Who thinks probability is just a number? A plea.

Many people think – perhaps they were taught it – that it is meaningful to talk about the unconditional probability of ‘Heads’ (I.e. P(Heads)) for a real coin, and even that there are logical or mathematical arguments to this effect. I have been collecting and commenting on works which have been – too widely – interpreted in this way, and quoting their authors in contradiction. De Finetti seemed to be the only example of a respected person who seemed to think that he had provided such an argument. But a friendly economist has just forwarded a link to a recent work that debunks this notion, based on wider  reading of his work.

So, am I done? Does anyone have any seeming mathematical sources for the view that ‘probability is just a number’ for me to consider?

I have already covered:

There are some more modern authors who make strong claims about probability, but – unless you know different – they rely on the above, and hence do not need to be addressed separately. I do also opine on a few less well known sources: you can search my blog to check.

Dave Marsay

Traffic bunching

In heavy traffic, such as on motorways in rush-hour, there is often oscillation in speed and there can even be mysterious ‘emergent’ halts. The use of variable speed limits can result in everyone getting along a given stretch of road quicker.

Soros (worth reading) has written an article that suggests that this is all to do with the humanity and ‘thinking’ of the drivers, and that something similar is the case for economic and financial booms and busts. This might seem to indicate that ‘mathematical models’ were a part of our problems, not solutions. So I suggest the following thought experiment:

Suppose a huge number of  identical driverless cars with deterministic control functions all try to go along the same road, seeking to optimise performance in terms of ‘progress’ and fuel economy. Will they necessarily succeed, or might there be some ‘tragedy of the commons’ that can only be resolved by some overall regulation? What are the critical factors? Is the nature of the ‘brains’ one of them?

Are these problems the preserve of psychologists, or does mathematics have anything useful to say?

Dave Marsay

Flood Risk Puzzle

This is a ‘Natural Hazards problem’ that has been used to explore people’s understanding of risk. As usual I question the supposedly ‘mathematical’ answer.

 Suppose that the probability that your house will be hit one or more times by the natural hazard during an exposure period of one year is .005. That is, if 1000 homes like yours were exposed to the natural hazard for one year, 5 of the homes would be damaged. Please estimate the probability that your home would avoid being hit by the natural hazard if exposed to the hazard for a period of 5/10/25/50 years.

You might like to ponder it for yourself.

In elementary probability theory the appropriate formula is 1-(1-p)n, where p=0.005 is the probability for one year and n is the number of years. But is this an appropriate calculation?

My in-laws were being charged a high property insurance premium because homes ‘like theirs’ were liable to flooding. They appealed, and are now paying a more modest premium. The problem is that a probability is rarely objective and individual, but experience-dependent on some classification. UK insurers rely on flood reports and surveys that are based on postcodes. Thus ‘properties like yours have a 0.5% risk of flood in any one year’ would really mean that some properties with your postcode have flooded, or have been assessed at being at risk from flooding. But, if like my in-laws your property is at the higher end of the postcode, this may not be at all appropriate. Just because the insurance company thinks that your risk of flooding is 0.5% does not mean that you should think the same.

Even from the insurance company’s perspective, the calculation is wrong. It would be correct if all post-codes were homogenous in terms of risk, but that clearly isn’t so. If a property hasn’t had a flood for 20 years then it is more likely to be at less risk (e.g., on higher ground) than those that have flooded. Hence its risk of flooding in the future is reduced. Taking more extreme figures, suppose that there are 2 houses on a river bank that flood every year and 8 on a hill that never flood. The chance of a house selected at random being flooded at some time in any long period is just 20%. And if you know that your house is an hill, then for you the probability may be 0%. In less extreme cases – typical of reality – the elementary formula also tends to overstate the risk. But the main point is that – contrary to the elementary theory – one shouldn’t just take probability estimates at face value. This could save you money!

See Also

Similar puzzles.

Dave Marsay

Disease

“You are suffering from a disease that, according to your manifest symptoms, is either A or B. For a variety of demographic reasons disease A happens to be nineteen times as common as B. The two diseases are equally fatal if untreated, but it is dangerous to combine the respectively appropriate treatments. Your physician orders a certain test which, through the operation of a fairly well understood causal process, always gives a unique diagnosis in such cases, and this diagnosis has been tried out on equal numbers of A- and B-patients and is known to be correct on 80% of those occasions. The tests report that you are suffering from disease B. Should you nevertheless opt for the treatment appropriate to A … ?”

