Kindness In Guizhou

You wouldn’t have guessed from the title and the opening story, but this post is about Bayesian reasoning.

My Guizhou Story

My wife and I love train rides. May last year, we travelled on China’s newest East-West high-speed rail. The high-speed train could cover the distance of 2,500 km from Kunming to Shanghai, with about 30 stops in between, in less than 12 hours。

Alas, I was unwell already at the start of the journey. When we arrived in Anshun, a small town in Guizhou, I was taken quite ill and was brought to a local clinic. The clinic was quite plain, very basic, facing a busy road, but I suppose that was as good a clinic as one could get in that small town.?

A small heater stove sat in the middle of the clinic’s waiting area. Naked fluorescent tubes mounted on the ceiling lit the room. My wife and I took our seats on foldable plastic chairs that were arranged around the stove.?

On the left were two examination beds separated from the waiting area by a light partition. One of the two lady doctors was examining a small boy. If I were to close my eyes and just listen to the doctor and patient, I would have imagined she was the mother consoling her sick child.?

On the right was a small room that served as a dispensary where the doctors would prepare medications.

As far as I could make out, these three areas, each not more than 10 square meters, represented the entire clinic.

The other doctor pulled up one of the foldable chairs and sat facing me. She checked my pulse on both wrists, checked my blood pressure, took my temperature and listened to my breathing with her stethoscope, first pressing it upon my chest then on my back and then on my chest again.?

The whole examination must have taken twice as long as normal because I communicated with her mostly through my phone’s?Google Translate. How to explain in Chinese that my cough started as a bad sore throat several weeks ago? What was Chinese for antibiotic? How to say that I had already completed one course of Amoxicillin? Or that I have had pneumococcal vaccination five years ago? There were hilarious moments too. Carelessly translated, feeling the chills sounded like a complaint about the lack of libido!

A brief discussion followed between the two doctors about me, about what exactly I couldn’t possibly understand. Then more checks with the stethoscope; breathe in, hold, breathe out; cough twice; stick out the tongue; pulse checked on each wrist; the whole gamut.?

The doctors came to their decision. They had better get me to the hospital.

?At the hospital, the nurse who registered me brought me to wait in queue to consult the resident respiratory doctor. As I sat and waited my turn, I studied other patients who were waiting ahead of me and I thought, “By their clothes, I would say they are poor. Their faces browned by long exposure to the sun. Farmers perhaps? Ah they all behave so civilised.”?

My name was called. The doctor and I went through the same process that I went through in the clinic. Then it was off to the radiology department. Another orderly queue. Another pleasant encounter, this time with an attentive and caring radiologist.

Back to the doctor. After studying the radiologist report, he asked about my medical history, what medication I was currently on, and so on. It wasn’t an easy consultation. Again I had to make frequent reference to?Google Translate: What is Chinese word for?Aspirin? How do I say “blood thinner” in Chinese? What is Chinese for antibiotics? But he was oh so patient! He would listen, nodding almost imperceptibly, as I explained how I felt.

A new patient, clearly unfamiliar with hospital procedures, walked in and took a seat next to me! The doctor looked at the intruder briefly, smiled, and gently said, “Take a seat outside the room. Come in when your name is called.”

The doctor then turned his attention back to me. He explained that he would prescribe an antibiotic to deal with my lung infection, medication to bring down the fever and a decongestant for my cough. And he asked the nurse to help me purchase the medicines at the hospital dispensary. For that I was truly glad because I would have had problem navigating through the Chinese hospital system myself.

It was past shift change when the nurse collected my medication. Hospital staffs were leaving, bidding each other goodbye. The doctor, now in his civilian clothes, came downstairs to the dispensary. He had brought a thermometer with him so that he could check my temperature! Told the nurse to bring me a cup of warm water for my medication. “Stay off cold drinks” he advised, “and keep your chest warm.” He, and the nurse who had assisted me since I arrived, stayed until my temperature had subsided before leaving. And he gave me the thermometer so that I could continue to monitor my temperature. (It was just an ordinary clinical thermometer. Not expensive, I’m sure, but that isn’t the point, is it?).

The total cost? The clinic refused to accept payment because they couldn’t treat me. The hospital charges, consultation, x-ray, and medicines came to about $50.

A Bayesian Interpretation Of My Guizhou Story

When I recounted my Guizhou experience to a friend he said I was lucky. There had been so many reports (and those were by reputable media) about medical malpractices in China. Hospitals held patients to ransom, doctors refused treatment unless they were bribed.

