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Varoufakis on corrupt statistics in respect of Greece debt April 24, 2014

Posted by larry (Hobbes) in Abuse of power, economics, Jefferson Airplane, Statistics.
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TruthFromLies

Varoufakis begins his post with this graphic, which reminds me of a song by Jefferson Airplane. Here they are appearing on The Smothers Brothers show. A hilarious TV program. It begins with White Rabbit but continues with (Do You Want) Somebody to Love whose first line is “When the truth is found to be lies”. The following line: “All the joy within you dies”. Sometimes I wonder whether members of the elite and their economic lackeys have been taking some of Alice’s pills.

[In case the video is not shown, here is the link: https://www.youtube.com/watch?v=hnP72uUt_pU]

Varoufakis argues that the statistics from Eurostat that show that Greek debt isn’t as great as it was thought to be is little more than a political ploy by the Euro elite in order not to upset the upcoming European elections in May. It is a short piece. You can read his argument for yourself.

http://yanisvaroufakis.eu/2014/04/24/greek-statistics-are-back-primary-deficit-presented-as-surplus-with-eurostats-seal-of-approval/

Happy listening and reading.

Bayes’ theorem: a comment on a comment March 10, 2014

Posted by larry (Hobbes) in Bayes' theorem, Logic, Philosophy, Statistics.
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Assume the standard axioms of set theory, say, the Zermelo-Fraenkel axioms.

Then provide a definition of conditional probability:

1. BayesEq1, which yields the identity

1a. BayesEq2, via simple algebraic cross multiplication.

Because set intersection is commutative, you can have this:

1b. BayesEq3, which equals

2. BayesEq4.

What we have here is a complex, contextual definition relating a term, P, from probability theory with a newly introduced stroke operator, |, read as “given”, so the locution becomes, for instance, the probability, P, of A given B. Effectively, the definition is a contextual definition of the stroke operator, |, “given”.

Although set intersection (equivalent in this context to conjunction) is commutative, conditional probability isn’t, which is due to the asymmetric character of the stroke operator, |. This means that, in general, P(A|B) ≠ P(B|A). If we consider the example of Data vs. Hypothesis, we can see that in general, for A = Hypothesis and B = Data, that P(Hypothesis|Data) ≠ P(Data|Hypothesis).

Now, from the definition of “conditional probability” and the standard axioms of set theory which have already been implicitly used, we obtained Bayes’ theorem trivially, mathematically speaking, via a couple of simple substitutions.

BayesEq5.

Or the Bayes-Laplace theorem, since Laplace discovered the rule independently. However, according to Stigler’s rule of eponymy in mathematics, theorems are invariably attributed to the wrong person (Stigler, “Who Discovered Bayes’ Theorem?”. In Stigler, Statistics on the Table, 1999).

Now, since we have seen that Bayes’ theorem follows from the axioms of set theory plus the definition of “conditional probability”, the following comments from a recent tutorial text on Bayes’ theorem can only be interpreted as being odd. The following quote is from James V. Stones’ Bayes’ Rule: A Tutorial Introduction to Bayesian Analysis (3rd printing, Jan. 2014).

If we had to establish the rules for calculating with probabilities, we would insist that the result of such calculations must tally with our everyday experience of the physical world, just as surely as we would insist that 1+1 = 2. Indeed, if we insist that probabilities must be combined with each other in accordance with certain common sense principles then Cox (1946) showed that this leads to a unique set of rules, a set which includes Bayes’ rule, which also appears as part of Kolmogorov’s (1933) (arguably, more rigorous) theory of probability (Stone: pp. 2-3).

Bayes’ theorem does not form part of Kolmogorov’s set of axioms. Strictly speaking, Bayes’ rule must be viewed as a logical consequence of the axioms of set theory, the Kolmogorov axioms of probability, and the definition of “conditional probability”.

Whether Kolmogorov’s axioms for probability tally with our experience of the real world is another question. The axioms are sometimes used as indications of non-rational thought processes in certain psychological experiments, such as the Linda experiment by Tversky and Kahneman.  (For an alternative interpretation of this experiment that brings into question the assumption that people either do or should reason according to a simple application of the Kolmogorov axioms, cf. Luc Bovens & Stephan Hartmann, Bayesian Epistemology, 2003: 85-88).

A matter of interpretation

In the discussion above, the particular set theory and the Kolmogorov axioms mentioned and used were interpreted  via the first-order extensional predicate calculus. This means that both theories can be viewed as not involving intensional contexts such as beliefs. The probability axioms in particular were understood by Kolomogorov and others using them as relating to objective frequencies and applicable to the real world, not to beliefs we might have about the world. For instance, an unbiased coin and die, in the ideal case admittedly, are considered to have a .5 and 1/6 (or .1666) probability for the side of the coin and a side of a six-sided die, respectively, on each flip or throw of the object in question. In these two particular cases, it is only via behavior observed over a long period of time that can produce data that will show whether in fact our assumption that the coin and the die are unbiased is true or not.

