire, it (melts and) flows. When the Yin and Yang act awry, heaven and earth are greatly perturbed; and on this comes the crash of thunder, and from the rain comes fire, which consumes great locust trees. (The case of men) is still worse. They are troubled between two pitfalls, from which they cannot escape. Chrysalis-like, they can accomplish nothing. Their minds are as if hung up between heaven and earth. Now comforted, now pitied, they are plunged in difficulties. The ideas of profit and of injury rub against each other, and produce in them a very great fire. The harmony (of the mind) is consumed in the mass of men. Their moonlike intelligence cannot overcome the (inward) fire. They thereupon fall away more and more, and the Course (which they should pursue) is altogether lost.

You’re trying to maximise the total utility of your practical gains while minimising danger (or rather, marginal utility of safety, which is less catchy but makes more sense in mathematical and construct validity terms).

However, this ends up placing you at a saddle point. Technically, you’ve optimised your total utility, but you’re not getting the maximum of either. You’re not a multibillionaire, and you’re not invulnerable to harm either.

However, this also has second order effects, because it’s fundamentally a psychological problem rather than an economic one. It isn’t a maximum or a minimum for the two variables we have here — danger and gains, but it does maximise a third, which is not visible —psychological labour. Someone who’s committed to either a high-risk/high-reward lifestyle or a low-risk/low-reward lifestyle has decided that one variable is unimportant to him and chucked it out. Every time the guy at the saddle point is faced with a new decision he has to worry about the effect his choice will have on his gains/danger coordinates. A guy who has chosen to place himself at one of the extremes has already made his choice, and he is psychologically at ease, though in constant physical danger or financial penury. The trick is not minding that it hurts.

There’s a great real-world example of this playing out at the moment. My company has an office in Indonesia and suppliers in China. The latter had some of the most extreme Covid restrictions on earth, the former had barely any at all. Both have functioned more or less as normal over the past two years, while Europe and America are in chaos: every decision is a new drama and no one is happy with any outcome. In other words, trying to steer a middle path between draconian lockdowns and telling people to sit back and pray produces the worst of both worlds, not the best. Go hard or go home, but don’t try to do both.

(The moon bit is supposed to be metaphorical according to the commentaries, but to me it just reads like meaning that a candle nearby looks brighter than the moon far away — i.e. perspective also has an effect — from within the system, it is almost impossible to distinguish between local and global minima. This remains a tough problem in machine learning.)

This extract relates to another one I mentioned in the original Zhuangzi stats thread on Twitter:

The ordinary men of the world all rejoice in men's agreeing with themselves, and dislike men's being different from themselves. This rejoicing and this dislike arise from their being bent on making themselves distinguished above all others. But have they who have this object at heart so risen out above all others? They depend on them to rest quietly (in the position which they desire), and their knowledge is not equal to the multitude of the arts of all those others! When they wish again to administer a state for its ruler, they proceed to employ all the methods which the kings of the three dynasties considered profitable without seeing the evils of such a course. This is to make the state depend on the peradventure of their luck. But how seldom it is that that peradventure does not issue in the ruin of the state! Not once in ten thousand instances will such men preserve a state. Not once will they succeed, and in more than ten thousand cases will they ruin it. Alas that the possessors of territory (the rulers of states) should not know the danger (of employing such men)!

This refers to over-reliance upon averages. Never forget:

That tweet’s kind of abstract and there are loads of different reasons to mistrust averages, so let’s take an example from politics calibrated to reflect Zhuangzi’s preoccupations here.

More people will vote for a candidate who shares more of their views. Thus politicians gravitate to the centre, however, the overall result is not to make them more electable, but to leave each individual subset of potential supporters vaguely unsatisfied while also having the second order effect of making them look like unprincipled panderers. (I’m looking at you, Macron.) Moreover, the second anything new comes up and they are unable to work out what the majority-approved way to deal with it might be, they are lost. While people say they want someone who agrees with their positions, they in fact want someone capable of coming up with new positions that are convincing enough to make them agree with him.

