Bayesian Reasoning: Updating Beliefs Under Uncertainty

Introduction

Clinicians make decisions under uncertainty. Every diagnosis, treatment choice, and prognosis involves reasoning through incomplete and uncertain information. Bayesian reasoning mirrors how clinicians naturally think—it provides a framework for updating beliefs as new evidence emerges, rather than treating each decision as an isolated statistical test.

Statistical inference is central to clinical decision-making, but frequentist and Bayesian methods answer different questions. Frequentist procedures evaluate how data and estimators behave under repeated sampling assumptions. Their results are often reduced in practice to a binary "significant" or "not significant" decision, although confidence intervals and effect estimates can communicate much more. Bayesian methods instead represent uncertainty about an unknown quantity directly and ask how a prior probability should change in light of new evidence.

Bayes' theorem can estimate the probability of a possible explanation given an observation, such as P(C|E). That inversion does not, by itself, establish that C causes E. Causal interpretation requires a defensible model of the relationships among causes, effects, alternatives, and selection processes. Bayesian updating can then revise probabilities within that model as evidence accumulates.

This distinction fits the critical realist framework introduced in Lesson 2. Observations belong to the empirical domain; events may occur in the actual domain whether or not we observe them; and causal powers, mechanisms, and structures belong to the real domain. Probability can express what we know about those domains, but probability alone does not collapse them into one.

Foundations of Bayesian Reasoning

Bayesian reasoning is based on Bayes’ Theorem, which describes how to update our beliefs when new information arrives. To understand Bayes’ Theorem you must first understand the notation of probability and conditional probability.

P(E): means the probability of E, whether E is an effect, evidence or an event. Sometimes I’ll use S as a “sign or symptom”.

P(E|X): means the probability of E given X, whether X is a cause or condition (C), a hypothesis (H), or a disease or disorder (D). The vertical bar means “given” or “conditional on”: restrict attention to situations in which X holds, then ask how often or how plausibly E occurs. This is not the same as the logical implication X → E. An implication asserts a truth-functional relationship; a conditional probability can take any value from 0 to 1.

Traditional classical propositional logic is bivalent: a proposition is assigned one of two truth values, true or false. In Boolean notation those truth values can be represented as 1 and 0, but they are not probabilities. Probability provides room between certainty and impossibility by representing uncertainty about propositions or events on a scale from 0 to 1.

Bayes’ Theorem, describes how to update our beliefs when new information arrives and more importantly enables us to invert the conditional probability. In the equation below, notice how P(E|H) (the probability of evidence given a hypothesis) allows for the calculation of P(H|E) (the probability of the Hypothesis given the Evidence).

P(H|E) = P(E|H) x P(H) / P(E)

Where:

• P(H|E) = Posterior probability (updated belief after new evidence)

• P(H) = Prior probability (initial belief before seeing new evidence)

• P(E|H) = Likelihood (probability of observing the evidence if the hypothesis is true)

• P(E) = Marginal probability (overall probability of the evidence occurring)

In summary terms, Bayesian reasoning does two things:

1. It revises our beliefs when new information is presented. Clinicians continually refine their diagnostic thinking based on test results, patient history, and response to interventions.

2. It inverts the conditional probability from P(E|H) to P(H|E). This shift is crucial in clinical reasoning because we are usually trying to infer the cause of an observed effect (e.g., determining the probability of a disease given a test result, rather than the probability of a test result given a disease).

Here, H and E remain general: H is a hypothesis that may or may not hold, and E is observed evidence. In Lesson 4’s diagnostic framing, H can represent a condition and E a finding. The same structure then expands into the familiar 2 × 2 table used to define sensitivity, specificity, and likelihood ratios.

From Hypothesis and Evidence to Cause and Effect

The concepts for which we use Bayes’ theorem are flexible - in fact the notation used in Bayes’ theorem reflect the context and doesn’t really matter other than for the people using it.