So how lucky was I? Was I extraordinarily lucky? Or just not too unlucky?

Well, my friend is a sensible chap. He knows that media sell stories. And readers don’t buy stories about ordinary doctors doing their ordinary things. So despite the horrific media stories, my friend would not conclude that most doctors in China are unethical. Nonetheless, that he should have thought that I was lucky reveals how media stories had insidiously shaped his beliefs about the medical profession in China.

The question is; should my friend revise his beliefs about the medical profession in China after hearing about my experience in Guizhou? Indeed also how should I revise mine? And what thinking process should I employ to revise my beliefs??

Well, I chose to revise mine through Bayesian reasoning.

At this point, logically I should state Bayes’ Theorem and present the proof for that theorem. In fact almost all articles about Bayesian reasoning starts this way. But I find that knowing how to prove Bayes’ Theorem adds little to our ability to use Bayesian reasoning.?

As an analogy, it is fun to learn the laws of thermodynamics butlearning the laws of thermodynamics won’t teach us how to drive. We don’t need to know those laws to drive a car.?

Nonetheless for the sake of completeness, I will derive Bayes’ Theorem, and it will be done without mathematics, but I will put that at the end of this post.

So this post is really about a Bayesian way of thinking. However we cannot think in a vacuum. There isn’t such a thing as a Bayesian analysis per se, only a Bayesian analysis about something. Specifically then, this post is about how to use Bayesian thinking to frame a belief and how that belief should be revised given fresh evidence. So in Bayesian reasoning you must start with a belief about something. For example, your Bayesian reasoning might be aboutArbitraging A Mispriceif you are betting in the financial markets, or?Preventing An Epidemic?if public health management is your calling, or?Farewell To Hunger?if you are into poverty reduction, or?Peace Among Nationsfor a politician, or?Hole In Onefor an avid golfer, or whatever happens to be your cup of tea.

Mine is about?Kindness In Guizhou.?And it is about how those encounters with the clinic and then the hospital modified my beliefs about China’s medical profession.?

Like my friend, I set store by a large dose of scepticism when reading media reports. So even though prior my Guizhou experience I had never sought medical treatment in China and therefore had no basis for my faith in China’s medical profession, I did not believe that China’s medical profession was as unethical as media made it out to be. I felt that it was irresponsible journalism to cast aspersions on an entire profession by cherry picking a few horrific stories of unsavoury practices by a few black sheep.

Nonetheless, for the purpose of illustrating Bayesian thinking, I shall assume that, given all the bad publicity, an impressionable person, especially one who has not spent much time in China, might believe that as much as 6 out of 10 clinics, and 4 out of 10 hospitals in China could not be trusted. Those then are the starting beliefs that we will use in our Bayesian reasoning.

Let’s start with clinics. Our starting belief is that 6 out of 10 clinics engage in questionable business practice. (My belief actually was closer to 1 out of 10, but let’s stick with 6 in 10 to be consistent with my friend’s belief that I was “lucky”).??

This belief is set out in the box designated as?Clinic Business Conductin Fig 1.??

Now we pose two questions??

• Given the initial belief how likely was I to have a satisfactory experience in my first visit to a clinic in China?
• If my visit turned out satisfactorily, how would that affect my original belief about clinics in China?

You’ve already read that I was entirely satisfied with my one encounter with a clinic in Guizhou. But what was my expectation before, given my original beliefs??

My original belief, for the purposes of illustrating Bayesian reasoning, was that 60% of the clinics couldn’t be trusted. That also means that there was a 40% chance that the clinic I went to was above board. Now, we humans are not infallible. Even if a clinic’s business conduct was entirely above board, there was always the possibility that I might have judged it wrongly. I might have misjudged an ethical clinic as unethical just because there were some hiccups or some miscommunication. And conversely, I could have mistakenly rated the clinic as ethical when actually it was not.?

So in addition to my overall belief about clinics’ business conduct there is another layer of beliefs about my ability to judge. For example, I might estimate that 80% of the time I would judge a clinic correctly when the clinic is ethical. However, there is a 5% chance that I might judge a clinic as ethical when actually it is not.

Given these original beliefs, the chances that I would have rated the clinic as ethical would have been only 35%. The proof of this result will be presented at the end of the article together with the proof of Bayes’ Theorem.

