A graphical explanation of Bayes theorem

A diagrammatic proof

If you take a test how do you work out how likely it is that you have the virus? Bayesian reasoning is one way (see “What are the chances of that”). Here is a graphical version of what that kind of reasoning is actually about.

If recent data shows that the virus currently affects one in 200 of the population, then it is reasonable to start with the assumption that the probability YOU have the virus is one in 200 (we call this the ‘prior probability’). Another way of saying that is that the prior probability is 0.5 per cent.

A better estimate

Suppose the probability a random person has the virus is 1 in 200 or 0.5 per cent. With no other evidence, your best guess that you have the virus is then also 0.5 per cent. You have also however taken a test and it was positive. However, for every 100 people taking the test, 2 will test positive when they actually do NOT have the virus. This means that the false positive rate is 2 per cent.

How? Bayes worked out a general equation for calculating this new, more accurate probability, called the ‘posterior’ probability (see page 8). It is based, here, on the probability of having the virus before testing (the original, prior probability) and any new evidence, which here is the test result.

A surprising result

How likely is it that you have the virus? With only this evidence, the probability you have the virus is still only 20 per cent.

– Norman Fenton, Queen Mary University of London, Spring 2021

Download Issue 27 of the cs4fn magazine on Smart Health here.

This post and issue 27 of the cs4fn magazine have been funded by EPSRC as part of the PAMBAYESIAN project.

What are the chances of that? The church minister’s hobby and clever machines

The hobby of a church minister over 250 years ago is helping computers make clever decisions.

Crowd of ghost people
Image by Free-photos from Pixabay

Thomas Bayes was an English church minister who died in 1761. His hobby was a type of maths that today we call probability and statistics, though his writings were never really recognised during his own lifetime. So, how is the hobby of this 18th century church minister driving computers to become smarter than ever? His work is now being used in applications as varied as: helping to diagnose and treat various diseases; deciding whether a suspect’s DNA was at a crime scene; accurately recommending which books and films we will like; setting insurance premiums for rare events; filtering out spam emails; and more.

How likely is that?

Bayes was interested in calculating how likely things were to happen (their probability) and particularly things that cannot be observed directly. Suppose, for example, you want to know the probability that you have an infectious virus, something you can’t just tell by looking. Perhaps you’re going to a concert of your favourite band – one for which you’ve already paid a lot of money. So you need to know you are not infected. If recent data shows that the virus currently affects one in 200 of the population, then it is reasonable to start with the assumption that the probability YOU have the virus is one in 200 (we call this the ‘prior probability’). Another way of saying that is that the prior probability is 0.5 per cent.

A better estimate

However, you can get a much better estimate of how likely it is that you have the virus if you can gather more evidence of your personal situation. With a virus you can get tested. If the test was always correct, then you would know for certain. Tests are never perfect though. Let’s suppose that for every 100 people taking the test, two will test positive when they actually do NOT have the virus. Scientists call this the false positive rate: here two per cent. You take the test and it is positive. You can use this information to get a better idea of the likelihood you have the virus.

How? Bayes worked out a general equation for calculating this new, more accurate probability, called the ‘posterior’ probability. It is based, here, on the probability of having the virus before testing (the original, prior probability) and any new evidence, which here is the test result.

A surprising result

If we assume in our example that every person who does have the virus is certain to test positive then, plugging the numbers into Bayes’ theorem, tells us there is actually a surprisingly low, one in five (i.e., 20 per cent) chance you have the virus after testing positive. See A Graphical Explanation of Bayes’ theorem for why the answer is correct. Although this is much higher than the probability of having the virus without testing (two per cent), it still means you are unlikely to have the virus despite the positive test result!

If you understand Bayes theorem, you might feel it unfair if your doctor still insists that you have the virus and must miss the trip. In fact, many people find the result very surprising; generally, doctors who do not know Bayes’ theorem massively overestimate the likelihood that patients have a disease after a positive test result. But that is why Bayes’ theorem is so important.

To go or not to go

Of course, no one knows which are the five concert goers that are the ones infected. If all 25 ignore their doctor that means there are five people mingling in the crowd, passing on the virus, which would mean lots more people catch the virus who pass it on to lots more, who … (see Ping pong vaccination).

