Cool look at risk - Part 2: What is stochastics?

A second post for the fourth edition of my book

[There is also a Greek version of the post—Υπάρχει και ελληνική έκδοση της ανάρτησης]

In this second post about my book Stochastics of Hydroclimatic Extremes - A Cool Look at Risk I highlight its Digression 1.A: The meaning of stochastics. Stochastics is the central theme of the book and is distinguished from deterministic mathematics. The meaning of stochastics is rich and has several facets, including those of being:

• probability theoretic;

• adept at quantifying the imprecise, the uncertain, or else the random;

• insightful—not superficial;

• capable of prediction in a probabilistic sense, using information from the past;

• suitable for the calculation of the mean, or expectation, of uncertain quantities.

Naturally, once we adopt a stochastic approach, we admit that uncertainty is inevitable and we deal with it by using probabilities and expectations of the processes of interest, and making inferences based on past observations of these processes.

A meeting of four wise men to discuss about stochastics: Plato, St. Basil the Great, Jacob Bernoulli, and Andrey Kolmogorov in a Grand Academic Hall (AI-generated image by ChatGPT / DALL·E, based on historical depictions.) Hopefully next generations of AI tools would be able to make images in which the depicted persons look like those they are supposed to depict…

Literally, stochastics is a term of Greek origin, stemming from the adjective ‘stochasticos’ (στοχαστικός), or better its feminine gender, ‘stochastice’ (στοχαστική). It is generated from the verb ‘stochazesthai’ (στοχάζεσθαι), which in turn comes from the noun ‘stochos’ (στόχος), meaning the target.

Aristotle, in his treatise Nicomachean Ethics (written ~350 BC) uses the term stochastice in its original meaning, related to the target, which, according to him, is the mean: “virtue, therefore, is a balance [‘mesotes’], in the sense that it is able to hit [as a target – ‘stochos’] the mean”¹. Furthermore, in his treatise Rhetoric he uses the term with a metaphorical meaning, which could be translated into English as guessing or guesswork: “men have a sufficient natural instinct for what is true, and usually do arrive at the truth. Hence the man who makes a good guess at truth is likely to make a good guess at probabilities [stochastically].”²

However, it was Plato who used the term with a meaning closer to the modern one, i.e., related to uncertainty. In his dialogue Philebus (written ~360 BC) he contrasts “arithmetic and the sciences of measurement” to stochastics and parallels the latter with music, which “attains harmony by guesswork […] so that the amount of uncertainty mixed up in it is great, and the amount of certainty small.” ³

The contrast between stochasticity and precision is made clear later by Galenus using the example of a city’s clock: “When a city is being built, let the following problem be set before those who will inhabit it: they want to expertly know, not stochastically but precisely, on an everyday basis, how much time has passed, and how much is left before sunset.”⁴

The connection of stochastics with prediction or forecast becomes evident in an excerpt from Basilius Caesariensis [St Basil the Great] who contrasts a prophet to a ‘stochastes’ (στοχαστής, a noun usually trans­lated incorrectly into English as diviner): “On the one hand, a prophet is he who foretells the future by revelation of the Spirit; on the other hand, a stochastes is he who infers the future by pru­dence, comparing similar states, and by the experience of forefathers.”⁵ It seems that this comment has influenced later scholars (e.g. Procopius) and perhaps determined the meaning of stochastic in modern Greek, which is imaginative, insightful, thoughtful, cogitative, contemplative, meditative.

The transplantation of stochastics, as an international scientific term, to the modern vocabulary is due to Jacob Bernoulli, evidently aware of the Greek language and literature, and in particular of the passage from Plato’s Philebus mentioned above. In his famous book Ars Conjectandi (written in Latin in 1684-89 but published after his death) he writes: “To conjecture about something is to measure its probability. Therefore we define the art of conjecture, or stochastics, as the art of measuring the probabilities of things as exactly as possible, to the end that, in our judgments and actions, we may always choose or follow that which has been found to be better, more satisfactory, safer, or more carefully considered. On this alone turns all the wisdom of the philosopher and all the practical judgment of the statesman.”⁶

