What are the odds of playing ClassJack?
Off the desk. The long journey of Probability Logic - Volume 1
This semester, following five prior attempts, I’m going to complete a progressive series on inference in my course titled “Clinical Inquiry I: Causation and inference.” The plan for this progressive series is to cover statistical inference; then Bayesian inference; and finally a relatively new inferential framework from computer science (artificial intelligence) called Probability Logic. I have never been able to get past a cursory introduction to Probability Logic. My prior approaches to statistical inference and Bayesian inference were less effective than my current approach. This required the class to take a more time on these two subjects. Which means I was never able to get to Probability Logic.
Therefore, the probability of getting to this final form of inference, given my history, was 0 [P(E) = 0, with E being the event of getting to that form of inference]. So the odds were also 0 [(Odds(E) = P(E) / 1 - P(E))]. I guess Hume, on this, was correct. The past cannot always be used to predict the future.
There are three forms of inference being progressively covered. Induction, made probabilistic by statistical inference; Abduction, made probabilistic by Bayesian inference (theorem, equation); and Deduction, made probabilistic by Probability Logic. The reason for making these three forms of inference probabilistic is that, for one reason or another, the inferences are not certain. They are uncertain. In terms of probability, a certain event is either P(E) = 0 (certain not to occur); or P(E) = 1 (certain to occur).
To utilize labels that people may be more familiar with:
Induction & statistical inference includes “research” as in studies to learn something about a population from a sample of observations.
Abduction & Bayesian inference is “diagnosis” or perhaps even prognosis, or a bit more generally classification. I place this form of inference second because it relies heavily on statistical inference and overlaps with it in many ways. But also because it is clinically utilized in specific situations in a similar way to Deduction (which is third). There is also a form of statistical inference that is called Bayesian in the sense that we have a belief about the population that we continue to adjust with the addition of more information (data). For a good review of the history of the debate about statistical inference between the “frequentists” and the “Bayesians” see McGrayne’s “The Theory that Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy“.
Deduction & probability logic is “clinical decision making” (deciding to “do X”). It comes last because in practice it requires both Induction and Abduction. To decide to “do X” - you must have an idea of what the effects of X are in the population of interest (Induction), and you must know whether the person you’re deciding to “do X” with is of the population of interest (Abduction, i.e. classification is putting someone into a population).
All three are necessary for “evidence” (or my preference, “knowledge”) based practice (EBP). There is, however, much more emphasis in the EBP literature (meaning, the literature about EBP) on research and diagnostic accuracy. And in my course, Probability Logic has been neglected but not forgotten.
This post kicks off a series on probability logic. It is my attempt to hash out ideas as I prepare, do, and reflect on teaching about probability logic in my course. The idea for the class activity on probability logic comes from a TA (we’ll call him Greg, because that’s his name). Greg’s idea was to use BlackJack as a game where you get better at making a decision (“do X” is get, or don’t get, another card, “hit” or “hold”). You get better by knowing the probability (or Odds) of drawing a card within or beyond a range of point values. Brilliant insight from Greg! So far most of the “hashing out” of Greg’s idea has occurred on white boards, in a moleskin notebook, and at the cards table with a colleague and TAs (Greg and Matt are getting better at card counting for sure). And we keep referring to the excellent book “Bringing down the house” (well, most people know it as the movie “21”) on this topic.
There should be some preliminaries about probability logic before talking too much about the transition from BlackJack to ClassJack as an activity. I learned about probability logic after (not during) taking a philosophy course called - “Introduction to Logic” that really introduced me to the formal difference between induction and deduction. Probability was commonly utilized with induction. But probability was not commonly utilized with deduction. Deduction, as I was taught in my logic class in the summer of 2004 was about “sure things”. When a deductive argument (syllogism) was a “valid” form, and the premises were true, then it was “sound” - that is correct and the conclusion was sure to be true.