My thoughts below …

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If, following Good, we use

P(A|B:C) to denote the odds of A, conditional on B in the context C, Odds(A1/A2|B:C) to denote the odds P(A1|B:C)/P(A2|B:C), and LR(B|A1/A2:C) to denote the likelihood ratio, P(B|A1:C)/P(B|A2:C).

then we want

Odds(A/B | diagnosis of B : you), given
Odds(A/B : population) and
P(diagnosis of B | B : test), and similarly for A.

This looks like a job for Bayes’ rule! In Odds form this is

Odds(A1/A2|B:C) = LR(B|A1/A2:C).Odds(A1/A2:C).

If we ignore the dependence on context, this would yield

Odds(A/B | diagnosis of B ) = LR(diagnosis of B | A/B ).Odds(A/B).

But are we justified in ignoring the differences? For simplicity, suppose that the tests were conducted on a representative sample of the population, so that we have Odds(A/B | diagnosis of B : population), but still need Odds(A/B | diagnosis of B : you). According to Blackburn’s population indifference principle (PIP) you ‘should’ use the whole population statistics, but his reasons seem doubtful. Suppose that:

  • You thought yourself in every way typical of the population as a whole.
  • The prevalence of diseases among those you know was consistent with the whole population data.

Then PIP seems more reasonable. But if you are of a minority ethnicity – for example – with many relatives, neighbours and friends who share your distinguishing characteristic, then it might be more reasonable to use an informal estimate based on a more appropriate population, rather than a better quality estimate based on a less appropriate estimate. (This is a kind of converse to the availability heuristic.)

See Also

My notes on Cohen for a discussion of alternatives.

Other, similar, Puzzles.

My notes on probability.

Dave Marsay

Cab accident

“In a certain town blue and green cabs operate in a ratio of 85 to 15, respectively. A witness identifies a cab in a crash as green, and the court is told [based on a test] that in the relevant light conditions he can distinguish blue cabs from green ones in 80% of cases. [What] is the probability (expressed as a percentage) that the cab involved in the accident was blue?” (See my notes on Cohen for a discussion of alternatives.)

For bonus points …. if you were involved , what questions might you reasonably ask before estimating the required percentage? Does your first answer imply some assumptions about the answers, and are they reasonable?

My thoughts below:

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If, following Good, we use

P(A|B:C) to denote the odds of A, conditional on B in the context C,
Odds(A1/A2|B:C) to denote the odds P(A1|B:C)/P(A2|B:C), and
LR(B|A1/A2:C) to denote the likelihood ratio, P(B|A1:C)/P(B|A2:C).

Then we want P(blue| witness: accident), which can be derived by normalisation from Odds(blue/green| witness : accident).
We have Odds(blue/green: city) and the statement that the witness “can distinguish blue cabs from green ones in 80% of cases”.

Let us suppose (as I think is the intention) that this means that we know Odds(witness| blue/green: test) under the test conditions. This looks like a job for Bayes’ rule! In Odds form this is

Odds(A1/A2|B:C) = LR(B|A1/A2:C).Odds(A1/A2:C),

as can be verified from the identity P(A|B:C) = P(A&B:C)/P(B:C) whenever P(B:C)≠0.

If we ignore the contexts, this would yield:

Odds(blue/green| witness) = LR(witness| blue/green).Odds(blue/green),

as required. But this would only be valid if the context made no difference. For example, suppose that:

  • Green cabs have many more accidents than blue ones.
  • The accident was in an area where green cabs were more common.
  •  The witness knew that blue cabs were much more common than green and yet was still confident that it was a green cab.

In each case, one would wish to re-assess the required odds. Would it be reasonable to assume that none of the above applied, if one didn’t ask?

See Also

Other Puzzles.

My notes on probability.

Dave Marsay

Unbirthday Paradox

Another puzzle, courtesy of a mathematics lecture I attended last night. It is  variant of the Birthday ‘Paradox‘. The original ‘paradox’ is that a typical group of people is much more likely to contain two people that share a birthday than most people would think. The variant was where 20 people were asked to pick an integer between 1 and 100 and it was found that two had picked ’42′. The mathematics is the same as for the birthday problem. But is it right?

