Author Archives: Stubborn Mule

Will vaccinated people end up in ICU?

When I posted about Australia’s slow vaccination progress we were ranked dead last among OECD countries in the full vaccination “race“. We have now nosed past South Korea into second last place. Sydney’s COVID outbreak and lockdown is certainly a factor, but whatever the reason, the increased rate of vaccination we’re seeing is good news.

The Sydney outbreak is also leading to growing numbers of COVID victims being admitted to intensive care units (ICUs) and, sadly, to a number of deaths. So far, while a small number of the patients in ICU have had one vaccination shot, none have been fully vaccinated. But based on international experience, it’s only a matter of time before that will change. We should also expect to see some deaths of fully vaccinated people. I am sure that the minute that happens there will be enormous media attention, but will it actually say anything about the effectiveness of the vaccine?

In fact, it won’t. It will simply be a statistical consequence of an increasingly vaccinated population. It’s sobering to recall that in the race to develop vaccines last year, 50% efficacy in trials was set as the hurdle for regulatory approval. As it turned out, the vaccines all did far better with trial efficacies around 80-90%. These are extremely good vaccines, but they are still not 100% effective at preventing disease, hospitalisation or even death. As more and more people are vaccinated, we should therefore expect to see a higher proportion of fully vaccinated patients in ICU or dying (see chart below). Of course, many people will still see this outcome as an indictment of the vaccines. Friend of the Mule, Dan, gave me this ironic analogy for this confusion: “Now that nearly everyone in Australia wears seatbelts, it turns out that nearly 100% of people who die in road accidents were wearing their seatbelts. Proving that seatbelts are not effective anyway in preventing road fatalities!”.

Expect to see more vaccinated people in ICU as vaccination rates rise

Focusing on the proportion of people in ICU or dying who are vaccinated misses the important point here: as vaccination rates increase, the total number of people in hospital or dying will be dropping. John Burn-Murdoch of the Financial Times illustrates this point nicely with a clever infographic.

How much of a reduction on ICU admissions should we expect to see as a result of increased rates of vaccination? It will depend on how effective the vaccines are at preventing severe disease. While current estimates of the effectiveness of the vaccines are around 80% (with some variation across the vaccines), there is some evidence that the effectiveness of preventing severe disease is over 90%. The impact of vaccines on hospitalisation will also depend on how much COVID is circulating in the community: when the risk of contracting COVID is low, even unvaccinated people are unlikely to end up in hospital.

The chart below shows the expected effect of vaccination on rates of admission to ICU for four different assumed effectiveness rates of the vaccine and for three different scenarios for the prevalence of COVID in the community. These scenarios are taken from the Australian Technical Advisory Group on Immunisation (ATAGI) paper “Weighing up the potential benefits against risk of harm from COVID-19 Vaccine AstraZeneca“. The “low” risk scenario corresponds to an infection rate similar to the first COVID wave in Australia (29 infections per 100,000 over 16 weeks), “medium” to an infection rate similar to the Victorian second wave (275 per 100,000 over 16 weeks) and “high” is similar to Europe in January 2021 (3,544 per 100,000 over 16 weeks).

When the risk of contracting COVID is high, higher vaccination rates will dramatically reduce hospitalisation. This is particularly evident in the UK experience, where COVID peaked well above the “high” scenario in this chart in January 2021 and then again in July. In the intervening six months Britain has had one of the more successful vaccine roll-outs and, as a result, hospitalisation and death rates in July were a fraction of the January experience.

I hope Sydney’s lockdown will be successful in containing the current outbreak, but if it’s not, the faster we roll out the vaccine, the fewer people will end up in ICU and the fewer people will die.

Vaccination of the Nation

To date Australia has fared relatively well by international standards in terms of its COVID-19 infection rates. Now, however, its vaccination progress does not compare so favourably to other countries. Charts showing Australia languishing at the bottom of a vaccination league table have been circulating widely online. The specimen below, which appeared in last weekend’s Saturday Paper, is a typical example.