We have seen that, with a little extra information (such as a test result), we can work out a more accurate probability and so have better information upon which to make decisions. In practice, there are many different kinds of information that we can use to improve our estimate of the real probability. There are symptoms such as lack of taste/smell which are quite specific to the virus. Others, like a cough, are common in people with the virus but also in people with flu. There are also factors that can cause a person to have the virus in the first place such as close contact with an infected relative. So, instead of just inferring the probability of having the virus from one piece of information, like the test result, we can consider lots of interconnected data, each with its own prior probability. This is where computers come in: to do all the calculations for us.

We first need to tell the computer about what causes what. A convenient way to do this is to draw a diagram of the connections and probabilities called a ‘Bayesian network’ (see A Simple Bayesian Network – to come). Once a computer has been given the Bayesian network, it can not only work out more accurate probabilities, but it can also use them to start making decisions for us. This is where all those applications come in. Deciding whether a suspect’s DNA was at a crime scene, for example, needs the same kind of reasoning as deciding whether you have the virus.

Obviously, it is more complex to apply Bayes’ theorem in realistic situations and, until quite recently, even the fastest computers weren’t able to do the calculations. However, breakthroughs by computer scientists developing new algorithms mean that very complex Bayesian networks, with lots of inter-connected causes, can now be computed efficiently. Because of this, Bayesian networks can now be applied to a multitude of important problems that were previously impossible to solve. And that is why, perhaps surprisingly, the ideas of Thomas Bayes, from over 250 years ago, are showing us how to build machines that make smarter decisions when things are uncertain.

– Norman Fenton, Queen Mary University of London, Spring 2021

Download Issue 27 of the cs4fn magazine on Smart Health here.

This post and issue 27 of the cs4fn magazine have been funded by EPSRC as part of the PAMBAYESIAN project.

The ping pong vaccination programming challenge

Vaccination programmes work best when the majority of the population are vaccinated. One way scientists simulate the effects of disease and vaccination programmes is by using computer simulations. But what is a computer simulation?

Lots of multi-coloured ping pong balls

You can visualise what a simulation is with ping pong balls bouncing around a crowd. Imagine having a large room full of people. A virus is represented by a ping pong ball, bouncing from person to person, infecting each person it touches. Each person who is hit by a ping pong ball and not already infected becomes infected. That means they toss that ping pong ball back into the crowd to infect more people, but they also toss an extra one too (and then they sit down: dead). Start with a few ping pong balls. Quickly the virus spreads everywhere and lots of people sit down (die). You have run a physical simulation of how a virus spreads!

Now start again but ‘vaccinate’ 80 per cent of the people first: give them a baseball cap to wear to show who is who. If those people get a ping pong ball, they just destroy it: they infect noone else. Start with the same number of ping pong balls. This time, the virus quickly dies out and only a few people sit down (die). Not only are the vaccinated people protected but they protect many of the un-protected people too who might have died.

Now (if you can program) you can write a program to do the same thing, and so simulate and explore the spread of infection, which is easier perhaps than getting a thousand people to chuck ping pong balls about. Create a grid (an array) of 1000 cells. Each represents a person. They can be infected or not. They can also be vaccinated or not. Start with five random cells (so people marked as infected). Run a series of rounds. After each round, a newly infected cell randomly chooses two others to infect. If not infected already and not vaccinated, then they become newly infected. If already infected or vaccinated, they do not pass the infection on.

You can run lots of different experiments with different conditions. For example, experiment with different proportions of people infected at the start or explore what percentage of people need to be vaccinated for the virus to quickly die out. Is 50 per cent enough? You could also change how many people one person infects, or for how long a person can infect others before dying. Perhaps they each keep causing new infections for three rounds before stopping instead of only one. In what situations does the virus infect lots of people and when does it die out quickly?

What you are doing here is computer modelling or simulating the effects of the virus in different scenarios, and that is essentially how computer scientists make the predictions that governments use to make decisions about lockdowns and mask wearing, if they are “following the science”. Of course, such models are only as good as the data that goes into them, such as how many other people does each person infect. In reality, this is data provided by surveys, experimental studies, and so on.

– Paul Curzon, Queen Mary University of London, Spring 2021

Download Issue 27 of the cs4fn magazine on Smart Health here.

This post and issue 27 of the cs4fn magazine have been funded by EPSRC as part of the PAMBAYESIAN project.