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Are there any unwarranted assumptions?

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The ‘official’ (Wikipedia) answer to the birthday paradox would be correct if people were randomly selected from a population whose birthdays were uniformly distributed through the year. But are they? This is not a mathematical question, so it cannot be that the answer provided is ‘mathematically’ correct, can it? But one could perhaps say is that the answer provided would be correct if the assumptions were true, and would be approximately correct if they were approximately true – but a sensitivity analysis would be revealing.

The variant brings in greater uncertainties. For example, before the experiment we all guessed the probability. We did much better than the previous audience. Could this be relevant? In any case, why should we expect the guesses to be evenly distributed? Might there not be lucky numbers or other special numbers – such as 42 – that would be chosen?

If numbers were clumped for any reason, the probability of a match significantly increases. I can imagine lots of reasons why numbers should be clumped, but none why they should be anti-clumped, so it seems to me that the ‘official’ probability is actually at the lower end of a range of possible probabilities. Thus if the official probability is 83% I would consider 91% (= (83%+100%)/2) a better guess, and [83%,100%] better still.

The calculations are simpler if we consider the possibility of a match when we toss a coin twice. If P(Heads) = 0.5+e then

P(Match) = (0.5+e)2 + (0.5-e)2 = 0.5 + 2.e2. Thus 0.5 is a lower bound on the probability of a match, provided that coin tosses are independent.

See Also

More Puzzles.

….Dave Marsay

JIC, Syria and Uncertainty

This page considers the case that the Assad regime used gas against the rebels on 21st August 2013 from a theory of evidence perspective. For a broader account, see Wikipedia.

The JIC Assessment

The JIC concluded on 27th that it was:

highly likely that the Syrian regime was responsible.

In the covering letter (29th) the chair said:

Against that background, the JIC concluded that it is highly likely that the regime was responsible for the CW attacks on 21 August. The JIC had high confidence in all of its assessments except in relation to the regime’s precise motivation for carrying out an attack of this scale at this time – though intelligence may increase our confidence in the future.

A cynic or pedant might note the caveat:

The paper’s key judgements, based on the information and intelligence available to us as of 25 August, are attached.

Mathematically-based analysis

From a mathematical point of view, the JIC report is an ‘utterance’, and one needs to consider the context in which it was produced. Hopefully, best practice would include identifying the key steps in the conclusion and seeking out and hastening any possible contrary reports. Thus one might reasonably suppose that the letter on the 29th reflected all obviously relevant information available up to the ends of the 28th, but perhaps not some other inputs, such as ‘big data’, that only yield intelligence after extensive processing and analysis.

But what is the chain of reasoning (29th)?

It is being claimed, including by the regime, that the attacks were either faked or undertaken by the Syrian Armed Opposition. We have tested this assertion using a wide range of intelligence and open sources, and invited HMG and outside experts to help us establish whether such a thing is possible. There is no credible intelligence or other evidence to substantiate the claims or the possession of CW by the opposition. The JIC has therefore concluded that there are no plausible alternative scenarios to regime responsibility.

The JIC had high confidence in all of its assessments except in relation to the regime’s precise motivation for carrying out an attack of this scale at this time – though intelligence may increase our confidence in the future.

The report of the 27th is more nuanced:

There is no credible evidence that any opposition group has used CW. A number continue to seek a CW capability, but none currently has the capability to conduct a CW attack on this scale.

Russia claims to have a ‘good degree of confidence’ that the attack was an ‘opposition provocation’ but has announced that they support an investigation into the incident. …

In contrast, concerning Iraqi WMD, we were told that “lack of evidence is not evidence of lack”. But mathematics is not so rigid: it depends on one’s intelligence sources and analysis. Presumably in 2003 we lacked the means to detect Iraqi CW, but now – having learnt the lesson – we would know almost as soon as any one of a number of disparate groups acquires CW.  Many outside the intelligence community might not find this credible, leading to a lack of confidence in the report. Others would take the JIC’s word for it. But while the JIC may have evidence that supports their rating, it seems to me that they have not even alluded to a key part of it.