(Click image to expand for full view)

The challenge when producing charts like this is that there are far too many countries to squeeze onto a bar chart, requiring a somewhat arbitrary choice of the countries to include. This chart suggests that Australia’s full vaccination rate is the worst and Chile’s is the best. Australia does not in fact have the lowest rate in the world, although it is the lowest among OECD countries. Chile, at the other end, is certainly doing very well, but is not the best in the world – that honour sits with Gibraltar (or Malta if we restrict to sovereign nations). Even considering only OECD countries, the chart omits Iceland and Israel which are both a little ahead of Chile.

So, how can we get a better picture covering all countries around the world? One approach is to use a map instead of a bar chart. The map below shows countries grouped into quintiles (five of them, of course), ranging from the 20% of countries with the lowest rates of vaccination through to the 20% of countries with the highest rates. On this basis, with a full vaccination rate of 7%, Australia just scrapes into the middle quintile along with countries with full vaccination rates ranging from 6-18%. So Australia may not be last in this “race”, but the countries with lower rates are all far poorer than Australia.

The striking band of top quintile blue between Russia and China is Mongolia, which has an interesting story behind its vaccination success (thanks to Dan for alerting me to that article).

World Map of Vaccination Rate Quintiles
(Click image to expand for full view)

Some notes on the data:

  1. The data here is all sourced from Our World in Data (OWID), which in turn sources data from national authorities. A significant number of countries do not appear in the dataset and these are likely to be countries with very low rates of vaccination. If this data were available it would push up Australia’s ranking.
  2. “Countries” includes dependencies (such as Gibraltar) and disputed territories not just widely recognised sovereign nations.
  3. OWID reports a vaccination rate for Gibraltar of over 100%. This appears to be because vaccination figures include guest workers and, given Gibraltar’s small population, this has a big impact on the rate. Since there continue to be some cases of COVID-19 among unvaccinated people in Gibraltar, the true rate must be below 100% but as I have seen some estimates that the figure is more like 90%, Gibraltar is probably still in the lead.

COVID-19 by Suburb in New South Wales

The New South Wales Department of Health has now released a breakdown of COVID-19 data by suburb. The website – and much of the media reporting of this data – displays this in the form of colour-coded maps, highlighting the hotspots. But the data is also available in more detail as part of the government’s open data initiative. This provides the opportunity to explore the data in different ways.

As an example, the chart below shows the evolution of confirmed COVID-19 cases over time for all postcodes with more than 20 total cases. This shows clearly the impact of the social distancing measures from early April. Many areas have seen no new cases since mid April. The large spot that pushed Caddens near the top of the list is result of the cases in the Anglicare Newmarch aged care nursing home.

Note that most of the postcodes in New South Wales include multiple suburbs. Here I have picked a single representative suburb to label the chart.

What’s Going On In Sweden?

Reportedly, Sweden has not gone into a COVID-19 lockdown, unlike its neighbours. While I am sure this is a deliberate policy choice, it will also serve as an interesting epidemiological experiment to test the effectiveness of different social response measures.

It is still early in the course of this international experiment, but a look at the growth in cases intially suggests that Sweden’s strategy is not leading to a significantly higher infection rate.

COVID-19 cases

However, confirmed cases are difficult to compare across countries as they can be heavily influenced by the testing regimes each country implements How many tests are performed? Do they target particular groups (international travellers, those in contact with confirmed cases or are they driven by symptoms? It is very hard to account for these factors. Instead it is easier to compare the number of deaths. While there can still be differences (Under what circumstances are post-mortem deaths tested for COVID-19? How are co-morbidities accounted for? Are there more older citizens?), I think it is reasonable to expect less significant variation across neighbouring countries.

Looking at deaths, Sweden looks far worse than its neighbours. Sweden has approximately the same number of confirmed cases as Norway, but more than five times the number of deaths.

COVID-19 deaths

Interestingly, Sweden’s case count is very close to Australia’s, but Australia has to date seen 23 deaths, compared to 239 in Sweden – almost 10 times as many.

This suggests either that Sweden’s confirmed case count is significantly understated – it would be understated everywhere but likely more so in Sweden than elsewhere – or Sweden is suffering a far higher mortality rate.

The experiment is not yet over, but so far Sweden’s no lockdown strategy does not seem to be working.