The summer of 2004 was a full year after defending my dissertation. I had gone through the existential crisis, burnout and depression that accompanies the completion of a nearly continuous 12 year period of higher education, while getting married, having 2 children, buying/selling/buying two homes and being in my fourth job - at a university, where I belonged all along but had to earn the right to be there. God saved me from the existential crisis - a testimony of those events is ready for any that would like to hear it. I was attending Church and meeting with several people (including Gary Moore - if you haven’t heard about Gary yet take a look at the wittiest of the comments to many of my posts and check out his page for deep thoughts, humor and photography). Gary et al were making me thinking differently and using a language I had only heard about - philosophy and logic. Therefore, naturally, that summer I enrolled in two “101” courses - Intro to Philosophy and Intro to Logic. As an UG student I had taken Intro to Ethics in 1991. At the time I was an atheist, but I didn’t want to be unethical ;) ; and I took a Rhetoric and Argument course in 1990 - I did like to argue. But logic had not made it on my course schedule until 12 years after earning my BS degree.
Having an ScD (doctorate in science) in ergonomics and epidemiology I was familiar with statistical inference - but had never heard it discussed as “induction” prior to that logic course. And I had never, before that course, fully appreciated the rich philosophical history and discussion on the difference between induction and deduction. I had heard the terms - just never made a complete connection. Abduction did not enter the picture for me until a few years later when reading Stephen Meyer’s “Signature in the Cell, which led me to read C.S. Peirce (late 19th century philosopher, spelled correctly pronounced “purse.”).
Deduction is what we do when we made a clinical decision to do something - “do X”. The basic format of the deductive inference utilized for the “do X” decision making is:
Premise 1: X implies Y (translate, not without further consideration, X causes Y)
Premise 2: X (that is, do X)
Conclusion: Y (get the outcome being sought after)
Premise 1 is rather universal. It is a statement about a relationship between X and Y. This is usually something we know from making observations (i.e. statistical inference). It can be modified from the above bare bones version. It can be that X implies Y when Z (meaning when Z is true). Perhaps airway clearance implies improved gas exchange when mucus is blocking the airways. In that such a conditional is pretty par for the course in a clinical situation.
The new deductive inference would then be expanded:
Premise 1: X implies Y given Z
Premise 2: Z
Premise 3: X implies Y
Premise 4: X
Conclusion: Y
In this new form, in clinical practice, premise 2 is obtained using Abductive inference. We can know Z because we are told Z by someone we trust (we are told by someone that there is mucus blocking the airways); or we do our own investigation. We could stick a camera down someone’s throat into their trachea and bronchi (flexible bronchoscopy - not likely), we could look at an x-ray, listen to lung sounds, consider the person’s medical history, their symptoms, etc. In all of these - aside from the bronchoscopy and actually “seeing” the mucus blocking the airways) we are abductively inferring that mucus is blocking the airways based on other signs. That is also an implication. For example: Mucus blocking airways implies crackles when listening to lung sounds. Even if we accept that implication, we invert the implication (using Bayesian inference) to be crackles implies mucus blocking the airways. The reason we need Bayesian inference is because the logical implication P implies Q is not equivalent to Q implies P. The point being - the abductive / Bayesian inference is part of this overall process:
Premise 1: X implies Y given Z
Premise 2: Z given S (conclude Z is true based on the existence of a or a set of Signs)
Premise 3: S
Premise 4: Therefore, Z
Premise 5: Therefore, X implies Y
Premise 6: do X
Conclusion: Therefore, Y
If you read last weeks post on PICO questions - you’ll see that this deductive inference has all of the pieces. P is the population or patient, it is the “Given Z” - when, in which situation, does X imply Y? I is intervention, it is the X in the previous notation. C is the comparison - it is part of the process to determine that X implies Y given Z and can be more explicitly added. O is the outcome, it is Y.
Re-written:
Premise 1: Intervention implies Outcome given Population as compared to Comparator
Premise 2: Population given Sign
Premise 3: Sign is present in this patient
Premise 4: Therefore, Patient is part of Population
Premise 5: Therefore, Intervention implies Outcome in this Patient
Premise 6: do the Intervention
Conclusion: Therefore, get the Outcome
All well and good. We have a format that builds from the logic to the clinical situation. It is general enough to be applied across many areas of clinical practice decision making. We have a deductive inference that pulls from knowledge we obtain from statistical inference; as well as from abductive (Bayesian) inference.