In addition to this, it is important to assess magnitude of sentiment, not merely its direction. Having 1000 people who agree with you intensely and will go out and do something about it is often more valuable than having 100,000 people who kind of agree with you but not to a point at which they’ll actually take action. We saw this in the 2016 US election, where Hillary’s positions likely had more general support, but it came from people who lacked the motivation to go to the polls. Result: Trump - whose positions enjoyed less mainstream backing but who was able to fire up his base - won.

There is a way round this, and it’s going to come up at the end of the chapter, creating some nice symmetry.

The next example is a lot easier to analyse, thankfully.

The family of Zhuang Zhou being poor, he went to ask the loan of some rice from the Marquis Superintendent of the He, who said, 'Yes, I shall be getting the (tax-) money from the people (soon), and I will then lend you three hundred ounces of silver - will that do?' Zhuang Zhou flushed with anger, and said, 'On the road yesterday, as I was coming here, I heard some one calling out. On looking round, I saw a goby in the carriage rut, and said to it, "Goby fish, what has brought you here?" The goby said, "I am Minister of Waves in the Eastern Sea. Have you, Sir, a gallon or a pint of water to keep me alive?" I replied, "Yes, I am going south to see the kings of Wu and Yue, and I will then lead a stream from the Western Kiang to meet you - will that do ?" The goby flushed with anger, and said, "I have lost my proper element, and I can here do nothing for myself; but if I could get a gallon or a pint of water, I should keep alive. Than do what you propose, you had better soon look for me in a stall of dry fish."'

Detrimental non-linearity. Zhuang Zhou’s need for money will stay roughly the same for little while longer and then crash when he starves to death, it won’t rise in a linear manner.

In other words, the Superintendent thinks that Zhuangzi’s marginal utility function for money is linear. Zhuangzi knows it is curved.

The government business promotion agencies I often deal with are extremely fond of a Superintendent-esque vision of money. Very often the support they offer requires large capital outlays on the part of the recipient, which will be reimbursed in 6–12 months. Obviously, no small business can afford this, but Deutsche Bank and Facebook get a nice opportunity to upgrade their PCs for next to nothing. (I’m being uncharitable and assuming this is accidental; it could simply be corruption.) Also relevant:

Appropriately, the next section covers beneficial non-linearity:

A son of the duke of Ren, having provided himself with a great hook, a powerful black line, and fifty steers to be used as bait, squatted down on (mount) Gui Ji, and threw the line into the Eastern Sea. Morning after morning he angled thus, and for a whole year caught nothing. At the end of that time, a great fish swallowed the bait, and dived down, dragging the great hook with him. Then it rose to the surface in a flurry, and flapped with its fins, till the white waves rose like hills, and the waters were lashed into fury. The noise was like that of imps and spirits, and spread terror for a thousand li. The prince having got such a fish, cut it in slices and dried them. From the Zhi river to the east, and from Cang-wu to the north, there was not one who did not eat his full from that fish.

In subsequent generations, story-tellers of small abilities have all repeated the story to one another with astonishment. (But) if the prince had taken his rod, with a fine line, and gone to pools and ditches, and watched for minnows and gobies, it would have been difficult for him to get a large fish. Those who dress up their small tales to obtain favour with the magistrates are far from being men of great understanding; and therefore one who has not heard the story of this scion of Ren is not fit to take any part in the government of the world - far is he from being so'.

If he’d produced the same amount of food from normal-sized fish no one would have found it remarkable, though they’d have received the exact same number of calories.

However, it is important to note that, being a prince, he presumably had enough money to finance a year’s worth of futile exertions, unlike Zhuang Zhou who needed rice right now. He had enough steady income to wait for a tail event that would make his name (which was bound to come sooner or later).

The second half is kind of obscure, but it seems to be referring to people who’ve never encountered a tail event and therefore have no conception of their existence. Their model is correct for the data points they have, but they’ve failed to account for the fact that they probably don’t have all relevant data.

Someone who has this data:

Will probably make a model that is radically different from someone who has this data:

They’re the equivalent of Lehman Brothers traders splurging on Christmas 2007 because they’re stoked about all their AAA CDOs. They are not dangerous because they’re smug over small victories — who isn’t? — but because the models they are selling are missing crucial information.