Often, of course, an argument may be technically flawed but still lead to a correct conclusion. To fix the argument one would want a much greater understanding of the situation. For example, the Russians seem to suggest that one opposition group would be prepared to gas another, presumably to draw the US and others into the war. Is the JIC saying that this is not plausible, or simply that no such group (yet) has the means? Without clarity, it is difficult for an outsider to asses the report and draw their own conclusion.

Finally, it is notable that regime responsibility for the attack of the 21st is rated ‘highly likely’, the same as their responsibility for previous attacks. Yet mathematically the rating should depend on what is called ‘the likelihood’, which one would normally expect to increase with time. Hence one would expect the rating to increase from possible (in the immediate aftermath) through likely to highly likely, as the kind of issues described above are dealt with. This unexpectedly high rating calls for an explanation, which would need to address the most relevant factors.

Anticipating the UN Inspectors

The UN weapons inspectors are expected to produce much relevant evidence. For example, it may be that even if an opposition group had CW an attack would necessarily lack some key signatures. But, from a mathematical point of view, one cannot claim that one explanation is ‘highly likely’ without considering all the alternatives and taking full account of how the evidence was obtained. It is quite true, as the PM argued, that there will always be gaps that require judgement to span. But we should strive to make the gap as slight as possible, and to be clear about what it is. While one would not want a JIC report to be phrased in terms of mathematics, it would seem that appropriate mathematics could be a valuable aid to critical thinking. Hopefully we shall soon have an assessment that genuinely rates ‘highly likely’ independently of any esoteric expertise, whether intelligence or mathematics.

Updates

30th August: US

The US assessment concludes that the attack was by Assad’s troops, using rockets to deliver a nerve agent, following their usual procedures. This ought to be confirmed or disconfirmed by the inspectors, with reasonable confidence. Further, the US claim ‘high confidence’ in their assessment, rather than very high confidence. Overall, the US assessment appears to be about what one would expect if Assad’s troops were responsible.

31st August: Blog

There is a good private-enterprise analysis of the open-source material. It makes a good case that the rockets’ payloads were not very dense, and probably a chemical gas. However, it points out that only the UN inspectors could determine if the payload was a prohibited substance, or some other substance such as is routinely used by respectable armies and police forces.

It makes no attribution of the rockets. The source material is clearly intended to show them being used by the Assad regime, but there is no discussion of whether or not any rebel groups could have made, captured or otherwise acquired them.

2nd September: France

The French have declassified a dossier. Again, it presents assertion and argumentation rather than evidence. The key points seem to be:

  • A ‘large’ amount of gas was used.
  • Rockets were probably used (presumably many).
  • No rebel group has the ability to fire rockets (unlike the Vietcong in Vietnam).

This falls short of a conclusive argument. Nothing seems to rule out the possibility of an anti-Assad outside agency loading up an ISO container (or a mule train) with CW (perhaps in rockets), and delivering them to an opposition group along with an adviser. (Not all the opposition groups all are allies.)

4th September: Germany

A German report includes:

  • Conjecture that the CW mix was stronger than intended, and hence lethal rather than temporarily disabling.
  • That a Hezbollah official said that Assad had ‘lost his nerve’ and ordered the attack.

It is not clear if the Hezbollah utterance was based on good grounds or was just speculation.

4th September: Experts

Some independent experts have given an analysis of the rockets that is similar in detail to that provided by Colin Powell to the UN in 2003, providing some support for the official dossiers. They asses that each warhead contained 50 litres (13 gallons) of agent. The assess that the rebels could have constructed the rockets, but not produced the large quantity of agents.

No figure is given for the number of rockets, but I have seen a figure of 100, which seems the right order of magnitude. This would imply 5,000 litres or 1,300 gallons, if all held the agent. A large tanker truck has a capacity of about 7 times this, so it does not seem impossible that such an amount could have been smuggled in.

This report essentially puts a little more detail on the blog of 31st August, and is seen as being more authoritative.

5th September: G20

The UK has confirmed that Sarin was used, but seems not to have commented on whether it was of typical ‘military quality’, or more home-made.

Russia has given the UN a 100 page dossier of its own, and I have yet to see a credible debunking (early days, and I haven’t found it on-line).