I’d like to now do the same process but using the concepts Cause and Effect.

P(C|E) = P(E|C) x P(C) / P(E)

This is saying that the Probability of the Cause given the Effect (notice the abduction), is based on the Probability of the Effect given the Cause (notice the induction); multiplied by the probability of the cause, and divided by the effect.

You may remember me saying either here, or in a course, that genuinely rare signs deserve our attention. A low marginal probability P(E) places a small denominator in Bayes’ theorem and, all else equal, can produce a substantial posterior shift. The important qualification is discrimination: the finding must be much more likely when C is present than when C is absent. Because P(E) includes the finding among people with C and among alternative explanations, a finding that is equally rare under both does not favor C. The lab above makes that distinction visible through P(E|C), P(E|not C), and their likelihood ratio.

Why This Matters for Statistical Inference and Clinical Reasoning

Bayesian reasoning provides a structured way to update beliefs based on new information and inverting conditional probabilities to reflect how clinicians refine their understanding over time. This aligns naturally with diagnostic reasoning, treatment selection, and prognostic estimation, reinforcing the role of inference in clinical decision-making.

However, despite its intuitive fit for clinical practice, Bayesian reasoning is not the dominant statistical framework used in research or taught in physical therapy education. Instead, most clinical studies rely on frequentist statistics, which treat probability as a measure of long-term frequency rather than an evolving degree of belief. This creates a disconnect between how research generates evidence and how clinicians apply it in practice.

To appreciate why this matters, we must compare the frequentist and Bayesian approaches, examining how they shape statistical inference, evidence interpretation, and ultimately, clinical decision-making.

Frequentist vs. Bayesian Thinking in Clinical Practice

Traditional frequentist methods define probability as the long-term frequency of an event occurring over repeated trials. In this framework, probability is not about belief or degrees of certainty but is instead tied strictly to how often something happens under repeated sampling. This leads to a strict interpretation where:

• A hypothesis is either true or false—there is no probability assigned to it (which is an unfortunate consequence of null hypothesis statistical testing - something I teach out of necessity because I exist in a system of evidence based practice that I reject - ironic isn’t it).

• Statistical significance (p-values) determines whether to reject the null hypothesis—without incorporating prior knowledge.

• Each study is treated as an isolated experiment, without formally integrating previous evidence (at least in terms of the the statistics performed in any particular study).

By contrast, Bayesian reasoning frames probability as a degree of belief that can be continuously updated in light of new evidence. It allows clinicians and researchers to:

• Incorporate prior knowledge (e.g., clinical experience, epidemiological data, previous studies).

• Quantify uncertainty using credible intervals rather than confidence intervals, providing direct probability estimates for clinical effects.

• Update probability estimates dynamically, rather than treating new evidence as completely independent from prior understanding.

Why This Matters for Clinical Decision-Making

Frequentist results are often interpreted as rigid, binary conclusions through questions such as:

• “Does this treatment work?”

• “Is this diagnostic test statistically significant?”

• “Should we reject the null hypothesis?”

That dichotomous interpretation discards information about effect size, precision, and uncertainty. It is a misuse to avoid, not an unavoidable property of every frequentist analysis.

Bayesian reasoning aligns naturally with how clinicians think in real-world practice, allowing us to ask:

• “How likely is this treatment to benefit this patient?”

• “How much does this test result change my belief about the diagnosis?”

• “Should I adjust my treatment strategy based on the evolving clinical picture?”

Unlike frequentist methods, Bayesian inference provides a more nuanced approach. It integrates prior knowledge with new evidence to estimate how likely something is to be true in a given context, making it especially relevant for individualized clinical decision-making.

This makes it clear that we are already engaging in Bayesian reasoning and inference—yet we are often constrained by a paradigm that does not explicitly recognize it. Many advocate for the frequentist paradigm not necessarily because it is better, but because it is dominant in research, education, and practice. Moreover, there is a widespread belief that students must first learn the frequentist paradigm before they can understand Bayesian methods. However, this approach merely extends the time required for training and reinforces a system that may not be the most clinically relevant.