The next section is a famous dunk on Confucius and his followers, but it’s also about overfitting:

Some literati, students of the Odes and Ceremonies, were breaking open a mound over a grave. The superior among them spoke down to the others, 'Day is breaking in the east; how is the thing going on?' The younger men replied, 'We have not yet opened his jacket and skirt, but there is a pearl in the mouth. As it is said in the Ode,
The bright, green grain
Is growing on the sides of the mound.
While living, he gave nothing away;
Why, when dead, should he hold a pearl in his mouth?"'
Thereupon they took hold of the whiskers and pulled at the beard, while the superior introduced a piece of fine steel into the chin, and gradually separated the jaws, so as not to injure the pearl in the mouth.

Confucius said that a good person would be punctilious about following even trivial rules. In this story Zhuangzi is arguing that a fixation on individual rules will tend to detract from overall virtue, rather than produce it, as you will overfit your model of goodness to individual rules, and find yourself neglecting the big picture. You will find yourself following the letter of the law and sacrificing its spirit — refusing to sit because the mat is not straight rather than sacrificing protocol to oblige your host.

The next section is another gratuitous dunk on Confucius, but it’s also a warning about the perils of seeing strict linear relationships where there are none. Not only will you continually be disappointed/frustrated, you won’t know why.

A disciple of Lao Lai-zi, while he was out gathering firewood, met with Zhongni. On his return, he told (his master), saying, 'There is a man there, the upper part of whose body is long and the lower part short. He is slightly hump-backed, and his ears are far back. When you look at him, he seems occupied with the cares of all within the four seas; I do not know whose son he is.' Lao Lai-zi said, 'It is Qiu call him here;' and when Zhongni came, he said to him, 'Qiu, put away your personal conceit, and airs of wisdom, and show yourself to be indeed a superior man.' Zhongni bowed and was retiring, when he abruptly changed his manner, and asked, 'Will the object I am pursuing be thereby advanced?' Lao Lai-zi replied, 'You cannot bear the sufferings of this one age, and are stubbornly regardless of the evils of a myriad ages: is it that you purposely make yourself thus unhappy? or is it that you have not the ability to comprehend the case? Your obstinate purpose to make men rejoice in a participation of your joy is your life-long shame, the procedure of a mediocre man. You would lead men by your fame; you would bind them to you by your secret art. Than be praising Yao and condemning Jie, you had better forget them both, and shut up your tendency to praise. If you reflect on it, it does nothing but injury; your action in it is entirely wrong. The sage is full of anxiety and indecision in undertaking anything, and so he is always successful. But what shall I say of your conduct? To the end it is all affectation.'

In this story, Laozi suggests that Confucius’ model of a world in which every additional unit of virtue accumulated will receive its corresponding unit of social recognition is clearly not applicable to the age in which they lived.

Moreover, this results in a temptation is to blame others for not living up to your model. Thus, in the years following the 2007 crash, Lehman brothers were apostrophised for their greed, but in reality all they had done was respond as best they could to the incentives that society gave them. If we wanted them to behave less irresponsibly, we should have pushed government to adjust their incentives. They did precisely what we paid them to. If we didn’t want this outcome, we should have anticipated it and paid for something else.

Next section:

The ruler Yuan of Song (once) dreamt at midnight that a man with dishevelled hair peeped in on him at a side door and said, 'I was coming from the abyss of Zai-lu, commissioned by the Clear Jiang to go to the place of the Earl of the He; but the fisherman Yu Qie has caught me.' When the ruler Yuan awoke, he caused a diviner to divine the meaning (of the dream), and was told, 'This is a marvellous tortoise.' The ruler asked if among the fishermen there was one called Yu Qie, and being told by his attendants that there was, he gave orders that he should be summoned to court. Accordingly the man next day appeared at court, and the ruler said, 'What have you caught (lately) in fishing?' The reply was, 'I have caught in my net a white tortoise, sievelike, and five cubits round.' 'Present the prodigy here,' said the ruler; and, when it came, once and again he wished to kill it, once and again he wished to keep it alive. Doubting in his mind (what to do), he had recourse to divination, and obtained the answer, 'To kill the tortoise for use in divining will be fortunate.' Accordingly they cut the creature open, and perforated its shell in seventy-two places, and there was not a single divining slip which failed.