Why does this matter. Simply because Bayes’ theorem has been interpreted in two distinct ways – as a descriptively objective statement about the character of the world and as a subjective statement about a users’ beliefs about the state of the world. The derivation above derives from two theories that are considered to be non-subjective in character. One can then reasonably ask: where does the subjective interpretation of Bayes’ theorem come from? Two answers suggest themselves, though these are not the only ones. One is that Bayes’ theorem is arrived at via a different derivation than the one I considered, relying, say, on a different notion of probability than that of Kolmogorov’s. The other is that Bayesian subjectivity is introduced by means of the stroke (or ‘given’) operator, |.

Personally, I see nothing subjective about statements concerning the probability of obtaining a H or a T on the flip of a coin as being .5 or that of obtaining one particular side of a 6-sided die being .166. These probabilities are about the objects themselves, and not about our beliefs concerning them. Of course, this leaves open the possibility of alternative interpretations of probabilities in other contexts, say the probability of guilt or non-guilt in a jury trial. Whether the notions of probability involving coins or dice are the same as those involving situations such as jury trials is a matter for further debate.

Replicability Vs Reanalysis in Macroeconomics April 17, 2013

Posted by larry (Hobbes) in economics, Logic & Theory of Theory Testing, Statistics.
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A recent furor has arisen about an analysis by Reinhart and Rogoff and a critique by Hernden et al. The discussion centers around data analysis and whether a piece of data analysis constitutes a reanalysis or is a replication of the original study. My response is to this issue. The relevant papers are (The first two are not links):

Reinhaart&Rogoff-GrowthInTimeOfDebt-AmericanEconRev-May2010 [arises out of their book];

HerndenAshPollin-DoesHighPublicDebtConsistentlyStifleEconomicGrowth_CritiqueReinhart&Rogoff-PERI-15Apr2013;

http://images.politico.com/global/2013/04/17/reinhart_rogoff_response_to_herndon_ash_and_pollen_april_17.html [Reinhart and Rogoff provide the wrong year for this response, dating it as 2012 when it is obviously 17 April 2013.].

There is a logical problem with replicability in macroeconomics. Let me contrast for this purpose economics with ecology. Ecological studies can be divided into roughly two types, experimental and field studies. With experimental studies in most disciplines, with the exception of advanced physics which I will come to later, it is clear what must be done in order to replicate the study. Experimental ecological studies generally satisfy these requirements.

But with field studies, it is not always so clear. Say you are studying a particular insect with respect to a set of attributes in a particular environment. To replicate this study, you need to find another environment that is as similar to the original environment as possible and at the same time of year with the temperature being similar, &c. You can use the same environment the following year for replicability purposes, but this has its own down sides. Strictly speaking, you will not be able to truly replicate the original study, but you may be able to get as close as the theories that are being tested need you to be. And that is what counts.

In experimental physics, you will not generally find true replications. This is because they don’t usually need them. What is sometimes referred to as a replication is actually an attempt to improve on the original experiment, usually in terms of some measure of precision. Exceptions occur when the results appear to be “off the wall”, as was felt to be the case by the physics community at large re the original cold fusion experiments.

What you have in both empirical ecological and physical studies are samples of the relevant populations with all the statistical analytical implications that that brings with it. In economics, however, you are not always presented with data samples. Sometimes you are presented with what I shall refer to as the entire data universe. For instance, say you want to know what banks have in reserve. Instead of a sample of banks qua their reserves, you may be presented with the reserves (many of which may be estimates) of every single bank. This is not a sample but the entire population of bank reserves. This procedure precludes certain standard statistical techniques being used. But as is usually the case in economics, since no statistics are employed, this difference generally makes no difference.

As for the critique of the Reinhart and Rogoff study, I think what we have here is not a true replication, but a reanalysis of more or less the same data. A reanalysis would be undertaken if you thought that either the data were poor or badly organized or that there was an error made in the original analysis. A replication, as opposed to a reanalysis, would involve gathering “new” data of the same kind and subjecting this “new” data to the same or an improved version of the original analysis.

It could be argued, and has been by some, that replications are impossible in disciplines like economics as too many factors change from one temporal interval to another, thereby making it impossible to replicate the same or similar conditions. But since replications (and reanalyses) are tests of relevant theories, if the theories in question are any good, they should be able to “tell” the researcher whether such a replication is possible or fruitful or not. On the other hand, there should exist, in principle, no theoretical obstacles to a reanalysis of the original data set. All such tests take place within a given theoretical context, including those where the theories under test are virtually completely contradictory. This latter circumstance renders the test environment more difficult to specify but not thereby impossible.