For example, in DPT education, students must be familiar with frequentist statistics to pass the National Physical Therapy Exam (NPTE), which evaluates statistical concepts solely through a frequentist lens. This forces educators to prioritize frequentist methods over Bayesian approaches, despite the fact that Bayesian reasoning more accurately reflects how clinicians make decisions in practice. And it puts Bayesian’s like me in a pickle!

The Challenge of Paradigm Shifts in Education

Bayesian inference provides a framework that better aligns with clinical reasoning—yet frequentist methods dominate research, education, and standardized testing. Many advocate for maintaining the frequentist approach simply because it is entrenched, rather than because it is superior.

One challenge of shifting paradigms is that we often feel compelled to teach the old system before introducing the new one. This creates an unnecessary barrier—Bayesian thinking is often more intuitive, yet students and clinicians that are trained in Bayesian methods (like me for instance) are first trained in frequentist concepts before they are introduced to Bayesian methods. In DPT education, this is particularly problematic:

• Students must learn frequentist statistics to prepare for standardized exams like the NPTE (National Physical Therapy Exam), where statistical questions are framed exclusively in frequentist terms.

• Educators are forced to teach methods that may not be the most clinically useful simply because they are required for credentialing.

• This makes it harder to introduce Bayesian reasoning as the primary framework for clinical decision-making.

This resistance to Bayesian thinking is not unique to DPT education. Sharon Bertsch McGrayne’s book, The Theory That Would Not Die, chronicles the long history of opposition to Bayes’ Rule—even in fields like cryptography, military strategy, and artificial intelligence, where it has repeatedly proven its value. The book illustrates how Bayes’ Rule has persisted despite skepticism, ultimately reshaping fields as diverse as submarine tracking, medical diagnostics, and machine learning. This historical reluctance to adopt Bayesian methods mirrors the challenge faced in clinical education today—where entrenched statistical conventions slow the adoption of a more intuitive and flexible reasoning framework.

If we acknowledge that Bayesian reasoning better reflects how clinicians make real-world decisions, the challenge is not just advocating for it, but finding ways to integrate it into education and practice despite existing constraints. Should Bayesian reasoning be introduced earlier in training? Should frequentist methods be taught only in the context of research interpretation rather than as the default paradigm?

While frequentist statistics will remain dominant for some time, clinicians do not have to remain trapped in this framework. Bayesian approaches allow for probabilistic, dynamic, and personalized decision-making, providing a more natural fit for the complexities of clinical care.

The Statistical Paradigm Trap Summary

Even if you recognize that Bayesian approaches better reflect real-world decision-making, frequentist methods remain dominant due to:

• Educational Constraints: DPT programs prioritize frequentist statistics to prepare students for standardized exams like the NPTE, leaving little room for Bayesian concepts.

• Publishing & Research Standards: Many journals primarily accept frequentist statistical analyses, making Bayesian research harder to publish.

• Professional Inertia: Frequentist methods persist in clinical guidelines, making Bayesian reasoning feel “unfamiliar” even when it aligns more closely with clinical reasoning.

Despite these barriers, Bayesian reasoning is already embedded in how clinicians think, and it can be formally integrated into both research and practice.

Bayesian Thinking and Clinical Uncertainty

Clinical decisions are rarely made with absolute certainty. Instead of proving or disproving a diagnosis, clinicians implicitly estimate probabilities based on patient presentation, test results, and prior knowledge. Bayesian reasoning can formalize this approach when possible, or even informally by helping to identify the reasons for belief adjustment by:

• Refining probability (subjective or objective) estimates based on accumulating data.

• Adjusting diagnostic confidence (subjective or objective) as new test results arrive.

• Optimizing treatment decisions in light of evolving patient responses.

