by Subjective probability — answering questions nobody asked Solve for x — give a single, unique number — in the following equation: x + y = 3. Of course, it cannot be done: under no rules of mathematics can a unique x be discovered; there are one too many unknowns. Nevertheless, someone holding to the subjective interpretation of probability could tell us, say, “1 feel x = 7.” Or he might say, “The following is my distribution for the possible values of x.” He’ll draw a picture, a curve of probability showing higher and lower chances for each possible x, maybe peaking somewhere near 3 and tailing off for very large and small numbers. He might say his curve is equivalent to one from the standard toolkit, such as the normal. Absurd? It shouldn’t sound
Topics:
Lars Pålsson Syll considers the following as important: Statistics & Econometrics
This could be interesting, too:
Lars Pålsson Syll writes Feynman’s trick (student stuff)
Lars Pålsson Syll writes Difference in Differences (student stuff)
Lars Pålsson Syll writes Vad ALLA bör veta om statistik
Lars Pålsson Syll writes Data analysis for social sciences (student stuff)
Subjective probability — answering questions nobody asked
Solve for x — give a single, unique number — in the following equation: x + y = 3. Of course, it cannot be done: under no rules of mathematics can a unique x be discovered; there are one too many unknowns. Nevertheless, someone holding to the subjective interpretation of probability could tell us, say, “1 feel x = 7.” Or he might say, “The following is my distribution for the possible values of x.” He’ll draw a picture, a curve of probability showing higher and lower chances for each possible x, maybe peaking somewhere near 3 and tailing off for very large and small numbers. He might say his curve is equivalent to one from the standard toolkit, such as the normal. Absurd?
It shouldn’t sound absurd. The situation is perfectly delineated. The open premise is that x = 3 — y, with a tacit premise that y must be something. The logical probability answer is that there is no probability: not enough information. (We don’t even know if y should be a real number!) But why not, a subjectivist might say, take a “maximal ignorance” position, which implies, he assumes, thaty can be any number, with none being preferred over any other. This leads to something like a “uniform distribution” over the real line; that being so, x is easily solved for, once for each value of y. Even if we allow the subjectivist free rein, this decision of uniformity is unfortunate because it leads to well known logical absurdities. There cannot be an equal probability for infinite alternatives because the sum of probabilities, no matter how small each of the infinite possibilities is, is always (in the limit) infinity; and indeed this particular uniform “distribution” is called “improper.” Giving the non¬ probability a label restores a level of comfort lost upon realizing the non-probability isn’t a probability, but it is a false comfort. Aiding the subjectivist is that the math using improper probabilities sometimes works out, and if the math works out, what’s to complain about?
To say we are “maximally ignorant” of y, or to say anything else about y (or x), is to add information or invent evidence which is not provided. Adding information that is not present or is not plausibly tacit is to change the problem. If we are allowed to arbitrarily change any problem so that it is more to our liking we shall, naturally, be able to solve these problems more easily. But we are not solving the stated problems. We are answering questions nobody asked.
Modern probabilistic econometrics relies on the notion of probability. To at all be amenable to econometric analysis, economic observations allegedly have to be conceived as random events. But is it really necessary to model the economic system as a system where randomness can only be analyzed and understood when based on an a priori notion of probability?
In probabilistic econometrics, events and observations are as a rule interpreted as random variables as if generated by an underlying probability density function, and a fortiori – since probability density functions are only definable in a probability context – consistent with a probability. As Haavelmo (1944:iii) has it:
For no tool developed in the theory of statistics has any meaning – except , perhaps for descriptive purposes – without being referred to some stochastic scheme.
When attempting to convince us of the necessity of founding empirical economic analysis on probability models, Haavelmo – building largely on the earlier Fisherian paradigm – actually forces econometrics to (implicitly) interpret events as random variables generated by an underlying probability density function.
This is at odds with reality. Randomness obviously is a fact of the real world. Probability, on the other hand, attaches to the world via intellectually constructed models, and a fortiori is only a fact of a probability-generating machine or a well-constructed experimental arrangement or “chance set-up”.