You might have noticed; according to Fig 1, the probability that I would rate the clinic?as?ethical was 38% (represented by the green bar), not 35%. Why is there this difference? There is this difference because I believed that the clinic’s setup would have had an influenced upon my judgment. Now a purist would point out that whether the clinic is run ethically or not has nothing to do with whether its setup is rudimentary or sophisticated. That of course is true if we are absolutely rational beings. Alas, our brain isn’t built that way. There is always the chance that irrelevant factors would creep into our judgment. But that, I think, turns out to be a plus point for Bayesian reasoning. Bayesian reasoning doesn’t assume away our inconvenient human fallibility.

So now we have to expand our list of beliefs. We have to include my beliefs about how a clinic’s setup might affect my judgment. I know that it gets quite tedious if I keep rattling off lists upon lists of beliefs. So I shan’t clutter this post with too much detail. Suffice to say that my Bayesian reasoning takes into account such additional layers of beliefs.

At the end of this post I will describe in detail my Bayesian reasoning for my clinic experience, at which point I will include a table of the different layers of beliefs pertaining to my clinic experience.

Fig 1 shows how my original beliefs about clinics’ setup and their business conduct would have shaped my expectation about a clinic in China.

So, given my initial beliefs, before my visit to the clinic, the probability that I would rate a clinic visit as “Good” would have been 38%.

That was before I have had any experience of visiting a clinic in China. After that visit in Guizhou, some of my beliefs would have crystalised. Depending on the outcome of my visit, my original beliefs would either be reinforced or they would be moderated.?

Fig 2 shows my clinic experience after my beliefs were crystalised. Recall that I found the clinic setup to be quite rudimentary. And that I judged that the clinic was run exceedingly ethically. These impressions are represented by a red bar and a green bar respectively that now stretch across the width of the boxes designated as?Clinic Setupand?My Experience.

Notice in the box designated?Clinic Business Conductthat now my belief about clinics in China has been revised.??According to my Bayesian reasoning, given my experience in Guizhou, now I should expect 91% of clinics to be operated ethically. This revised belief reflects the fact that, despite my initial reservations when I saw how basic the clinic’s setup was, I found the two doctors in Guizhou to be honest and kind. (You will find out how I arrive at the likelihood of 91% at the end of this post).

Next we turn to consider my experience at the hospital and the implications that had upon my beliefs.

So what were my prior beliefs about hospitals in China? The prior belief was that probably 40% of hospitals engaged in questionable practices. (I have to reiterate that my actual belief was never that unfavourable. I use 40% as the probability just to be consistent with the comment that I was “lucky”).?

Here is a fact of life. Humans are not meant to possess objective absolute knowledge. It just is not possible for me to say categorically whether that Guizhou hospital’s business practice was ethical or not. However, notwithstanding my human limitations I do formed opinions all the time. So in Guizhou I formed my opinion about the hospital based on my interactions with some of its staffs. What I experienced was how the doctor, the nurse and other staffs treated me. And that experience, augmented by my impression about the general ambience in the hospital, formed the basis for a judgment about the hospital as a whole.?

Fig 3 is a schematic of how my judgment might have been shaped. On the one hand, if the hospital’s business practice was ethical, I expected that most (but maybe not all) of the doctors and staffs would have acted ethically. On the other hand, if the hospital’s practice had been unethical, then more likely than not the doctors and staffs would not have treated me with sincerity. And yet even in the most ethical hospital one cannot rule out that there also might be a few black sheep, while even in the most unethical hospital, some doctors still might observe their Hippocratic oath.

Taking into account these different layers of beliefs, my Bayesian reasoning places my likelihood of a pleasant encounter at the hospital at 55%. This is represented as a green bar across slightly more than half the width of the box designated?My Experiencein Fig 3.

Now we return to the two questions that I posed:

• Given my initial beliefs how likely was I to have a satisfactory experience in my first visits to both a clinic and a hospital in China?
• If my visits turned out satisfactorily, how would that affect my original beliefs about clinics and hospitals in China?

Given my prior beliefs, the chances of me finding a clinic with caring doctors was only 37% and the chances of finding a hospital with committed doctors and staffs was 55%. But I had the good fortune of going to such a clinic and then after that to go to such a hospital.??According to Bayesian reasoning, which is depicted as a schematic in Fig 4, there was only a 2 in 10 chance of me finding kindness on both occasions.