COVID-19 data

There is no shortage of commentary on COVID-19 online and off. There is also an abundance of data available, which is as good a reason as any for the first Mule post of 2020.

One of the best data resources online is the Johns Hopkins dashboard created by the Center for Systems Science and Engineering (CSSE). For those interested in performing their own analysis, the CSSE has also made the underlying data available in a Github repository.

It has become commonplace to refer to the “exponential growth” of the disease. For many, this is just short-hand for “really” fast. Others may recognise an exponential curve in the charts below, with the exception of South Korea.

Confirmed COVID-19 cases

However, displaying the case data in this way does not give a very clear picture of what is actually going on. A better way to display the data (New York Times) is to use a logarithmic scale. In these charts, the values on the vertical axis increase exponentially (the labels here are successive powers of 10) and pure exponential growth would appear as a straight line. Of course, the real world is never pure, so the COVID-19 data do not appear straight lines, but for a number of countries – including Australia – there are periods where it comes very close.

Confirmed COVID-19 cases (log scale)

When plotted on a logarithmic scale, the slope of the line corresponds to rate of growth. In late February, the slope of the curve for South Korea turns sharply up, as the number of cases exploded, only to flatten again as drastic measures began to slow the growth rate. In contrast, the slope of the curve for the USA is becoming steeper – disturbingly the rate on growth in confirmed cases is increasing.

To get a better sense of these growth rates, the charts below show the daily growth rate in confirmed cases. The data is noisy, so smoothed curves are added to give a sense of the trend. In mid-February, the number of confirmed cases was growing extremely rapidly. As containment measures were introduced, the growth rate was quickly reduced, but not before cases had reached the thousands. After that, the gains became harder fought and, while the growth rate is still falling, but only slowly and is currently around 12-13% daily. In contrast, South Korea has managed to bring its growth rate down to only 1%.

Growth rate of confirmed COVID-19 cases

Although the case count in Australia is still only in the hundreds, it is growing at a similar rate to that seen in Italy in mid-March. Confirmed cases inevitably lag actual cases, as detection takes time, and as a result there will be a lag in the impact of any containment measures. So, it is too early to tell how much impact the recently imposed travel and social distancing restrictions will have.

Everyone will be hoping these measures will help, but if they do not and the current growth rate persists in Australia, the confirmed case count in New South Wales will be over 7,000 in two weeks, around 30,000 in three weeks and heading towards 200,000 in a month. The prospects in the USA look scarier still: there are already over 5,000 cases and the daily growth rate is around 30%. If that rate doesn’t slow, in a month there will be around 50 million confirmed cases.

Here is hoping those log scale curves start to flatten.

Alive and kicking

It has been almost two years since there has been a new post here at the Mule, so you would be forgiven for thinking that the blog was defunct. But, I have now been prodded into action by the need to change my hosting provider.

For over 10 years a friend has very generously hosted the Mule on his servers but he is now shutting down the operation. So a big thank you to Brendan for those years of hosting.

Now that I have successfully moved the site to its new home, I will have to prod myself into action and start posting again more regularly.

Bitcoin and the Blockchain

It’s hard to believe that a whole year has passed since I last wrote on the topic of bitcoin, and my remaining 1 bitcoin is worth rather less than it was back then. During the week I presented at the Sydney Financial Mathematics Workshop on the topic of bitcoin, taking a rather more technical look at the mechanics of the blockchain than in my previous posts here on the Mule. For those who are interested in how Satoshi Nakamoto solved the “double spend” problem, here are the slides from that presentation.

Bitcoin and the Blockchain

As part of my preparation for the presentation, I read Bitcon: The Naked Truth About Bitcoin. If you are a bitcoin sceptic, you should enjoy the book. If you are a Bitcoin true believer, you will probably hate it. It is over-blown in parts and gets a few technical details wrong, but I am increasingly convinced by the core argument of the book: the blockchain is an extraordinary innovation which may well change the way money moves around the world, but bitcoin the currency will prove to be a fad.