Just as there is no such thing as a “free lunch,” there is — as forcefully argued by Briggs — no such thing as a “free probability.” To be able at all to talk about probabilities, you have to specify a model. If there is no chance set-up or model that generates the probabilistic outcomes or events – in statistics, one refers to any process where you observe or measure as an experiment (rolling a die) and the results obtained as the outcomes or events (number of points rolled with the die, being e. g. 3 or 5) of the experiment –there strictly seen is no event at all.
Probability is a relational element. It always must come with a specification of the model from which it is calculated. And then to be of any empirical scientific value it has to be shown to coincide with (or at least converge to) real data generating processes or structures – something seldom or never done!
And this is the basic problem with economic data. If you have a fair roulette wheel, you can arguably specify probabilities and probability density distributions. But how do you conceive of the analogous nomological machines for prices, gross domestic product, income distribution etc? Only by a leap of faith. And that does not suffice. You have to come up with some really good arguments if you want to persuade people into believing in the existence of socioeconomic structures that generate data with characteristics conceivable as stochastic events portrayed by probabilistic density distributions!
From a realistic point of view we really have to admit that the socio-economic states of nature that we talk of in most social sciences – and certainly in econometrics – are not amenable to analyze as probabilities, simply because in the real-world open systems that social sciences – including econometrics – analyze, there are no probabilities to be had!
The processes that generate socio-economic data in the real world cannot just be assumed to always be adequately captured by a probability measure. And, so, it cannot really be maintained – as in the Haavelmo paradigm of probabilistic econometrics – that it even should be mandatory to treat observations and data – whether cross-section, time series or panel data – as events generated by some probability model. The important activities of most economic agents do not usually include throwing dice or spinning roulette wheels. Data generating processes – at least outside of nomological machines like dice and roulette wheels – are not self-evidently best modelled with probability measures.
If we agree on this, we also have to admit that probabilistic econometrics lacks a sound justification. I would even go further and argue that there really is no justifiable rationale at all for this belief that all economically relevant data can be adequately captured by a probability measure. In most real-world contexts one has to argue one’s case. And that is obviously something seldom or never done by practitioners of probabilistic econometrics.
Econometrics and probability are intermingled with randomness. But what is randomness?
In probabilistic econometrics it is often defined with the help of independent trials – two events are said to be independent if the occurrence or nonoccurrence of either one has no effect on the probability of the occurrence of the other – as drawing cards from a deck, picking balls from an urn, spinning a roulette wheel or tossing coins – trials which are only definable if somehow set in a probabilistic context.
But if we pick a sequence of prices – say 2, 4, 3, 8, 5, 6, 6 – that we want to use in an econometric regression analysis, how do we know the sequence of prices is random and a fortiori being able to treat as generated by an underlying probability density function? How can we argue that the sequence is a sequence of probabilistically independent random prices? And are they really random in the sense that is most often applied in probabilistic econometrics – where X is called a random variable only if there is a sample space S with a probability measure and X is a real-valued function over the elements of S?
Bypassing the scientific challenge of going from describable randomness to calculable probability by just assuming it, is of course not an acceptable procedure. Since a probability density function is a “Gedanken” object that does not exist in a natural sense, it has to come with an export license to our real target system if it is to be considered usable.
Among those who at least honestly try to face the problem – the usual procedure is to refer to some artificial mechanism operating in some “games of chance” of the kind mentioned above and which generates the sequence. But then we still have to show that the real sequence somehow coincides with the ideal sequence that defines independence and randomness within our – to speak with science philosopher Nancy Cartwright (1999) – “nomological machine”, our chance set-up, our probabilistic model.
As the originator of the Kalman filter, Rudolf Kalman (1994:143), notes:
Not being able to test a sequence for ‘independent randomness’ (without being told how it was generated) is the same thing as accepting that reasoning about an “independent random sequence” is not operationally useful.
So why should we define randomness with probability? If we do, we have to accept that to speak of randomness we also have to presuppose the existence of nomological probability machines, since probabilities cannot be spoken of – and actually, to be strict, do not at all exist – without specifying such system-contexts (how many sides do the dice have, are the cards unmarked, etc)
If we do adhere to the Fisher-Haavelmo paradigm of probabilistic econometrics we also have to assume that all noise in our data is probabilistic and that errors are well-behaving, something that is hard to justifiably argue for as a real phenomenon, and not just an operationally and pragmatically tractable assumption.