For example, when diagnosing deep vein thrombosis (DVT):

• The prior probability is informed by clinical factors (e.g., leg swelling, risk factors).

• A D-dimer test modifies that probability based on its sensitivity and specificity.

• If the test is positive, the probability of DVT increases; if negative, it decreases.

• If probability remains uncertain, additional testing (e.g., ultrasound) further refines the estimate.

Addressing the Subjectivity Critique of Bayesian Inference

One common critique of Bayesian methods is that priors introduce subjectivity—two different clinicians might start with different priors. However:

1. All reasoning is based on prior beliefs. Bayesian inference simply makes these priors explicit and structured.

2. Priors can be evidence-based. Epidemiological data, mechanistic studies, and expert consensus can inform reasonable priors.

3. As more evidence accumulates, the influence of the prior diminishes. The posterior probability converges toward the true value, making Bayesian inference increasingly objective over time.

4. Priors are - like all probabilities - conditional. Even the supposedly “non conditional probability of X (P(X))” is actually the “conditional probability of X given not knowing anything (P(X| not knowing anything))”. This underscores that even frequentist probabilities depend on assumptions about what is known and unknown.

Hidden Subjectivity in Frequentist Statistics

While Bayesian methods explicitly define priors, frequentist statistics also contain subjective choices, even though they are often framed as objective. These include:

• Selecting significance thresholds (e.g., p < 0.05). Why 0.05? This threshold is an arbitrary convention rather than a mathematically justified boundary between “significant” and “not significant.” Results just above or below this cutoff are treated dramatically differently, despite minimal difference in evidence.

• Choosing which statistical test to use. Different tests make different assumptions (e.g., t-tests assume normality, while nonparametric tests do not). The choice of test often depends on researcher preference or convenience rather than a strict rule.

• Deciding on inclusion/exclusion criteria for studies. What data should be included in an analysis? Which studies should be included in a meta-analysis? These decisions shape results and can introduce bias based on the chosen criteria.

• Data-dependent stopping rules. Some studies are stopped early if p-values cross a chosen threshold, but this introduces subjectivity in defining what is “statistically significant” and may inflate reported effects.

• P-hacking and selective reporting. Researchers may run multiple analyses and report only those that produce p < 0.05, leading to false-positive results and publication bias.

However, the subjectivity of frequentist methods does not stop at the research level—it extends into clinical decision-making itself, where statistical results must be interpreted and applied to patient care.

The Subjectivity of Applying Frequentist Statistics in Clinical Practice

Even when clinicians rely on frequentist-based research, they must subjectively decide how to act on the evidence:

• What constitutes “clinical significance” vs. statistical significance? A treatment may be statistically significant but have a minimal real-world effect. Clinicians must determine if an effect size is meaningful for their patients.

• How should population-level evidence be applied to an individual patient? Frequentist methods generate conclusions for populations, but individual patient factors may influence treatment outcomes.

• What if a study is inconclusive? When a p-value is “not significant,” does that mean the treatment is ineffective, or was the study underpowered? Clinicians must navigate this uncertainty without guidance from frequentist statistics.

• How should multiple conflicting studies be weighted? In Bayesian inference, conflicting studies would update our probability estimate, but frequentist methods do not provide a formal process for reconciling contradictory findings.

The Advantage of Bayesian Transparency

While Bayesian inference is sometimes criticized for allowing subjectivity in the choice of priors, frequentist statistics embed subjectivity at multiple levels—in study design, statistical analysis, and clinical interpretation. The difference is that Bayesian reasoning makes the inferential process transparent and structured, allowing for dynamic updates as new evidence emerges.

This approach aligns far better with real-world clinical decision-making, where uncertainty is the norm, evidence is constantly evolving, and individualized patient factors must be considered.

What’s Next?

Next, we will make Bayesian updating concrete. Lesson 4 applies prior probability, diagnostic information, and posterior probability to diagnosis and prognosis while keeping the limits of population-to- individual inference visible.