Smart health: decisions, decisions, decisions

The trouble with healthcare is that it’s becoming ever more expensive: new drugs, new treatments, more patients, the ever-increasing time needed with experts. Smart healthcare might be able to help.

Cover of cs4fn issue 27 on smart health - a spiders web covered in droplets of dew

We want everyone to get the care they need, but the costs are growing. Perhaps computer scientists can help? Research groups worldwide are exploring ways to create computing technology to improve healthcare, and intelligent programs that can support patients at home, helping monitor them and make decisions about what to do.

For example, say you are on powerful drugs to manage a long term illness: should you have the vaccine? Can you have a baby? Is a flare up of your disease about to hit you and how can you avoid it? Is that new ache a side effect of the drugs? Do you need to change medicines? Do you need to see a specialist?

If smart programs can help support patients then the doctors and nurses can spend more time with those who actually need it, hospitals can save on expensive drugs that aren’t working, and patients can have better lives. But what kind of technology can deliver this sort of service?

In the current issue of cs4fn magazine, we explore one particular way being developed on the EPSRC funded PAMBAYESIAN project at Queen Mary University of London, based on an area of computing called Bayesian networks, that might just be the answer. We also look at other ways computers can help deliver better healthcare for all and other uses of Bayesian networks.

We will be blogging each article here over the coming days or you can download Issue 27 of the cs4fn magazine on Smart Health here and read it all now.

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This post and issue 27 of the cs4fn magazine have been funded by EPSRC as part of the PAMBAYESIAN project. UK schools that subscribe will be sent copies in the coming weeks.

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Sabine Hauert: Swarm Engineer

Based on a 2016 talk by Sabine Hauert at the Royal Society

Sabine Hauert is a swarm engineer. She is fascinated by the idea of making use of swarms of robots. Watch a flock of birds and you see that they have both complex and beautiful behaviours. It helps them avoid predators very effectively, for example, so much so that many animals behave in a similar way. Predators struggle to fix on any one bird in all the chaotic swirling. Sabine’s team at the University of Bristol are exploring how we can solve our own engineering problems: from providing communication networks in a disaster zone to helping treat cancer, all based on the behaviours of swarms of animals.

A murmuration  - a flock of starlings

Sabine realised that flocks of birds have properties that are really interesting to an engineer. Their ability to scale is one. It is often easy to come up with solutions to problems that work in a small ‘toy’ system, but when you want to use it for real, the size of the problem defeats you. With a flock, birds just keep arriving, and the flock keeps working, getting bigger and bigger. It is common to see thousands of Starlings behaving like this – around Brighton Pier most winter evenings, for example. Flocks can even be of millions of birds all swooping and swirling together, never colliding, always staying as a flock. It is an engineering solution that scales up to massive problems. If you can build a system to work like a flock, you will have a similar ability to scale.

Flocks of birds are also very robust. If one bird falls out of the sky, perhaps because it is caught by a predator, the flock itself doesn’t fail, it continues as if nothing happened. Compare that to most systems humans create. Remove one component from a car engine and it’s likely that you won’t be going anywhere. This kind of robustness from failure is often really important.

Swarms are an example of emergent behaviour. If you look at just one bird you can’t tell how the flock works as a whole. In fact, each is just following very simple rules. Each bird just tracks the positions of a few nearest neighbours using that information to make simple decisions about how to move. That is enough for the whole complex behaviour of the flock to emerge. Despite all that fast and furious movement, the birds never crash into each other. Fascinated, Sabine started to explore how swarms of robots might be used to solve problems for people.

Her first idea was to create swarms of flying robots to work as a communications network, providing wi-fi coverage in places it would otherwise be hard to set up a network. This might be a good solution in a disaster area, for example, where there is no other infrastructure, but communication is vital. You want it to scale over the whole disaster area quickly and easily, and it has to be robust. She set about creating a system to achieve this.

The robots she designed were very simple, fixed wing, propellor-powered model planes. Each had a compass so it knew which direction it was pointing and was able to talk to those nearest using wi-fi signals. It could also tell who its nearest neighbours were. The trick was to work out how to design the behaviour of one bird so that appropriate swarming behaviour emerged. At any time each had to decide how much to turn to avoid crashing into another but to maintain the flock, and coverage. You could try to work out the best rules by hand. Instead, Sabine turned to machine learning.