Zhongni said, 'The spirit-like tortoise could show itself in a dream to the ruler Yuan, and yet it could not avoid the net of Yu Qie. Its wisdom could respond on seventy-two perforations without failing in a single divination, and yet it could not avoid the agony of having its bowels all scooped out. We see from this that wisdom is not without its perils, and spirit-like intelligence does not reach to everything. A man may have the greatest wisdom, but there are a myriad men scheming against him. Fishes do not fear the net, though they fear the pelican. Put away your small wisdom, and your great wisdom will be bright; discard your skilfulness, and you will become naturally skilful. A child when it is born needs no great master, and yet it becomes able to speak, living (as it does) among those who are able to speak.'

On the face of it, this looks like the standard issue daoist_midwit_meme.jpg:

And it is, but there are other things going on too.

This is something I’m really interested in, being a general intelligence enthusiast, but unfortunately intelligence is a statistic that’s subject to every weird effect in the Weird Effects Book, and covering all of them in detail would make this article way too long. Nassim Taleb summarises the key ones here. The main target here seems to be one of underspecification due to poor definition of the constructs under investigation — intelligence in one area does not necessarily translate well to another, so these should be regarded as two separate variables. It may also be implying that gaining more domain-specific knowledge reduces adaptability. (Overfitting and neglecting tail effects again.)

Huizi said to Zhuangzi, 'You speak, Sir, of what is of no use.' The reply was, 'When a man knows what is not useful, you can then begin to speak to him of what is useful. The earth for instance is certainly spacious and great; but what a man uses of it is only sufficient ground for his feet. If, however, a rent were made by the side of his feet, down to the yellow springs, could the man still make use of it?' Huizi said, 'He could not use it,' and Zhuangzi rejoined, 'Then the usefulness of what is of no use is clear.'

Or, as Warren Buffet put it, “In order to succeed, you must first survive.” As the fish story above shows, if you play long enough you’re pretty much guaranteed to win big sooner or later. If you are dead or bankrupt, the game is over. This is why, for example, a martingale strategy is only a great system if you have infinite reserves.

(Behavioural economists never take any risks and receive nice monthly salaries with taxes deducted at source, so they do not understand this, and call loss aversion a cognitive bias. Ignore them.)

I’m not a fan of the translation of the next one, so I’ve done my own. Here’s the official version:

Zhuangzi said, 'If a man have the power to enjoy himself (in any pursuit), can he be kept from doing so? If he have not the power, can he so enjoy himself? There are those whose aim is bent on concealing themselves, and those who are determined that their doings shall leave no trace. Alas! they both shirk the obligations of perfect knowledge and great virtue. The (latter) fall, and cannot recover themselves; the (former) rush on like fire, and do not consider (what they are doing). Though men may stand to each other in the relation of ruler and minister, that is but for a time. In a changed age, the one of them would not be able to look down on the other. Hence it is said, "The Perfect man leaves no traces of his conduct." 'To honour antiquity and despise the present time is the characteristic of learners; but even the disciples of Khih-wei have to look at the present age; and who can avoid being carried along by its course? It is only the Perfect man who is able to enjoy himself in the world, and not be deflected from the right, to accommodate himself to others and not lose himself. He does not learn their lessons; he only takes their ideas into consideration, and does not discard them as different from his own.

Here’s mine:

Master Zhuang said: “If a person can do as he likes, what can prevent him from doing so? If a person cannot do as he likes, what can allow him to do so? There are those who are determined to follow the current, and those who decide to resist and pursue their own path. Ah, they will never attain the position of one who attains a full understanding of the excellent qualities. They sink and cannot rise or gallop onwards furious and heedless. Dominate or submit, it is only for a time. In different times, there would be no cause for disdain between them. Therefore I say: he who has attained completion is not set in his ways. Even honouring antiquity and holding the present in contempt is just how scholars follow the herd. Even those who follow Xi Wei must look to modern times, so how can they avoid being carried along by the current? Only he who has attained completion can do as he likes within the world without running up against others, or follow others without himself being swept away. Do not study their teachings; receive their ideas without assimilating yourself to them.

