I’m all about the Bayes, ’bout the Bayes, no treble

Bayes stemlyns #FOAMed

Who knew that the Reverend Thomas Bayes would be such a prominent figure in pop culture?! Ignore the spelling error in the title of the song, we know who she is really singing about.  I haven’t gone crazy, and hopefully I can explain the inspiration behind this post in due course.  Thomas Bayes was a Presbyterian minister back in the days (1700s) when church and science began to butt heads.  Despite writing some mathematical papers, he never published his eponymous theory.  It was a friend, Richard Price, who discovered Bayes’ manuscript after his death, rewrote the document and submitted it for publication.  It was further elaborated upon by Laplace (of Laplace’s Law fame) who got wind of the Theorem when Price visited France.  Over the years it has been shunned and been the subject of intense scrutiny but nonetheless has had a huge impact, especially during wartime (Turing, in part, used Bayesian techniques to crack Enigma!).


Laplace described his version of Bayes’ Theorem as “inverse probability”.  By this he meant that the Theorem looked at effects to infer the causes.  Bayes’ Theorem uses the terms prior, likelihood and posterior to describe probabilities.  The prior is the probability of an initial belief, the likelihood is the probability of other hypotheses and the posterior is probability of the revised belief:

Prior times likelihood is proportional to the posterior.

Prior probability x likelihood  α posterior probability

Or very simply interpreted: Initial belief + new objective data = new and improved belief

There are very few things, if any, that we can have 100% certainty about, and as McGrayne says in her excellent book (The Theory That Would Not Die): ‘probability is the mathematical expression of our ignorance’.  I would really recommend reading the book!

Why is any of this relevant?

One could argue that a lot of medicine is a practical example of Bayes’ Theorem, but none more so than Emergency Medicine.  We have a prior probability, which we update as and when new information is found and develop a posterior probability.  We do this all day every day, albeit without realising it, on a subconscious level.  I think the real importance is being aware of how this theorem applies to us and how it can help us to avoid pitfalls.  I am going to propose that by not using this theorem, or using it incorrectly and incompletely leads to missed diagnoses and other errors.

Let’s imagine you see a patient with chest pain.  You take a brief history and by the end of doing so have applied probabilities to what you believe the cause to be; let’s say ACS for the sake of argument (essentially your differential diagnosis is a list of Bayesian priors!).  You then examine the patient and increase or decrease the probabilities depending on your clinical findings, in this case there are none.  You send a Troponin, fortunately it is within an appropriate timeframe for your particular assay, and maybe a couple of other bloods and eagerly await the results.  The results all come back normal….bingo! You can discharge the patient confidently having ruled out your differential diagnosis.

This is is an all too familiar scenario, and if scenarios like the above are happening often then it is no wonder that we miss other pathologies (i.e aortic dissections).  I believe an all too common pitfall in EM, amongst other specialities, is a failure to update our probabilities once we have our posterior.  I am not endorsing the over-investigation of every single patient with chest pain, but rather challenging whoever is reading this to take an extra minute or two and re-examine their probabilities and differentials when receiving a bundle of negative results.



Bayesians and Frequentists have a long-lasting feud over the superior approach to probabilities.  The NEJM recently published an article, around breast cancer therapies, but also had a discussion around the future of clinical trials and the applications, strengths and weaknesses of using Bayesian and Frequentist study designs.  The I-SPY 2 Trial uses Bayesian statistics to identify which combination of breast cancer drugs has the highest probability of being efficacious, and using this outcome will progress certain drugs to the next round of clinical trialling.  This is an efficient means of research, in that it reduces the unnecessary cost and time in investigating all the potential medications in all stages of clinical trial, and limiting it to those that are the most likely to be efficacious.  The data to form the initial probabilities is based on small study populations.

A great clinical application of Bayes’ Theorem is work done by the Centre for Trauma Sciences that used Bayesian modelling to predict the risk of individuals developing acute traumatic coagulopathy (ATC).  You can see and use the model here.  The model combines existing data from studies looking at the causes of ATC with expert knowledge.

You can also read lots of Bayes related work in the #FOAMed world – Casey Parker at Broome docs is a great place to start.

This article is intended as a very inadequate primer to Bayes’ Theorem.  I believe it is integral to our practice and is gaining a position in our research activities.  My take home message:  ‘Always update your priors’, i.e. don’t ignore new information!





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  1. caseyparker207

    Thanks Rich. Great article. If you want to hear how much I like Bayes – then stay tuned firmly SMACCDUB talk when it is released
    Love the video

  2. Junior ISBA

    We are pleased to invite you to join the Facebook group “j-ISBA”. The purpose of this group is to bring together “young” (not-so-young are welcome as well) Bayesian statisticians (not-so Bayesian are welcome as well) in order to foster interactions, tell people about interesting workshops, conferences, jobs, scholarships, or anything related to the Bayesian world.



  3. Derek Louey

    The practical difficulty of course is that we do not always have known statistical information about the accuracy of different symptoms and signs to generate accurate pre-test probabilities. It is easy for well researched areas such as Wells score and PE. Even the published sensitivity and specificity data on investigations (which are supposed to be population independent) can be biased by the disease prevalence in the original populations from which they are derived e.g. PERC rule

    Another interesting effect is how the sequence in which testing occurs alters the interpretation of known performance data e.g. the significance of a subsequent negative 2-point bedside ultrasound for DVT following a previously negative scan and positive d-dimer.

    Lastly Bayes Theorem, isn’t sufficient to explain the ultimate influence on clinical decision making which is the risk aversiveness or risk tolerance of the individual clinician given the same clinical information.

  4. caseyparker207

    HI Derek
    I struggled with all of these points over the last few years… I think my SMACC talk does go some way to resolving these “troubles with Bayes” in the real world. Especially the concept of risk tolerance is key to ensuring we do not over treat or miss too much in clinical decision making

  5. Derek Louey

    I think risk tolerance is socially determined. Our practice gradually falls into line with your peer group even though the evidence for its superiority is questionable e.g. Liberal use of panscan in trauma

  6. caseyparker207

    Agree. Doctors are less risk tolerant than the societal average. We tend to perfection which is often the enemy of good

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