With respect to the presentation of data, whether of the entire population or of a sample thereof, it used to be relatively common practice, for instance in the thirties, to include with the data an error estimate, often plus or minus some percentage. This no longer seems to be the case. Statistics has moved on from that period, but confidence intervals or their equivalent were common then and are now. Yet they do not seem to find their way into a good number of economic analyses, whether presented in tabular or in graphical form. And this is as true of the Reinhart and Rogoff studies as many others, although it is less applicable to the figures in the Hernden et al. study. Additional statistical issues related to the nature of the data themselves are more relevant in this case.

Hazards of Black Swans April 3, 2013

Posted by larry (Hobbes) in economics, Philosophy, Statistics.
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Black Swan land = massive uncertainty = the domain of ignorance.

BlackSwanHazardExtremestan

Black Swan event (or idea): highly improbable, massive impact, rationalized afterward.  A Black Swan is so extreme that its probability distribution is unknown hence there is no ability to assess the risk of encountering it, and hence something of which we are ignorant until it occurs.  It is possible to create a financial instrument whose behavior is completely unpredictable and hence whose risk is incalculable.  A sufficient number of these in circulation could, under certain circumstances that can not be predicted in advance, trigger a Black Swan event.

Thin-tailed probability distributions give the likelihood of an extreme event too low a probability.  Thick-tailed probability distributions provide a greater probability to the likelihood of extreme events.  However, even the latter can not adequately handle Black Swan events.  Here, we are at the limits of statistics.  Thin-tailed: mild volatility at the extremes of the distribution; thick-tailed: harsh to unknown volatility at the extremes of the distribution.

Normal stocks reside in Mediocristan.  The derivatives that put us in the red zone are the complex derivatives invented after the early 1970s subsequent to the initial round of deregulation.  After the further deregulation of 1999, these types of derivatives went off the scale in terms of risk assessment and potential volatility. (“Mediocristan” must be understood in a Galtonian sense, that of the average.  So, in Mediocristan, we are in average-land.  Galton’s term, “regression to mediocrity” we interpret as “regression to the mean”.)

The derivative payoff domain lies of course in Extremistan.  The payoffs are incredibly complicated and often impossible to assess before they are “traded”.  Hence, I have dubbed them Schrödinger derivatives.
SchrodingerDerivative

Distributions are descriptive and reflect real processes, so any planned alterations must be carried out in terms of the payoff matrix.  Therefore, to eliminate as much as possible the hazardous effects of Black Swans, complex financial instruments, such as complex derivatives, could be banned completely or strictly limited in some way.

The only justification of a complex derivative and related synthetic financial instruments is income generation for the few.  They do not, and can not, benefit the many, contrary to the claims of some of their advocates, for which there is no evidence. In fact, evidence exists to the contrary – the recent volatility in the price of oil, the run on the pound in the ’90s, the present global financial crisis. All were considered highly unlikely, all had massive impact. And in the last case, where there was, and still is, massive fraud, virtually no one has gone to jail, largely because the event has been rationalized for the benefit of the well-off. Were there many successful prosecutions, many wealthy investors would lose their shirts.  Therefore, so far, everything is being done in order to ensure that this does not happen. Even to the disbenefit of society as a whole.

Hypotheses and Corroboration and Data Variation I May 4, 2011

Posted by larry (Hobbes) in data, Duesenberry, economics, interpretation, Lakatos, Logic, nature of science, Statistics, Suppes.
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Duesenberry has an excellent discussion about the relationship between a theory or hypothesis and a test of that theory or hypothesis.  He correctly notes that one can never prove a hypothesis or theory but fails to give a reason why this should be so.  He also does not mention the Duhem-Quine problem in the testing of hypotheses.

To simplify the discussion, I will consider the testing of a single hypothesis, but what I say applies to theories as well.  The reason that a given hypothesis, H, can not be proved, or verified, is for logical reasons.  Most scientific hypotheses are in the form of universal generalizations.  For instance, for all x, Ax implies Bx.   Now, in order to prove or verify that all As are Bs, one would have to be able to inspect, in principle, every thing that is an A and/or a B, past, present and future.  This is impossible.  Hence, general laws that are in the form of universal generalizations can never be verified.  But they can be falsified.  Again, for logical reasons.  All you need to falsify the hypothesis that for all x, Ax implies Bx, that is, that all As are Bs, is to find an A that is not a B.  A simple example of this is the eponymous generalization that was once believed, that all swans are white, that is, that for all x, if x is a swan, then x is white.  To falsify this, you need to find one black swan, that is, one thing x that is a swan and is not white.  Not only is this possible in principle, such swans were discovered in Australia.  The major difference between a hypothesis H and a theory T is this – that a theory can be seen as a conjunction of related hypotheses.  Therefore, a hypothesis H can be viewed as a smallest theory.