“Throwing those flying robots

and seeing them flock

was truly magical”

The idea of machine learning is that instead of trying to devise algorithms that solve problems yourself, you write an algorithm for how to learn. The program then learns for itself by trial and error the best solution. Sabine created a simple first program for her robots that gave them fairly random behaviour. The machine learning program then used a process modelled on evolution to gradually improve. After all evolution worked for animals! The way this is done is that variations on the initial behaviour are trialled in simulators and only the most successful are kept. Further random changes are made to those and the new versions trialled again. This is continued over thousands of generations, each generation getting that little bit better at flocking until eventually a behaviour of individual robots results that leads to them swarming together.

Sabine has now moved on to to thinking about a situation where swarms of trillions of individuals are needed: nanomedicine. She wants to create nanobots that are each smaller than the width of a strand of hair and can be injected into cancer patients. Once inside the body they will search out and stick themselves to tumour cells. The tumour cells gobble them up, at which point they deliver drugs directly inside the rogue cell. How do you make them behave in a way that gives the best cancer treatment though? For example, how do you stop them all just sticking to the same outer cancer cells? One way might be to give them a simple swarm behaviour that allows them to go to different depths and only then switch on their stickiness, allowing them to destroy all the cancer cells. This is the sort of thing Sabine’s team are experimenting with.

Swarm engineering has all sorts of other practical applications, and while Sabine is leading the way, some time soon we may need lots more swarm engineers, able to design swarm systems to solve specific problems. Might that be you?

Explore swarm behaviour using the Oxford Turtle system [EXTERNAL] (click the play button top centre) to see how to run a flocking simulation as well as program your own swarms.

Paul Curzon, Queen Mary University of London

What’s on your mind?

Telepathy is the supposed Extra Sensory Perception ability to read someone else’s mind at a distance. Whilst humans do not have that ability, brain-computer interaction researchers at Stanford have just made the high tech version a virtual reality.

Image by Andrei Cássia from Pixabay

It has long been know that by using brain implants or electrodes on a person’s head it is possible to tell the difference between simple thoughts. Thinking about moving parts of the body gives particularly useful brain signals. Thinking about moving your right arm, generates different signals to thinking about moving your left leg, for example, even if you are paralysed so cannot actually move at all. Telling two different things apart is enough to communicate – it is the basis of binary and so how all computer-to-computer communication is done. This led to the idea of the brain-computer interface where people communicate with and control a computer with their mind alone.

Stanford researchers made a big step forward in 2017, when they demonstrated that paralysed people could move a cursor on a screen by thinking of moving their hands in the appropriate direction. This created a point and click interface – a mind mouse – for the paralysed. Impressively, the speed and accuracy was as good as for people using keyboard applications

Stanford researchers have now gone a step even further and used the same idea to turn mental handwriting into actual typing. The person just thinks of writing letters with an imagined pen on imagined paper, the brain-computer interface then picks up the thoughts of subtle movements and the computer converts them into actual letters. Again the speed and accuracy is as good as most people can type. The paralysed participant concerned could communicate 18 words a minute and made virtually no mistakes at all: when the system was combined with auto-correction software, as we now all can use to correct our typing mistakes, it got letters right 99% of the time.

The system has been made possible by advances in both neuroscience and computer science. Recognising the letters being mind-written involves distinguishing very subtle differences in patterns of neurons firing in the brain. Recognising patterns is however, exactly what Machine Learning algorithms do. They are trained on lots of data and pick out patterns of similar data. If told what letter the person was actually trying to communicate then they can link that letter to the pattern detected. Here each letter will not lead to exactly the same pattern of brain signals firing each time, but they will largely clump together,. Other letters will also group but with slightly different patterns of firings. Once trained, the system works by taking the pattern of brain signals just seen and matching it to the nearest clumping pattern. The computer then guesses that the nearest clumping is the letter being communicated. If the system is highly accurate, as this one was at 94% (before autocorrection), then it means the patterns of most letters are very distinct. A letter being mind-written rarely fell into a brain pattern gap, which would have meant that letter could as easily have been the pattern of one letter as the other.

So a computer based “telepathy” is possible. But don’t expect us all to be able to communicate by mind alone over the internet any time soon. The approach involves having implants surgically inserted into the brain: in this case two computer chips connecting to your brain via 100 electrodes. The operation is a massive risk to take, and while perhaps justifiable for someone with a problem as severe as total paralysis, it is less obvious it is a good idea for anyone else. However, this shows at least it is possible to communicate written messages by mind alone, and once developed further could make life far better for severely disabled people in the future.