Last year I wrote on a couple of occasions about the Sleeping Beauty problem. The problem raises some tricky questions and I did promise to attempt to answer the questions, which I am yet to do. Only last week, I was discussing the problem again with my friend Giulio, whose paper on the subject I published here. That discussion prompted me to go back to the inspiration for the original post: a series of posts on the Bob Walter’s blog. I re-read all of his posts, including his fourth post on the topic, which began:

I have been waiting to see some conclusions coming from discussions of the problem at the Stubborn Mule blog, however the discussion seems to have petered out without a final statement.

Sadly, even if I do get to my conclusions, I will not be able to get Bob’s reaction, because last week he died and the world has lost a great, inspirational mathematician.

Bob was my supervisor in the Honours year of my degree in mathematics and he also supervised Giulio for his PhD. Exchanging emails with Giulio this week, we both have vivid memories of an early experience of Bob’s inspirational approach to mathematics. This story may not resonate for everyone, but I can assure you that there are not many lectures from over 25 years ago that I can still recall.

The scene was a 3rd year lecture on Categories in Computer Science. Bob started talking about stacks, a very basic data structure used in computing. You should think of a stack of plates: you can put a plate on the top of the stack, or you can take one off. Importantly, you only push on or pop off plates from the top of the stack (unless you want your stack to crash to the floor). And how should a mathematician think about a stack? As Bob explained it, from the perspective of a category theorist, the key to understanding stacks is to think about pushing and popping as inverse functions. Bear with me, and I will take you through his explanation.

Rather than being a stack of plates, we will work with a stack of a particular type of data and I will denote by X the set of possible data elements (X could denote integers, strings, Booleans, or whatever data type you like). Stacks of type X will then be denoted by S. Our two key operations are push and pop.

The push operation takes an element of X and a stack and returns a new stack, which is just the old stack with the element of X added on the top. So, it’s a function push: X ×  → S. Conversely, pop is a function  → X ×  which takes a stack and returns the top element and a stack, which is everything that’s left after you pop the top.

So far, so good, but there are some edge cases to worry about. We should be able to deal with an empty stack, but what if we try to pop an element from the empty stack? That doesn’t work, but we can deal with this by returning an error state. This means that we should really think of pop as a function pop → X × I, where I is a single element set, say {ERR}. Here the + is a (disjoint) union of sets, which means that the pop function will either return a pair (an element of X and a stack) or an error state. This might be a bit confusing, so to make it concrete, imagine I have a stack s = (x1, x2, x3) then

pop((x1, x2, x3)) = (x1, (x2, x3))

and this ordered pair of data element xand (shorter) stack (x2, x3) is an element of X × S. Now if I want to pop an empty stack (), I have

pop(()) = ERR

which is in I. So pop will always either return an element of X × S or an element of I (in fact, the only element there is).

This should prompt us to revisit push as well, which should really be considered as a function push: X × S + I → S which, given an element of X and a stack will combine them, but given the single element of I will return an empty stack, so push(ERR) = ().

The key insight now is that pop and push are inverses of each other. If I push an element onto a stack and pop it again, I get back my element and the original stack. If I pop an element from a stack and push it back on, I get back my original stack. Extending these functions X × ensures that this holds true even for the edge cases.

But if push and pop are inverses then X × S + I  and S must essentially be the same—mathematically they are isomorphic. This is where the magic begins. As Bob said in is lecture, “let’s be bold like Gauss“, and he proceeded with the following calculation:

X × S + I = S

I = SX × S = S × (I – X)

S = I / (I – X)

and so

S = I + X + X2 + X3 + …

The last couple of steps are the bold ones, but actually make sense. The last equation basically says that a stack is either an empty stack, a single element of X, an ordered pair of elements of X, an ordered triple of elements of X and so on.

I’d known since high school that 1/(1-x) could be expanded to 1 +  + x2 + x3 + …, but applying this to a data structure like stacks was uncharted territory. I was hooked, and the following year I had the privilege of working with Bob on my honours thesis and that work ultimately made it into a couple of joint papers with Bob.

I haven’t seen Bob face to face for many years now, but we have occasionally kept in touch through topics of mutual interest on our blogs. While I have not kept up with his academic work, I have always considered him more than just a brilliant mathematician. He was a creative, inspirational, radical thinker who had an enormous influence on me and, I know, many others.