As far as I can tell, this is an alternative take on the optimisation problem described above, and the two should be read as companion pieces. The original version has the individual acting in isolation, while in this version he’s dealing with other actors in a situation of Pareto optimality. Every extra unit of individual satisfaction he gains must be clawed away from the group, with the stress of obtaining it being equal to the enjoyment of having it. In such a situation even the optimum solution kind of sucks — whether you choose to fight ferociously for your individuality, acquiesce to the group entirely, or search for a middle path, you will gain no more than you lose.

Essentially, it’s an Edgeworth box where O represents you and A the rest of the world, with Ωy being personal fulfilment and Ωx relaxation.

While Zhuangzi formulates this as a personal fulfilment problem, the trade-off is probably easier to imagine from an economic perspective. Imagine you’re a prehistoric shepherd; every night wolves come and grab another of your sheep. You can stay up every night fighting wolves, or you can give in and resign yourself to losing your flock. Neither is an appealing survival strategy. To resolve it, you need to step outside the rules of the game. What if you were to set aside 10% of every meal you have and leave it out for the wolves? They’re getting less mutton, but they’re getting it on a reliable basis and without a side order of crossbow bolts. Turns out dependability is a third factor that was not being considered until now. For greater reliability, both humans and wolves will be willing to accept fewer gains/more effort. A few generations later:

In other words, this is an under-specification problem. The only way to solve it is to add a third dimension. Instead of a trade-off between gains and relaxation, it’s now a trade off between gains, relaxation and reliability of outcomes.

It’s kind of hard to visualise (not least because conventional indifference curve graphs are actually 3D phenomena represented in two dimensions), but this gives an idea. Just imagine multiple planes.

The important part is that curves that do not intersect in two dimensions may well do so in three. If so, you and the wolves can now transact in a mutually beneficial way:

(This is also related to the issue of construct validity — whether you consider the appropriate variables to be “sheep/death”, “pleasure/pain”, “personal fulfilment/relaxation” or something else, but that’s a whole other rabbit hole. The second half touches on this in a way, by arguing that those who believe themselves to be fierce individualists are often merely a different variety of conformist, while those who claim mainstream status are actually positioning themselves in opposition to the changing spirit of the age. Being able to see what each is actually gaining and losing helps to clarify the transaction and work out how to circumvent the dichotomy.)

In the original problem an unseen third variable turns an optimal solution into a bad one. In this case, it turns an insoluble problem into a soluble one.

A nice real-world example of this approach that’s come up lately on Twitter would be the Dujiangyan irrigation project from the third century BC. If the Qin engineers had let nature take its course, they wouldn’t have been able to turn Sichuan into the breadbasket of the empire. On the other hand, if they’d imposed their wishes upon it regardless and built modern irrigation systems with complex pipe arrangement, concrete-lined reservoirs and AI-based pressure differential trackers, the friction caused by the resistance to this would mean that it’d require replacing within 50 years. Because they found the point at which their indifference curve and the river’s overlapped - rather than turning the problem into a zero-sum battle for dominance - it’s still working now, and if you didn’t know you wouldn’t guess anything had been changed. The original engineers died long ago, but the optimal course for both the ecosystem and the current population is to keep things just as they are.

Incidentally, this was the most difficult section to work through and it’s not even close. I’m pretty sure I had some kind of shamanic experience while trying to imagine 4D Edgeworth boxes on the train.

Moving on.

'It is the penetrating eye that gives clear vision, the acute ear that gives quick hearing, the discriminating nose that gives discernment of odours, the practised mouth that gives the enjoyment of flavours, the active mind that acquires knowledge, and the far-reaching knowledge that constitutes virtue. In no case does the connexion with what is without like to be obstructed; obstruction produces stoppage; stoppage, continuing without intermission, arrests all progress; and with this all injurious effects spring up. The knowledge of all creatures depends on their breathing. But if their breath be not abundant, it is not the fault of Heaven, which tries to penetrate them with it, day and night without ceasing; but men notwithstanding shut their pores against it. The womb encloses a large and empty space; the heart has its spontaneous and enjoyable movements. If their apartment be not roomy, wife and mother-in-law will be bickering; if the heart have not its spontaneous and enjoyable movements, the six faculties of perception will be in mutual collision. That the great forests, the heights and hills, are pleasant to men, is because their spirits cannot overcome (those distracting influences).