The term was revived by Bortkiewicz (1917;⁷ Russian economist and statistician of Polish ancestry) and also by Slutsky (1925, 1928a,b, 1929;⁸ Ukrainian/Russian/Soviet mathematical statistician and economist). It appears that the prevalence in USSR of the more sophisticated term stochastic (over the term random) must have been related to political and ideological reasons (incongruence of randomness with the dialectical materialism: models beyond strict determinism were considered with a priori suspicion; see Mazliak 2018).⁹

But it was Kolmogorov (1931)¹⁰ who made the term popular and widespread, as he introduced the term stochastic process, also clarifying that process means change of a certain system. Additionally, he used the term stationary to describe a probability density function that is unchanged in time (while at the same time the system state changes). Soon after, Kolmogorov (1933)¹¹ introduced the modern and consistent definition of probability in an axiomatic manner, based on the measure theory (see section 2.1).

1. «μεσότης τις ἄρα ἐστὶν ἡ ἀρετή, στοχαστική γε οὖσα τοῦ μέσου» (Aristot. Nic. Eth. 1106b, translation into English adapted from that by H. Rackham. Cambridge, MA, Harvard University Press; London, William Heinemann Ltd. 1934). The notion of ‘mesotes’ (μεσότης), loosely translated as balance, middle, mean between a respective ‘too much’ and ‘too little’, is a key concept in Aristotle’s ethical philosophy and thus to hit it as a target is important for him.

2. «οἱ ἄνθρωποι πρὸς τὸ ἀληθὲς πεφύκασιν ἱκανῶς καὶ τὰ πλείω τυγχάνουσι τῆς ἀληθείας: διὸ πρὸς τὰ ἔνδοξα στοχαστικῶς ἔχειν τοῦ ὁμοίως ἔχοντος καὶ πρὸς τὴν ἀλήθειάν ἐστιν» (Aristot. Rh. 1.1, translation into English by W. Rhys Roberts, http://classics.mit.edu/Aristotle/rhetoric.1.i.html).

3. The complete passage is: ΣΩΚΡΑΤΗΣ: «οἷον πασῶν που τεχνῶν ἄν τις ἀριθμητικὴν χωρίζῃ καὶ μετρητικὴν καὶ στατικήν, ὡς ἔπος εἰπεῖν φαῦλον τὸ καταλειπόμενον ἑκάστης ἂν γίγνοιτο. […] τὸ γοῦν μετὰ ταῦτ᾽ εἰκάζειν λείποιτ᾽ ἂν καὶ τὰς αἰσθήσεις καταμελετᾶν ἐμπειρίᾳ καί τινι τριβῇ, ταῖς τῆς στοχαστικῆς προσχρωμένους δυνάμεσιν ἃς πολλοὶ τέχνας ἐπονομάζουσι, μελέτῃ καὶ πόνῳ τὴν ῥώμην ἀπειργασμένας. […] οὐκοῦν μεστὴ μέν που μουσικὴ πρῶτον, τὸ σύμφωνον ἁρμόττουσα οὐ μέτρῳ ἀλλὰ μελέτης στοχασμῷ, καὶ σύμπασα αὐτῆς αὐλητική, τὸ μέτρον ἑκάστης χορδῆς τῷ στοχάζεσθαι φερομένης θηρεύουσα, ὥστε πολὺ μεμειγμένον ἔχειν τὸ μὴ σαφές, σμικρὸν δὲ τὸ βέβαιον.»

(SOCRATES: “For example, if arithmetic and the sciences of measurement and weighing were taken away from all arts, what was left of any of them would be, so to speak, pretty worthless. […] All that would be left for us would be to conjecture and to drill the perceptions by practice and experience, with the additional use of the powers of guessing, which are commonly called arts and acquire their efficacy by practice and toil. […] Take music first; it is full of this; it attains harmony by guesswork based on practice, not by measurement; and flute music throughout tries to find the pitch of each note as it is produced by guess, so that the amount of uncertainty mixed up in it is great, and the amount of certainty small” (Plat. Phileb. 55e, translation by Harold N. Fowler; Cambridge, MA, Harvard University Press; 1925.)