The squabbles continue. The UN wants to wait for its inspectors.

6th September: Veteran Intelligence Professionals for Sanity

An alternative, unofficial narrative. Can this be shown to be incredible? Will it be countered?

9th September: German

German secret sources indicate that Assad had no involvement in the CW attack (although others in the regime might have).

9th September: FCO news conference

John Kerry, at a UK FCO news conference, gives very convincing account of the evidenced for CW use, but without indicating any evidence that the chemicals were delivered by rocket. He is asked about Assad’s involvement, but notes that all that is claimed is senior regime culpability.

UN Inspectors’ Report

21st September. The long-awaited report concludes that rockets were used to deliver Sarin. The report, at first read, seems professional and credible. It is similar in character to the evidence that Colin Powell presented to the UN in 2003, but without the questionable ‘judgments’. It provides some key details (type of rocket, trajectory) which – one hopes – could be tied to the Assad regime, especially given US claims to have monitored rocket launches. Otherwise, they appear to be of  type that the rebels could have used.

The report does not discuss the possibility, raised by the regime, that conventional rockets had accidentally hit a rebel chemical store, but the technical details do seem to rule it out. There is an interesting point here. Psychologically, the fact that the regime raised a possibility in their defence which has been shown to be false increases our scepticism about them. But mathematically, if they are innocent then we would not expect them to know what happened, and hence we would not expect their conjectures to be correct. Such a false conjecture could even be counted as evidence in their favour, particularly if we thought them competent enough to realise that such an invention would easily be falsified by the inspectors.

Reaction

Initial formal reactions

Initial reactions from the US, UK and French are that the technical details, including the trajectory, rule out rebel responsibility. They appear to be a good position to make such a determination, and it would normally be a conclusion that I would take at face value. But given the experience of Iraq and their previous dossiers, it seems quite possible that they would say what they said even without any specific evidence. A typical response, from US ambassador to the UN Samantha Power was:

The technical details of the UN report make clear that only the regime could have carried out this large-scale chemical weapons attack.”

Being just a little pedantic, this statement is literally false: one would at least have to take the technical details to a map showing rebel and regime positions, and have some idea of the range of the rockets. From the Russian comments, it would seem they have not been convinced.

Media reaction

A Telegraph report includes:

Whether the rebels have captured these delivery systems – along with sarin gas – from government armouries is unknown. Even if they have, experts said that operating these weapons successfully would be exceptionally difficult.

”It’s hard to say with certainty that the rebels don’t have access to these delivery systems. But even if they do, using them in such a way as to ensure that the attack was successful is the bit the rebels won’t know how to do,” said Dina Esfandiary, an expert on chemical weapons at the International Institute for Strategic Studies.

The investigators had enough evidence to trace the trajectories followed by two of the five rockets. If the data they provide is enough to pinpoint the locations from which the weapons were launched, this should help to settle the question of responsibility.

John Kerry, the US secretary of state, says the rockets were fired from areas of Damascus under the regime’s control, a claim that strongly implicates Mr Assad’s forces.

This suggests that there might be a strong case against the regime. But it is not clear that the government would be the only source of weapons for the rebels, that the rebels would need sophisticated launchers (rather than sticks) or that they would lack advice. Next, given the information on type, timing and bearing it should be possible to identify the rockets, if the US was monitoring their trajectories at the time, and hence it might be possible to determine where they came from, in which case the evidence trail would lead strongly to the regime. (Elsewhere it has been asserted that one of the rockets was fired from within the main Syrian Army base, in which case one would have thought they would have noticed a rebel group firing out.)

17 September: Human Rights Watch

Human Rights Watch has marked the UN estimate of the trajectories on a map, clearly showing tha- they could have been fired from the Republican Guard 104 Brigade area.

Connecting the dots provided by these numbers allows us to see for ourselves where the rockets were likely launched from and who was responsible.

This isn’t conclusive, given the limited data available to the UN team, but it is highly suggestive and another piece of the puzzle.

This seems a reasonable analysis. The BBC has said of it:

Human Rights Watch says the document reveals details of the attack that strongly
suggest government forces were behind the attack.

