The Art of Uncertainty

1. Uncertainty is a Personal Relationship

Uncertainty is a relationship between someone (perhaps ‘you’) and the outside world, so it depends on the subjective perspective and knowledge of the observer.

Subjectivity of Uncertainty. Uncertainty isn't an inherent property of the world, but rather a reflection of our individual knowledge and perspective. What's uncertain to one person may be clear to another, depending on their experiences and information. This personal aspect means that uncertainty is not a fixed state, but a dynamic one that evolves as we learn and adapt.

Components of Uncertainty. This relationship comprises a subject (the one experiencing uncertainty), an object (the thing they're uncertain about), a context (the known information), a source (the reason for uncertainty), an expression (how uncertainty is communicated), and an emotional response. Consider a coin flip: your uncertainty about the outcome is different from mine if I've already seen the result.

Embracing the Subjective. Recognizing the subjective nature of uncertainty allows us to be more understanding of differing viewpoints and more critical of claims presented as absolute truths. It encourages us to explore the basis of our own uncertainties and to seek out information that can reduce them. This perspective is crucial for navigating the complexities of life, from personal decisions to scientific inquiry.

2. Quantifying Uncertainty with Numbers

We demand rigidly defined areas of doubt and uncertainty!

Words vs. Numbers. While everyday language uses terms like "likely" or "possible," these are often vague and open to misinterpretation. Putting uncertainty into numbers, expressing it as probabilities, provides a more precise and standardized way to communicate degrees of belief or confidence. This is especially important in fields like intelligence analysis, medicine, and climate science, where clear communication is essential.

Scoring Rules and Honesty. Quizzes and scoring rules can help quantify our uncertainty and reveal overconfidence. Proper scoring rules, like the quadratic score, encourage honesty by penalizing exaggerated claims and rewarding accurate assessments. These tools can be used to identify individuals with well-calibrated judgment and to improve decision-making in various contexts.

Foxes vs. Hedgehogs. Good forecasters tend to be "foxes," who are skeptical of grand theories and willing to adjust their ideas in light of new evidence, rather than "hedgehogs," who cling to a single overarching view of the world. Good forecasters use multiple sources of information, are open to new knowledge, and are happy to work in teams.

3. Probability Tames Chance

It was not until the Renaissance that any attempt was made to analyse what was going on when dice were thrown, and then, like the release from a pent-up dam, the field exploded into applications in pensions and annuities, astronomy and the law, as well as, of course, gambling.

From Games to Science. The formal study of probability emerged relatively late in history, despite millennia of gambling and games of chance. It wasn't until the Renaissance that mathematicians like Cardano and Pascal began to analyze the underlying principles, leading to a revolution in fields ranging from finance to astronomy. Probability provides a framework for understanding and quantifying randomness.

Basic Rules of Probability. The rules of probability, such as the addition rule (OR) and the multiplication rule (AND), can be intuitively grasped through examples like coin flips and sock-drawing problems. These rules allow us to calculate the probabilities of complex events by breaking them down into simpler components. Probability trees are a useful tool for visualizing these calculations.

Beyond Equally Likely Outcomes. While early probability theory focused on situations with equally likely outcomes, modern probability extends to more complex scenarios where outcomes have different probabilities. The binomial distribution, for example, allows us to calculate the probability of a certain number of successes in a series of independent trials, even when the probability of success is not 50%.

4. Coincidences Happen More Than You Think

Coincidence: ‘a surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection’.

Defining Coincidence. Coincidences are surprising concurrences of events that seem meaningfully related but lack an obvious causal link. They often involve unexpected connections, grab our attention, and defy easy explanation. Examples include meeting someone with a shared connection in an unlikely place or an object reappearing unexpectedly.

The Law of Truly Large Numbers. The Law of Truly Large Numbers explains why coincidences happen more often than we might expect. With enough opportunities, even very rare events will eventually occur. This principle can be used to analyze seemingly improbable events, such as shared birthdays or matching phone numbers.

The Illusion of Meaning. While coincidences can be fascinating, it's important to avoid attributing them to external forces or mystical connections. Often, they are simply the result of chance and our tendency to seek patterns, even where none exist. The key is to appreciate the surprise without overinterpreting its significance.

5. Luck: Control vs. the Uncontrollable

Whether an exposed subject does or does not develop a cancer is largely a matter of luck; bad luck if the several necessary changes all occur in the same stem cell when there are several thousand such cells at risk, good luck if they don’t.

Defining Luck. Luck is the operation of chance, taken personally. It involves events that are beyond our control, often perceived as unlikely, that have a significant impact on our lives. Luck can be good or bad, and it can manifest in various forms.

Types of Luck. Luck can be categorized into resultant luck (outcomes despite similar situations), circumstantial luck (being in the right place at the right time), and constitutive luck (inherent traits and circumstances). Constitutive luck, in particular, plays a major role in shaping our opportunities and life trajectories.

Beyond Superstition. While some people believe in luck as an external force, it's more useful to focus on what we can control. Developing skills, seizing opportunities, maintaining a positive attitude, and building resilience can all increase our chances of "good luck" by putting us in a better position to navigate the uncertainties of life.

6. Randomness: A Useful Illusion

We demand rigidly defined areas of doubt and uncertainty!

Effective Randomness. While the universe may or may not be fundamentally random, "effective randomness" – unpredictability that is practically indistinguishable from true randomness – is extraordinarily useful. It ensures fairness, balances groups, and enables complex simulations.