In other words if the prior beliefs were true, I was indeed lucky. However, it was more likely that my prior beliefs were wrong. It could be that perhaps I had been too cynical, too pessimistic about the people’s capacity for kindness. That means that perhaps I ought to revise my beliefs.?

How should I revise my beliefs?

The way to revise prior beliefs is to crystalise my expectations, based on what I actually experienced. So here is a list of the expectations that became crystalised.

• Given my knowledge that Anshun was a small town and that Guizhou was one of the poorest provinces in China, I had expected the clinic to be quite basic. That expectation was confirmed. In other words my expectation was crystalised and that fact is represented with a red bar across the full width of the box denoting?Clinic Setup.
• However, contrary to my expectations, I came away from the clinic with very favourable impression about the clinic, and especially about the two lady doctors. That experience is represented with a green bar across the full width of the box denoting?My Experience.
• The general ambience of the hospital was good. There was a national campaign at that time exhorting people to act civilised. Perhaps that had an effect on how patients behaved in the hospital. A full green bar.
• In my opinion the conduct of the hospital’s respiratory doctor, the nurse and other staffs was exemplary. Another full green bar.

Given these experiences, my beliefs about hospitals in China improved appreciably. The green bar now takes up 72% of the width of the box denoting?Hospital Business Practice.

But of course my real original beliefs were not those shown in Fig 1 and Fig 3. I think I have more faith in human decency. So let’s recast my Bayesian analysis, but starting with my real original beliefs.

I didn’t think that the reputation of a noble profession should be so badly tarnished just because a few black sheep fail to heed their conscience. So despite horrific stories about hospital practices, I held the belief that 90% (not 60%) of hospitals could be trusted. Pause for a moment to reflect upon what 90% means. It means that I thought that 1 in 10 hospitals could not be trusted. That is still not a very comforting thought!

If my original belief was that 1 in 10 hospitals could not be trusted, then given my good experience with the clinic and hospital in Anshun, Guizhou, I would say that now I believe that I have trust in 97% of hospitals in China. Fig 6 is the schematic of my Bayesian reasoning but based on my true beliefs.

So have we arrived at a definitive true answer about hospital ethics in China? Is it true that 97% of hospitals are ethical in their business practice? Of course not. One anecdotal experience isn’t going to give us a definitive true answer, if ever there is one.

Let’s suppose that after Guizhou I went to another hospital, say, in Changsha. The experience again was excellent. My beliefs about hospitals would have to be revised again. That in essence is Bayesian reasoning. One updates one’s beliefs once fresh evidence becomes available.

Should you carry out a thorough Bayesian analysis for every trivial event? Obviously not. But should you run your investment model through a Bayesian algorithm if your million dollars bets hinge on very thin spreads? Absolutely.

So let’s find out more about Bayes Theorem.

Bayes Theorem

The equation for Bayes Theorem is simplicity itself.

In plain language it says, the likelihood of an event A given that an event B has occurred can be found by multiplying the likelihood of an event B given that an event A has occurred multiplied by the probability of event A and divided by the probability of event B.

I have never found any use for this equation but here is my proof of the theorem anyway. I will provide the proof through colours, just two; yellow and blue.

We all know this about paints. If we mix yellow and blue paints we get a green paint. In other words,?Green is both Yellow AND Blue.?

We paint two overlapping circles, one Blue and one Yellow. What is the colour where the two circles overlap? Green, of course. We restate with these three statements:

• Green is part of Yellow
• Green is part of Blue
• Green is both Yellow AND Blue.

We throw a dart. It lands within the Yellow circle. What are the chances that the dart is also within the Blue circle? That could only happen if the dart falls in the Green portion of the Yellow circle. And obviously that Green portion is just a fraction of the full Yellow circle.?

So we say that the likelihood of the dart being within the Blue circle given that it is in the Yellow circle is equal to the Green portion as a fraction of the Yellow circle. That is quite a mouthful. So we write this as

We throw another dart and it lands within the Blue circle. Now what is the chance that the dart is also within the Yellow circle? Obviously that is the Green portion of the Blue circle. And the Green portion is only a fraction of the full Blue circle. We write this as?

We can rewrite this as?

Recall that:

So, replacing?

we get a colourful version of Bayes Theorem

Replace?

and we obtain the classic Bayes Theorem

It is an elegant theorem but I have never found much use for the equation as is.