But this seems to exaggerate the strength of the evidence. One would at least want to see if the trajectories are consistent with the rockets having been launched from rebel held areas (map, anyone?) It also seems a little odd that a salvo of M14 rockets appear to have been fired over the presidential palace. Was the Syrian Army that desperate? Depending on the view that one takes of these questions, the evidence could favour the rebel hypothesis. On the other hand, if the US could confirm that the only rockets fired at that time to those sites came from government areas, that would seem conclusive.

(Wikipedia gives technical details of rockets. It notes use by the Taliban, and quotes its normal maximum range as 9.8km. The Human Rights Watch analysis seems to be assuming that this will not be significantly reduced by the ad-hoc adaptation to carry gas. Is this credible? My point here is that the lack of explicit discussion of such aspects in the official dossiers leaves room for doubt, which could be dispelled if their ‘very high confidence’ is justified.)

18 September: Syrian “proof”

The BBC has reported that the Syrians have provide what they consider proof to the Russia that the rebels were responsible for the CW attack, and that the Russians are evaluating it. I doubt that this will be proof, but perhaps it will reduce our confidence in  the ‘very high’ likelihood that the regime was responsible. (Probably not!) It may, though, flush out more conclusive evidence, either way.

19 September: Forgery?

Assad has claimed that the materials recovered by the UN inspectors were forged. The report talks about rebels moving material, and it is not immediately clear, as the official dossiers claim, that this hypothesis is not credible, particularly if the rebels had technical support.

Putin has confirmed that the rockets used were obsolete Soviet-era ones, no longer in use by the Syrian Army.

December: US Intelligence?

Hersh claims that US had intelligence that the Syrian rebels had chemical weapons, and that the US administration  deliberately ‘adjusted’ the intelligence to make it appear much more damning of the Syrian regime. (This is disputed.)

Comment

The UN Inspectors report is clear about what it has found. It is careful not to make deductive leaps, but provides ample material to support further analysis. For example, while it finds that Sarin was delivered by rockets that could have been launched from a regime area, it does not rule out rebel responsibility. But it does give details of type, time and direction, such that if – as appears to be the case from their dossier – the US were monitoring the area, it should be possible to conclude that the rocket was actually fired by the regime. Maybe someone will assemble the pieces for us.

My own view is not that Assad did not do it or that we should not attack, but that any attack based on the grounds that Assad used CW should be supported by clear, specific evidence, which the dossiers prior to the UN report did not provide. Even now, we lack a complete case. Maybe the UN should have its own intelligence capability? Or could we attack on purely humanitarian grounds, not basing the justification on the possible events on 21 Aug? Or share our intelligence with the Russians and Chinese?

Maybe no-one is interested any more?

See Also

Telegraph on anti-spy cynicism. Letters. More controversially: inconclusive allegations. and an attempted debunking.

Discussion of weakness of case that Assad was personally involved. Speculation on UN findings.

A feature of the debate seems to be that those who think that ‘something must be done’ tend to be critical of those who question the various dossiers, and those who object to military action tend to throw mud at the dossiers, justified or not. So maybe my main point should be that, irrespective of the validity of the JIC assessment, we need a much better quality of debate, engaging the public and those countries with different views, not just our traditional allies.

A notable exception was a private blog, which looked very credible, but fell short claiming “high likelihood”. It gives details of two candidate delivery rockets, and hoped that the UN inspectors will have got evidence from them, as they did. Neither rocket was known to have been used, but neither do they appear to be beyond the ability of rebel groups to use (with support). The comments are also interesting, e.g.:

There is compelling evidence that the Saudi terrorists operating in Syria, some having had training from an SAS mercenary working out of Dubai who is reporting back to me, are responsible for the chemical attack in the Ghouta area of Damascus.

The AIPAC derived ‘red line’ little game and frame-up was orchestrated at the highest levels of the American administration and liquid sarin binary precursors mainly DMMP were supplied by Israeli handled Saudi terrorists to a Jabhat al-Nusra Front chemist and fabricator.

Israel received supplies of the controlled substance DMMP from Solkatronic Chemicals of Morrisville, Pa.

This at least has some detail, although not such as can be easily checked.

Finally, I am beginning to get annoyed by the media’s use of scare quotes around Russian “evidence”.

Dave Marsay

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