4. «πόλεως κτιζομένης προκείσθω τοῖς οἰκήσουσιν αὐτὴν ἐπίστασθαι βούλεσθαι, μὴ στοχαστικῶς ἀλλ’ ἀκριβῶς, ἐφ’ ἑκάστης ἡμέρας, ὁπόσον τε παρελήλυθεν ἤδη τοῦ χρόνου τοῦ κατ’ αὐτήν, ὁπόσον θ’ ὑπόλοιπόν ἐστιν ἄχρι δύσεως ἡλίου.» (Γαληνοῦ Περὶ Διαγνώσεως καὶ Θεραπείας τῶν ἐν τῇ ἑκάστου Ψυχῇ Ἁμαρτημάτων — De Dignotione et Curatione cujusque Animi Peccatorum, 80, http://www.poesialatina.it/_ns/greek/testi/Galenus/De_animi_cuiuslibet_peccatorum_dignotione_et_curatione.html).

5. «Οὐκοῦν Προφήτης μέν ἐστιν, ὁ κατὰ ἀποκάλυψιν τοῦ Πνεύματος προαγορεύων τὸ μέλλον· στοχαστὴς δὲ, ὁ διὰ σύνεσιν ἐκ τῆς τοῦ ὁμοίου παραθέσεως, διὰ τὴν πεῖραν τῶν προλαβόντων, τὸ μέλλον συντεκμαιρόμενος.» (Basilius, Ερμηνεία εις τον προφήτην Ησαΐαν —Enarratio in prophetam Isaiam, 3.102.26).

6. “Conjicere rem aliquam est metiri illius probabilitatem: ideoque Ars Conjectandi sive Stochastice nobis definitur ars metiendi quàm fieri potest exactissimè probabilitates rerum, eo fine, ut in judiciis & actionibus nostris semper eligere vel sequi possimus id, quod melius, satius, tutius aut consultius fuerit deprehensum; in quo solo omnis Philosophi sapientia & Politici prudential versatur.”

Bernoulli, J., 1713. Ars Conjectandi, Opus Posthumum. Accedit Tractatus de Seriebus Infinitis, et Epistola Gallicé Scripta de Ludo Pilae Reticularis. Basileae, Impensis Thurnisiorum, Fratrum [English translation: The art Of Conjecturing, Together With Letter to a Friend on Sets in Court Tennis. Translated with an Introduction and Notes by Edith Dudley Sylla; The Johns Hopkins University Press, Baltimore, Maryland, USA, 2006.]

7. Bortkiewicz, L., 1917. Die Iterationen — Ein Beitrag zur Wahrscheinlichkeitstheorie. Springer, Berlin.

8. Slutsky, E.,1925, Über stochastische Asymptoten und Grenzwerte, Metron, 5(3), 3-89.

Slutsky, E., 1927. Slozhenie sluchainykh prichin, kak istochnik tsiklicheskikh protsessov. Voprosy kon’’yunktury, 3, 34 – 64. 1927 (English edition: Slutzky, E., 1937. The summation of random causes as the source of cyclic processes. Econometrica: Journal of the Econometric Society, 105-146).

Slutsky, E., 1928a. Sur ùn critérium de la convergence stochastique des ensembles des valeurs éventuelles). Comptes rendus Acad. Sci., 187, 370.

Slutsky, E., 1928b. Sur les fonctions éventuelles continues, intégrables et dérivables dans le sens stochastiques. Comptes rendus des séances de l’Académie des Sciences, 187, 878.

Slutsky, E.,1929. Quelques propositions sur les limites stochastiques éventuelles. Comptes rendus des séances de l’Académie des Sciences, 189, 384.

9. Mazliak, L., 2018. The beginnings of the Soviet Encyclopedia. The utopia and misery of mathematics in the political turmoil of the 1920s. Centaurus, 60(1-2), 25-51.

10. Kolmogorov, A.N., 1931. Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann., 104, 415-458. (English translation: On analytical methods in probability theory, In: Kolmogorov, A.N., Selected Works of A. N. Kolmogorov - Volume 2, Probability Theory and Mathematical Statistics, ed. by A.N. Shiryayev, Kluwer, Dordrecht, The Netherlands, 62-108, 1992).

11. Kolmogorov, A.N., 1933. Grundbegriffe der Wahrscheinlichkeitsrechnung, Ergebnisse der Math. (2), Berlin (2nd English Edition: Foundations of the Theory of Probability, 84 pp. Chelsea Publishing Company, New York, 1956).