The medical/qigong stuff is where I nope out. I feel like the background necessary to interpret this has been definitively lost, and anyone who claims to understand these sections is a charlatan.

Next.

'Virtue overflows into (the love of) fame; (the love of) fame overflows into violence; schemes originate in the urgency (of circumstances); (the show of) wisdom comes from rivalry; the fuel (of strife) is produced from the obstinate maintenance (of one's own views); the business of offices should be apportioned in accordance with the approval of all. In spring, when the rain and the sunshine come seasonably, vegetation grows luxuriantly, and sickles and hoes begin to be prepared. More than half of what had fallen down becomes straight, and we do not know how.

That is to say, beware of confusing the unit of measurement for the thing being measured (eg. IQ and intelligence), and — related to this — mission creep: clever kids get good test scores, therefore we should train kids to pass tests and this will make them clever. It also ties these in with the concept of confused causality mentioned above — spring causes plants to grow, not the inverse. If you’re reasonably intelligent you’ll probably score over 100 on an IQ test, but it’s not scoring over 100 that makes you intelligent.

The next one is also relatively intuitive:

'Stillness and silence are helpful to those who are ill; rubbing the corners of the eyes is helpful to the aged; rest serves to calm agitation; but they are the toiled and troubled who have recourse to these things. Those who are at ease, and have not had such experiences, do not care to ask about them. The spirit-like man has had no experience of how it is that the sagely man keeps the world in awe, and so he does not inquire about it; the sagely man has had no experience of how it is that the man of ability and virtue keeps his age in awe, and so he does not inquire about it; the man of ability and virtue has had no experience of how it is that the superior man keeps his state in awe, and so he does not inquire about it. The superior man has had no experience of how it is that the small man keeps himself in agreement with his times that he should inquire about it.'

Your position will determine the data sample to which you have access. This will enable you to create a reasonable method for dealing with your own situation, but it won’t necessarily be applicable to anyone else’s. Just because you’re feeling an urge to rub your own eyes, it doesn’t necessarily mean that the guy sitting next to you on the train will thank you for extending the service to him.

This is also related to the scale on which you’re focusing, because — as mentioned above — big enough quantitative differences become qualitative differences. This is why we distinguish between micro- and macro-economics, or Newtonian and quantum physics. A guy might be great at running a fried chicken shop, but he couldn’t transfer his model to running the country. Conversely, a guy whose model was developed for running the country would fail if he transferred it to the fried chicken business. Moreover, neither of then could replace a theoretical physicist, and he couldn’t sub in for them. Consider this:

All of these images are of the same thing, just at a different zoom level. The information you extrapolate from each will differ.

The next section goes back to non-linearity strictu sensu:

The keeper of the Yan Gate, on the death of his father, showed so much skill in emaciating his person that he received the rank of 'Pattern for Officers.' Half the people of his neighbourhood (in consequence) carried their emaciation to such a point that they died. When Yao wished to resign the throne to Xu You, the latter ran away. When Thang offered his to Wu Guang, Wu Guang became angry. When Ji Ta heard it, he led his disciples, and withdrew to the river Kuan, where the feudal princes came and condoled with him, and after three years, Shen Tu-di threw himself into the water.

Losing weight when in formal mourning is good (until you die). The second part is less clear — it seems like it’s aiming for a parallelism (suggesting that some or all of these people were just pretending to refuse power out of a sense of personal integrity), but it’s hard to tell without more detailed biographies. They were conventionally regarded as paragons, but Zhuangzi and his followers enjoyed puncturing such reputations.

Finally, we return to the original Twitter thread:

Fishing-stakes are employed to catch fish; but when the fish are got, the men forget the stakes. Snares are employed to catch hares, but when the hares are got, men forget the snares. Words are employed to convey ideas; but when the ideas are apprehended, men forget the words. Fain would I talk with such a man who has forgot the words!

You use the data points you have to build a model, and if the model is good enough, then you will no longer need the data points.

P.S. I should give the standard disclaimer that all of these stories were written to have multiple interpretations, and this is just one set. If this thread has one message it’s that every outcome is a distribution rather than a point.