RFC Walters, rest in peace.

Musical Education

Musical EducationOn our longer family drives I take an old iPod crammed with even older music. Usually I take requests, and almost inevitably the children choose They Might Be Giants, and preferably the tracks Fingertips and Particle Man. But, our last trip was different. Instead I took the opportunity to the children some exposure to artists formative in the history of popular music. There is nothing like a grand plan to pass the time on the freeway.

Skimming through the albums, I decided that the best of The Jam would be a good place to start. It went down surprisingly well. Even our eldest, who generally prefers electronica, responded well to Eton Rifles. Marking that up as a success, the next choice was the best of Madness. This was more familiar territory, as they already knew (and loved) I Like Driving in My Car. Again it was successful.

Although this was a good start, it was not systematic, depending as it did on swift scanning through the albums on the iPod. So I have now begun to assemble a playlist on Spotify with a name as grandiose as its aim: Musical Education. The rules are simple but tough:

  1. Four representative tracks each (no more) are selected from major artists in the history of popular music.
  2. Each track must be from a different studio album. If the artist does not have at least four albums, refer step three. Singles not released on an album are also eligible.
  3. Single tracks can be included for important artists lacking the catalogue breadth for four essential tracks.

The playlist has nearly reached 150 tracks and includes artists such as The Doors, The Animals, James Brown and Prince. Inevitably, some choices reflect my own interests. My taste in Krautrock ensures the appearance of Kraftwerk, but in their defence I point to their appearance at the Tate and MOMA in recent years. Other choices may not have the endorsement of the artworld, but surely the sheer persistence of Mark E. Smith in continuing his post-punk aesthetic justifies a place for The Fall (Update: also The New Yorker rates The Fall highly too). As for XTC, well my own obsessions may be tilting the scales of significance. But perhaps not.

For some artists, choosing only four tracks is extremely difficult. Four David Bowie tracks…how? But rules are rules. Fortunately the toughest choice is taken away from me. The Beatles are not on Spotify, so they are ruled out on a technicality.

I have been road testing the list and there have been some surprises. The middle child has developed a strong interest in The Beach Boys, particularly God Only Knows (and that’s not just because of the BBC version), while the eldest has expressed a visceral dislike for James Brown. I did expect some bumps in the road of this musical journey: after all the boys refuse to let me play Nick Drake in the car (maybe one day they will learn they are wrong). Still, I am now getting requests for Hit the North, so something must be working.

This musical education is a work in progress, so I need help from all of you. Are there any big names I have missed? Let me know in the comments. Not all of the lists in the list are my own favourites, so I may have missed an essential track. Comments are open below, so please jump in!


Sleeping Beauty – a “halfer” approach

If you read the last post on the Sleeping Beauty problem, you may recall I did not pledge allegiance to either the “halfer” or the “thirder” camp, because I was still thinking my position through. More than a month later, I still can’t say I am satisfied. Mathematically, the thirder position seems to be the most coherent, but intuitively, it doesn’t seem quite right.

Mathematically the thirder position works well because it is the same as a simpler problem. Imagine the director of the research lab drops in to see how things are going. The director knows all of the details of the Sleeping Beauty experiment, but does not know whether today is day one or two of the experiment. Looking in, she sees Sleeping Beauty awake. To what degree should she believe that the coin toss was Heads? Here there is no memory-wiping and the problem fits neatly into standard applications of probability and the answer is 1/3.

My intuitive difficulty with the thirder is better expressed with a more extreme version of the Sleeping Beauty problem. Instead of flipping the coin once, the experimenters flip the coin 19 times. If there are 19 tails in a row (which has a probability of 1 in 524,288), Sleeping Beauty will be woken 1 million times. Otherwise (i.e. if there was at least one Heads tossed), she will only be woken once. Following the standard argument of the thirders, when Sleeping Beauty is awoken and asked for her degree of belief that the coin tosses turned up at least one Heads, she should say approximately 1/3 (or more precisely, 524287/1524287). Intuitively, this doesn’t seem right. Notwithstanding the potential for 1 million awakenings, I would find it hard to bet against something that started off as a 524287/524288 chance. Surely when Sleeping Beauty wakes up, she would be quite confident that at least one Heads came up and she is in the single awakening scenario.