Maybe Kalman’s (1994:147) verdict that
Haavelmo’s error that randomness = (conventional) probability is just another example of scientific prejudice
is, from this perspective seen, not far-fetched.
Accepting Haavelmo’s domain of probability theory and sample space of infinite populations– just as Fisher’s (1922:311) “hypothetical infinite population, of which the actual data are regarded as constituting a random sample”, von Mises’ “collective” or Gibbs’ ”ensemble” – also implies that judgments are made on the basis of observations that are actually never made!
Infinitely repeated trials or samplings never take place in the real world. So that cannot be a sound inductive basis for science with aspirations of explaining real-world socio-economic processes, structures or events. It’s not tenable.
As David Salsburg (2001:146) notes on probability theory:
[W]e assume there is an abstract space of elementary things called ‘events’ … If a measure on the abstract space of events fulfills certain axioms, then it is a probability. To use probability in real life, we have to identify this space of events and do so with sufficient specificity to allow us to actually calculate probability measurements on that space … Unless we can identify [this] abstract space, the probability statements that emerge from statistical analyses will have many different and sometimes contrary meanings.
Just as e. g. Keynes (1921) and Georgescu-Roegen (1971), Salsburg (2001:301f) is very critical of the way social scientists – including economists and econometricians – uncritically and without arguments have come to simply assume that one can apply probability distributions from statistical theory on their own area of research:
Probability is a measure of sets in an abstract space of events. All the mathematical properties of probability can be derived from this definition. When we wish to apply probability to real life, we need to identify that abstract space of events for the particular problem at hand … It is not well established when statistical methods are used for observational studies … If we cannot identify the space of events that generate the probabilities being calculated, then one model is no more valid than another … As statistical models are used more and more for observational studies to assist in social decisions by government and advocacy groups, this fundamental failure to be able to derive probabilities without ambiguity will cast doubt on the usefulness of these methods.
Some wise words that ought to be taken seriously by probabilistic econometricians are also given by mathematical statistician Gunnar Blom (2004:389):
If the demands for randomness are not at all fulfilled, you only bring damage to your analysis using statistical methods. The analysis gets an air of science around it, that it does not at all deserve.
Richard von Mises (1957:103) noted that
Probabilities exist only in collectives … This idea, which is a deliberate restriction of the calculus of probabilities to the investigation of relations between distributions, has not been clearly carried through in any of the former theories of probability.
And obviously not in Haavelmo’s paradigm of probabilistic econometrics either. It would have been better if one had heeded von Mises warning (1957:172) that
the field of application of the theory of errors should not be extended too far.
This importantly also means that if you cannot show that data satisfies all the conditions of the probabilistic nomological machine – including randomness – then the statistical inferences used, lack sound foundations!
References
Gunnar Blom et al: Sannolikhetsteori och statistikteori med tillämpningar, Lund: Studentlitteratur.
Cartwright, Nancy (1999), The Dappled World. Cambridge: Cambridge University Press.
Fisher, Ronald (1922), On the mathematical foundations of theoretical statistics. Philosophical Transactions of The Royal Society A, 222.
Georgescu-Roegen, Nicholas (1971), The Entropy Law and the Economic Process. Harvard University Press.
Haavelmo, Trygve (1944), The probability approach in econometrics. Supplement to Econometrica 12:1-115.
Kalman, Rudolf (1994), Randomness Reexamined. Modeling, Identification and Control 3:141-151.
Keynes, John Maynard (1973 (1921)), A Treatise on Probability. Volume VIII of The Collected Writings of John Maynard Keynes, London: Macmillan.
Pålsson Syll, Lars (2007), John Maynard Keynes. Stockholm: SNS Förlag.
Salsburg, David (2001), The Lady Tasting Tea. Henry Holt.
von Mises, Richard (1957), Probability, Statistics and Truth. New York: Dover Publications.