In classical logic with deduction using a valid form (and the above is a valid form), the conclusions follow necessarily from the assumed to be true premises. The conclusion, “get the outcome” is guaranteed to be true if the premises are all true. Cool. But wait. We know that many of these premises include uncertainty - that is have a certain probability of being true. We do not know with certainty that the premises are true.
This is where probability logic steps in. I came to probability logic from two directions that both led me to studying artificial intelligence as a branch of computer science. The first path was through my friend Ned. Ned was the father of my son’s best friend in elementary school and our families would frequently get together. Ned was a computer engineer. One day I was explaining to Ned my grand plan to use graphs to help untangle the complexity of diagnosing the set of problems in a complicated case. Ned went to his study and got a book for me - Gries & Schneider’s system of equational logic called: A Logical Approach to Discrete Math (LADM). He told me how learning to do the proofs in that book would equip me to think logically and clearly about graphs - to make the complicated less complicated. I could not just jump into this book though. It was dense. But I could read parts of it - and the parts that I read enabled me to see that logicians (as in my logic course) were talking about logic; but that computer scientists were regularly using logic in very discrete and concrete ways. I enrolled in a course called “Discrete Structures” in the math department where I was teaching. It did not use the LADM approach, but it bridged my knowledge between the philosopher’s approach to logic and the computer scientists approach to logic. Discrete Structures was the foundational math course for the computer science program at the university. With this course behind me I then dove into LADM. Still difficult. So I found an online, self paced course on “Formal Methods” that utilized LADM as the core textbook. It was an on campus course at Pepperdine University offered by Dr. Stan Warford and he - well before COVID - recorded his lectures and made them, the problem sets and exams all freely available. Thank you Dr. Warford! (I have thanked him by email, he responded and is a genuinely great guy.) The course is still online. In addition to Dr. Warford’s course I purchased - with my professor status - the solution manual to LADM so that I could grade my assignments on my own. I then spent time - over the course of a few years - working through the book with the help of Dr Warford’s video lectures and assignments (which basically emphasized particular problems and proofs from the LADM exercises). With LADM behind me as an equational approach to logic, including propositional (deduction) and predicate (deduction with quantifies and sets), I was ready to take on further study of computer science and programming theory for artificial intelligence…. A road that led to probability logic.
But I directly learned about probability logic while learning more about Bayesian inference. It turns out a lot of great work on Bayesian inference has been done in the field of computer science and artificial intelligence. There is a uniquely Bayesian network approach to causality that is both simple and dense. There’s a larger story to be told about the causality journey I have taken by reading and studying Dr. Judea Pearl’s writings for another time. Artificial intelligence includes a wide set of disciplines and is a fascinating area of thought about thought. It is a model of reasoning and as such requires us to consider how we reason, learn and know. The study of AI led me to the study of language and linguistics within those fields - and the work of Earnest Adams. In particular, his A Primer on….. Probability Logic.
This backstory and reflection has helped me organize my thoughts. The journey to now teaching about probability logic by using ClassJack has been long and I wanted to get it out before diving deeper into what ClassJack is prior to exploring what it can do, and then actually testing whether it accomplishes its goal.
To wrap up this first volume on this Probability Logic Series:
The probability of a conclusion from a deduction cannot be greater than the probability of any of the premises. This makes intuitive sense. And yet it is a profoundly under discussed topic that leads clinicians to ponder the usefulness and utility of evidence based practice; and everyone to be worried about the potential impact of bias as a cause of uncertainty. Bias is a cause for uncertainty, for sure. But it is not the only cause. To truly understand the impact of bias as a cause for uncertainty we must first understand the causes for uncertainty built into our reasoning. Knowing the probabilities helps to make “informed” decisions. Even if what you know is that the probability is 0.5 (50/50), at least the decision is informed and that odds of success when doing the intervention as equal to the odds of success when not doing the intervention. It might not be the “information” you’d like to have - but it may be the only information that you can have. And to me - the real bias occurs when you assume certainty when there is no certainty.
In Volume 2 of the series I will describe the game - ClassJack - that we’re going to utilize to study probability logic in class.