Despite the concerns my intuition throws up, the typical thirder argues that Sleeping Beauty should assign 1/3 to Heads on the basis that she and the director have identical information. For example, here is an excerpt from a comment by RSM on the original post:

I want to know if halfers believe that two people with identical information about a problem, and with an identical set of priors, should assign identical probabilities to a hypothesis. I see the following possibilities:

  1. The answer is no -> could be a halfer (but not necessarily).
  2. The answer is yes, but the person holds that conditionalization is not a valid procedure –> could be a halfer.
  3. The answer is yes and the person accepts conditionalization, but does not accept that the priors for the four possibilities in the Sleeping Beauty puzzle should be equal –> could be a halfer.
  4. Otherwise, must be a thirder.

My intuition suggests, in a way I struggle to make precise, that Sleeping Beauty and the director do not in fact have identical information. All I can say is that Sleeping Beauty knows she will be awake on Monday (even if she subsequently forgets the experience), but the director may not observe Sleeping Beauty on Monday at all.

Nevertheless, option 2 raises interesting possibilities, on that have been explored in a number of papers. For example in D.J. Bradley’s “Self-location is no problem for conditionalization“, Synthese 182, 393–411 (2011), it is argued that learning about temporal information involves “belief mutation”, which requires a different approach to updating beliefs than “discovery” of non-temporal information, which makes use of conditionalisation.

All of this serves as a somewhat lengthy introduction to an interesting approach to the problem developed by Giulio Katis, who first introduced me to the problem. The Stubborn Mule may not be a well-known mathematical imprint, but I am pleased to be able to publish his paper, Sleeping Beauty, the probability of an experiment being in a state, and composing experiments, here on this site. In this post I will include excerpts from the paper, but encourage those interested in a mathematical framing of a halfer’s approach to the problem. I am sure that Giulio will welcome comments on the paper.

Giulio begins:

The view taken in this note is that the contention between halfers and thirders over the Sleeping Beauty (SB) problem arises primarily for two reasons. The first reason relates to exactly what experiment or frame of reference is being considered: the perspective of SB inside the experiment, or the perspective of an external observer who chooses to randomly inspect the state of the experiment. The second reason is that confusion persists because most thirders and halfers have not explicitly described their approach in terms of generally defining a concept such as “the probability of an experiment being in a state satisfying a property P conditional on the state satisfying property C”.

Here Giulio harks back to Bob Walters’ distinction between experiments and states. In the context of the Sleeping Beauty problem, the “experiment” is a full run from coin toss, through Monday and Tuesday, states are a particular point in the experiment and as an example, P could be a state with the coin toss being Heads and C being a state in which Sleeping Beauty is awake.

From here, Giulio goes on to describe two possible “probability” calculations. The first would be familiar to thirders and Giulio notes:

What thirders appear to be calculating is the probability that an external observer randomly inspecting the state of an experiment finds the state to be satisfying P . Indeed, someone coming to randomly inspect this modified SB problem (not knowing on what day it started) is twice as likely to find the experiment in the case where tails was tossed. This reflects the fact that the reference frame or ‘time­frame’ of this external observer is different to that of (or, shall we say, to that ‘inside’) the experiment they have come to observe. To formally model this situation would seem to require modelling an experiment being run within another experiment.

The halfer approach is then characterised as follows:

The halfers are effectively calculating as follows: first calculate for each complete behaviour of the experiment the probability that the behaviour is in a state satisfying property P; and then take the expected value of this quantity with respect to the probability measure on the space of behaviours of the experiment. Denote this quantity by ΠX(P) .

An interesting observation about this definition follows:

Note that even though at the level of each behaviour the ‘probability of being in a state satisfying P’ is a genuine probability measure, the quantity ΠX(P) is not in general a probability measure on the set of states of X . Rather, it is an expected value of such probabilities. Mathematically, it fails in general to be a probability measure because the normalization denominators n(p) may vary for each path. Even though this is technically not a probability measure, I will, perhaps wrongly, continue to call ΠX(P) a probability.