Thinking In Squares

If you don’t want to think in circles then a good way to think Bayesian is to count squares. And we are going to count squares for my clinic experience as an example.

We start by summarising my beliefs. This is best set out as a table. However, first I shall describe the beliefs, but you might find the description rather convoluted. Just read it through quickly. It will help you understand the table that follows.

Before Guizhou, influenced by media stories, I started with the belief that 60% of clinics in China were unethical. (Well, I didn’t, but I’m using 60% for the purposes of this exercise). And I believed that most (80%) of the clinics had quite rudimentary (basic) setups. And I also believed that if the clinic’s setup was rudimentary, resulting in a bad experience, there was a 20% chance that I might misjudge an ethical clinic as unethical. However I would most likely (90%) have a good experience and would be less likely to misjudge an ethical clinic if its setup was sophisticated (advanced). I believed it was very unlikely (only 5% chance) for me to have a good experience if the clinic was both unethical and basic in setup. However, even if a clinic’s practice were unethical, there was still a 20% chance that I thought I had a good experience just because the clinic was so sophisticatedly set up.

Now we are in the position to draw some logical implications from these beliefs.

To do that, we draw a diagram that contains 100 columns and 100 rows. In other words, we draw a diagram with 10,000 squares.

We paint the first 40 rows Blue. And the Blue area represents my belief that 40% of the clinics was ethical.?

We line the last 20 columns with Red grids. The Red gridded area represents my belief that 20% of the clinics had sophisticated setup.?

So we have divided the surface into four sections, designated as #1, #2, #3, and #4 (refer to Fig 8). (Note: Red grids with Blue squares give #2 the impression that it is Magenta)

Now we incorporate Yellow to represent?Good Experience(refer to Fig 9).

• We overlay 32 (80% of 40) of the Blue rows in #1 with Yellow paint, turning that?portion(i.e. 32 rows)Green.
• We overlay 36 (90% of 40) of the Blue rows in #2 with Yellow paint, turning that portion Green.
• We paint 2 (5% of 60) rows of #3 Yellow.
• And we paint 12 (20% of 60) rows of #4 Yellow.

Recall that

• Green is a fraction of Blue
• Green is also a fraction of Yellow

Given my beliefs, what was the likelihood that I would have had a good experience? The likelihood is sum of all Yellow squares (remember that Yellow includes Green squares) as a fraction of all possible squares.

That works out to be 3,760 squares out of 10,000 squares. In other words, the likelihood that I would have had a good clinic experience, given my beliefs, was only 37.6%.

And next, we crystalise my clinic experience. And this was my experience; the clinic’s setup was quite basic but my experience was surprisingly good. That means that the sections with Red grids, i.e. #2 and #4 collapse to zero. Notice there is no longer any sections with Red grids in Fig 10.

What is my new belief about clinics in China? That is like asking, “If a dart lands in a Yellow square, what is the likelihood that it is also in a Blue square?” That is equivalent to asking, “How many Blue squares are within the Yellow squares?” The answer is found by counting the number of Green squares as a fraction of all Yellow squares. (Remember that Green is part of Yellow and also part of Blue).

There are 3,200 Green squares. There are a total of 3,500 Yellow (plus Green) squares. The likelihood of a dart being in a Green square given that the dart landed in the Yellow area is 3,200 out of 3,500, i.e. 91.4%.

That is tantamount to saying that there is a 91% likelihood of clinics in China being ethical given that my experience was good, despite the clinic being?quite?rudimentary.

We could also count squares for my hospital experience but that would be terribly tedious because there are too many intervening layers of beliefs.?

Do I ever?really?plot a bell curve distribution and count the squares (and parts of squares) under the curve? No, I would just run my data through a statistics software. Similarly, did I really count those thousands of squares to arrive at my conclusions about hospitals and clinics in China? No, I just ran my tables of beliefs through a Bayesian algorithm.

The long and short of it all is that given the admirable conduct of the people that I encountered in the clinic and the hospital, my faith in the medical profession in China is even stronger than before.

And finally, lest you misread this as endorsement of all things Chinese, I hasten to add that during that Kunming-to-Shanghai trip there also were occasions when my wife and I witnessed behaviours that we frowned upon.

Parting Shot

Would I count squares for every everyday event, like I did for?My Guizhou Story? Obviously not. Would I run scenarios through a Bayesian algorithm if a million dollars bet hinges on an ultra thin spread? Absolutely.