Yet again science fiction is no longer fantasy, it is possible, just not in the way the science fiction writers perhaps originally imagined by the power of a person’s mind alone.

Paul Curzon, Queen Mary University of London, Spring 2021.

Sick tattoos

Image by Anand Kumar from Pixabay

Researchers at MIT and Harvard have new skin in the game when it comes to monitoring people’s bodily health. They have developed a new wearable technology in the form of colour- and shape-changing tattoos. These tattoos work by using bio-sensitive inks, changing colour, fading away or appearing under different coloured illumination, depending on your body chemistry. They could, for example, change their colour, or shape as their parts fade away, depending on your blood glucose levels.

This kind of constantly on, constantly working body monitoring ensures that there is nothing to fall off, get broken or run out of power. That’s important in chronic conditions like diabetes where monitoring and controlling blood glucose levels is crucial to the person’s health. The project, called Dermal Abyss, brings together scientists and artists in a new way to create a data interface on your skin.

There are still lots of questions to answer, like how long will the tattoos last and would people be happy displaying their health status to anyone who catches a glimpse of their body art? How would you feel having your body stats displayed on your tats? It’s a future question for researchers to draw out the answer to.

– Peter W. McOwan, Queen Mary University of London, Autumn 2018

One in the eye for wearable tech

Contact lenses, normally used to simply, but usefully, correct people’s vision, could in the future do far more.

Tiny microelectronic circuits, antennae and sensors can now be fabricated and set in the plastic of contact lenses. Researchers are looking at the possibility of using such sensors to sample and transmit the glucose level in the eye moisture: useful information for diabetics. Others are looking at lenses that can change your focus, or even project data onto the lens, allowing new forms of augmented and virtual reality.

Conveniently, you can turn the frequent natural motion from the blinks of your eye into enough power to run the sensors and transmitter, doing away with the need for charging. All this means that smart contact lenses could be a real eye opener for wearable tech.

– Peter W. McOwan, Queen Mary University of London, Autumn 2018

Smart tablets (to swallow)

The first ever smart pill has been approved for use. It’s like any other pill except that this one has a sensor inside it and it comes with a tracking device patch you wear to make sure you take it.

A big problem with medicine is remembering to take it. It’s common for people to be unsure whether they did take today’s tablet or not. Getting it wrong regularly can make a difference to how quickly you recover from illness. Many medicines are also very, very expensive. Mass-produced electronics, on the other hand, are cheap. So could the smart pill be a new, potentially useful, solution? The pill contains a sensor that is triggered when the pill dissolves and the sensor meets your stomach acids. When it does, the patch you wear detects its signal and sends a message to your phone to record the fact. The specially made sensor itself is harmless and safe to swallow. Your phone’s app can then, if you allow it, tell your doctor so that they know whether you are taking the pills correctly or not.

Smart pills could also be invaluable for medical researchers. In medical trials of new drugs, knowing whether patients took the pills correctly is important but difficult to know. If a large number of patients don’t, that could be a reason why the drugs appeared less effective than expected. Smart pills could allow researchers to better work out how regularly a drug needs to be taken to still work. 

More futuristically still, such pills may form part of a future health artificial intelligence system that is personalised to you. It would collect data about you and your condition from a wide range of sensors recording anything relevant: from whether you’ve taken pills to how active you’ve been, your heart rate, blood pressure and so on: in fact anything useful that can be sensed. Then, using big data techniques to crunch all that data about you, it will tailor your treatment. For example, such a system may be better able to work out how a drug affects you personally, and so be better able to match doses to your body. It may be able to give you personalised advice about what to eat and drink, even predicting when your condition could be about to get better or worse. This could make a massive difference to life for those with long term illnesses like rheumatoid arthritis or multiple sclerosis, where symptoms flare up and die away unpredictably. It could also help the doctors who currently must find the right drug and dose for each person by trial and error.

Computing in future could be looking after your health personally, as long as you are willing to wear it both inside and out.

– Paul Curzon, Queen Mary University of London, Spring 2021

Download Issue 27 of the cs4fn magazine on Smart Health here.

This post and issue 27 of the cs4fn magazine have been funded by EPSRC as part of the PAMBAYESIAN project.