There is, therefore, an asymmetry between verification and falsification – universal scientific generalizations (scientific laws) can not be verified though they can be falsified.  On the other hand, existential generalizations, of the form there is an x such that Ax, i.e., there is at least one swan, can be verified but not falsified.  It is possible to show that there is a swan by finding one, but impossible, for logical reasons similar to those above, to prove or verify that there are none on the basis that one has yet to be found.  The situation is even more complicated than I have described, involving other factors, but that is for another time. (But it is recommended that the works of Imre Lakatos, such as the methodology of scientific research programs, be consulted.  His conceptual scheme is non-trivial and more than just interesting.  And Patrick Suppes’ article on models of data (http://suppes-corpus.stanford.edu/articles/mpm/41.pdf).)  The important lesson to take away from this is that for a hypothesis to be scientific, it must be falsifiable in principle, although adhering to this requirement involves considerable complexity and is not without its difficulties.

The Duhem-Quine problem is more complicated. This is known as the method of saving the hypothesis.  According to the Duhem-Quine principle, it is always possible to save a hypothesis from falsification or refutation.  This is due to the logical nature of the testing process.  To show this, a little technical detail is required.  When a hypothesis is tested, the conditions in which the test is conducted such as the experimental or field conditions, the assumptions of the influence of the observer, and the like are also under test.  The experimental apparatus may be wrong, or the investigator may be unconsciously influencing the experiment or observation, or the test apparatus may be faulty.  This list can be extended ad infnitum, but for all practical purposes it is inevitably finite and small.  The logic of the situation is this.  Suppose you have a hypothesis H and from it you can deduce a proposition concerning an event E.  In the testing scenario illustrated above, you assume H to be true and look to see whether E is true or not.  If you find E, while you have not proved or verified that H is the case, you have, as Popper would have said, corroborated E.  That is, you have made the truth of E appear more likely.

Now, let us suppose that on the assumption that H is the case, you fail to observe E.  One can infer from this that H or something else being assumed is not the case.  The assumptions consist of H & C &B & Q, where C denotes the experimental or observational conditions, B the influence or bias of the observer, and Q any additional factors that might be influencing the outcome of the test.  So, if E is not observed, instead of falsifying H, you can save the hypothesis by rejecting C or B or Q.  You can then claim that E really does follow from H;  it is just that this test failed to substantiate this particular outcome because it was flawed.

As Duesenberry discusses, there is another factor involved in the testing of a hypothesis.  And this is that even should you succeed in corroborating H, all you have shown is that for the data at hand and under the conditions of the test, H seems to explain the data better than a set of alternatives, not that it is true simpliciter.  This state of affairs can, however, change and another hypothesis can take the place of H as the favored one.  This process of replacement can be highly contentious.

As Duesenberry himself notes, the data available to economists is often not very good.  Not only that but the variation inherent in such data remains unanalyzed more often than not.  Economists often present data in the absence of error coefficients and the like.  They also do not conduct statistical hypothesis tests of data even when it is not obvious, from ‘eye-balling’ the data, that H0 explains the data better than some alternative from a set of alternative hypotheses, H1, …, Hn, under the conditions of such an informal test scenario.  They appear to assume that the data ‘speak for themselves’, which they do not.  Data, to make sense, must be interpreted and that means placing the data in an interpretive context, that is, a theoretical context.  Otherwise,  there is no difference between a set of data and a list of numbers or names in a phone book.  In saying this, I am not arguing that statistical hypothesis testing is essential, only that it is not carried out even when it would appear to be helpful.  Irrespective of this, data should never be presented in the absence of error coefficients, unless the data differences obviously swamp any inevitable errors the data set may contain.  But how often is this going to be the case?

I must mention that this is not always the case in the present nor in the past.  Duesenberry (1949) himself cites references with statistical content – notably Keynes’ ‘A Statistical Testing of Business Cycle Theories’ (1939), Trygve Haavelmo’s’ The Probability Approach in Econometrics’ (1944), and G. Udny Yules’ ‘Why Do We Sometimes Get Nonsense Correlations’ (1926), along with eminent social psychologists such as Abram Kardner and Leon Festinger, the latter of whom’s Theory of Cognitive Dissonance has influenced Akerlof, the psychoanalyst Karen Horney (The Neurotic Personality of Our Time, 1937), and the social scientist Thorsten Veblen (Theory of the Leisure Class, 1934).  There is no reference to Talcott Parsons, who was probably one of the most famous Harvard sociologists (in the Department of Social Relations) with an economic background at the time of the publication of Duesenberry’s Income, Saving and the Theory of Consumer Behavior (1949).  It may be that, although both were at Harvard at this time, Duesenberry felt that Parsons’ approach, which was rather idiosyncratic, was rather tangential to his own.  I will come back to this issue regarding the different and possibly not easily reconcilable approaches of sociologists, anthropologists and economists to the fields of economics and political economy.

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