I think that this is an important observation. As I noted at the outset, the mathematics of the thirder position “works”, but typically halfers end up facing all sorts of nasty side-effects. For example, an incautious halfer may be forced to conclude that, if the experimenters tell Sleeping Beauty that today is Monday then she should update her degree of belief that the coin toss came up Heads to 2/3. In the literature there are some highly inelegant attempts to avoid these kinds of conclusions. Giulio’s avoids these issues by embracing the idea that, for the Sleeping Beauty problem, something other than a probability measure may be more appropriate for modelling “credence”:

I should say at this point that, even though ΠX(P) is not technically a probability, I am a halfer in that I believe it is the right quantity SB needs to calculate to inform her degree of ‘credence’ in being in a state where heads had been tossed. It does not seem ΞX(P) [the thirders probability] reflects the temporal or behavioural properties of the experiment. To see this, imagine a mild modification of the SB experiment (one where the institute in which the experiment is carried out is under cost pressures): if Heads is tossed then the experiment ends after the Monday (so the bed may now be used for some other experiment on the Tuesday). This experiment now runs for one day less if Heads was tossed. There are two behaviours of the experiment: one we denote by pTails which involves passing through two states S1 = (Mon, Tails), S2 = (Tue, Tails) ; and the other we denote by pHeads which involves passing through one state S3 = (Mon,Heads). Let P = {S3}, which corresponds to the behaviour pHeads . That is, to say the experiment is in P is the same as saying it is is in the behaviour pHeads. Note π(pHeads) = 1/2 , but ΞX(P) = 1/3 . So the thirders view is that the probability of the experiment being in the state corresponding to the behaviour pHeads (i.e. the probability of the experiment being in the behaviour pHeads) is actually different to the probability of pHeads occurring!

This halfer “probability” has some interesting characteristics:

There are some consequences of the definition for ΠX(P) above that relate to what some thirders claim are inconsistencies in the halfers’ position (to do with conditioning). In fact, in the context of calculating such probabilities, a form of ‘interference’ can arise for the series composite of two experiments (i.e. the experiment constructed as ‘first do experiment 1, then do experiment 2’), which does not arise for the probabilistic join of two experiments (i.e. the experiment constructed as ‘with probability p do experiment 1, with probability 1-­p do experiment 2’).

In a purely formal manner (and, of course, not in a deeper physical sense) this ‘non­locality’, and the importance of defining the starting and ending states of an experiment when calculating probabilities, reminds me of the interference of quantum mechanical experiments (as, say, described by Feynman in the gem of a book QED). I have no idea if this formal similarity has any significance at all or is completely superficial.

Giulio goes on to make an interesting conjecture about composition of Sleeping Beauty experiments:

We could describe this limiting case of a composite experiment as follows. You wake up in a room with a white glow. A voice speaks to you. “You have died, and you are now in eternity. Since you spent so much of your life thinking about probability puzzles, I have decided you will spend eternity mostly asleep and only be awoken in the following situations. Every Sunday I will toss a fair coin. If the toss is tails, I will wake you only on Monday and on Tuesday that week. If the toss is heads, I will only wake you on Monday that week. When you are awoken, I will say exactly the same words to you, namely what I am saying now. Shortly after I have finished speaking to you, I will put you back to sleep and erase the memory of your waking time.” The voice stops. Despite your sins, you can’t help yourself, and in the few moments you have before being put back to sleep you try to work out the probability that the last toss was heads. What do you decide it is?

In this limit, Giulio argues that a halfer progresses to the thirder position, assigning 1/3 to the probability that the last toss was heads!

These brief excerpts don’t do full justice to the framework Giulio has developed, but I do consider it a serious attempt to encompass all of the temporal/non-temporal, in-experiment/out-of-experiment subtleties that the Sleeping Beauty problem throws up. This paper is only for the mathematically inclined and, like so much written on this subject, I doubt it will convince many thirders, but if nothing else I hope it will put Giulio’s mind at rest having the paper published here on the Mule. Over recent weeks, his thoughts have been as plagued by this problem as have mine.