Random Number Generators. Modern random number generators are often deterministic algorithms that produce pseudo-random sequences. These sequences are unpredictable and pass statistical tests for randomness, making them suitable for various applications, from gaming to cryptography.

The Clumpiness of Randomness. Our intuition about randomness is often flawed. We tend to expect regularity and evenness, but true randomness is often clumpy, with clusters and patterns that can seem surprising. Understanding this can help us avoid misinterpreting random events as meaningful signals.

7. Bayesian Thinking: Updating Beliefs with Evidence

Once we accept a personal, subjective view of probability and uncertainty, we are led naturally to Bayesian analysis, in which we use the theory of probability to revise our beliefs in the light of new evidence.

Bayes' Theorem as a Learning Tool. Bayesian analysis provides a framework for updating our beliefs in light of new evidence. It starts with a prior belief, incorporates new data through a likelihood function, and produces a posterior belief that reflects the combined information. This process allows us to learn from experience and refine our understanding of the world.

Conditional Probability and Independence. Bayesian thinking relies on conditional probability, where the probability of an event depends on prior knowledge or assumptions. Conditional independence, where events are independent given a common factor, is a powerful tool for simplifying complex analyses.

Humility and Open-Mindedness. A key aspect of Bayesian thinking is the willingness to revise our beliefs in light of new evidence. This requires humility and a recognition that our initial assumptions may be wrong. By assigning a small probability to alternative possibilities, we can remain open to surprises and avoid becoming trapped in rigid belief systems.

8. Science: Embracing Uncertainty

We demand rigidly defined areas of doubt and uncertainty!

Beyond Settled Science. While science provides a foundation of established knowledge, active scientific research is characterized by uncertainty. Scientists grapple with questions about the magnitude of physical quantities, the number of things, the nature of virtual quantities, and the causes of past events.

Statistical Models and Assumptions. Scientific investigations often rely on statistical models to analyze data and draw conclusions. However, these models are based on assumptions that may not be entirely true. It's crucial to acknowledge these limitations and to assess the sensitivity of results to different assumptions.

The Importance of Humility. Scientific progress depends on acknowledging uncertainty and being willing to revise our understanding in light of new evidence. This requires humility and a recognition that our current knowledge is always incomplete. By embracing uncertainty, scientists can avoid overconfidence and make more robust claims.

9. Expressing Confidence in Analysis

There is no such thing as absolute certainty, but there is assurance sufficient for the purposes of human life.

Beyond Numerical Estimates. While numerical estimates of probability are valuable, they don't always capture the full extent of our uncertainty. Factors such as the quality of evidence, the rigor of the analytic process, and the complexity of the situation can all influence our confidence in a particular analysis.

Qualitative Confidence Scales. To address this, many organizations use qualitative confidence scales to supplement numerical estimates. These scales provide a way to express the level of trust we have in our analysis, based on the strength of the underlying evidence and the potential for future revisions.

The Value of Transparency. By being transparent about our confidence levels, we can help audiences better understand the limitations of our knowledge and avoid overinterpreting our findings. This can lead to more informed decision-making and a greater appreciation for the complexities of the world.

10. Causality: Untangling Cause and Effect

I can as easily throw one, three, five as two, four, six.

General vs. Specific Causation. Assessing causality involves two distinct questions: whether an action or exposure tends to cause an outcome in general (general causation) and whether a specific action or exposure led to a particular outcome (specific causation). Both types of causal claims are fraught with uncertainty.

Randomization and Causality. Randomized trials provide the strongest evidence for causality by creating balanced groups and minimizing the influence of confounding factors. However, in many situations, randomized trials are not feasible, and we must rely on observational studies.

Attributable Fraction and Legal Rulings. The attributable fraction, which estimates the proportion of cases caused by a particular exposure, can be used in legal rulings to determine liability. However, it's important to use caution when applying population-level statistics to individual cases.

11. Prediction: A Glimpse into the Murky Future

We demand rigidly defined areas of doubt and uncertainty!

The Limits of Prediction. Predicting the future is inherently uncertain, as it involves projecting current trends and patterns into an unknown and potentially volatile world. While mathematical models can be useful tools for forecasting, they are always based on assumptions that may not hold true.

Sources of Uncertainty. Uncertainty in prediction models arises from various sources, including unavoidable variability, epistemic uncertainty about parameters, limitations in model structure, and the influence of human behavior. Longer-term predictions are particularly susceptible to these uncertainties.

Embracing Humility. Given the inherent limitations of prediction, it's crucial to approach forecasts with humility and to avoid overconfidence. Acknowledging uncertainty, exploring multiple scenarios, and seeking diverse perspectives can help us make more informed decisions in the face of an uncertain future.

12. Risk Management: Preparing for the Worst

We demand rigidly defined areas of doubt and uncertainty!

Assessing and Comparing Risks. Risk management involves identifying potential threats, assessing their likelihood and impact, and developing strategies to mitigate them. This process often involves comparing different types of risks, from environmental hazards to financial crises.

Quantitative Risk Analysis. Quantitative risk analysis uses mathematical models to estimate the probabilities and consequences of extreme events. However, these models are based on assumptions that may not be accurate, and they often struggle to capture the full range of potential outcomes.

The Importance of Resilience. In the face of deep uncertainty, it's more important to focus on building resilience than on making precise predictions. This involves developing flexible strategies that can adapt to a wide range of potential futures and that can help us recover quickly from unexpected shocks.

Last updated: April 19, 2025