PDF Summary:Bernoulli's Fallacy, by Aubrey Clayton

Probability underpins all observational science and data-based decision making. But as Bernoulli's Fallacy by Aubrey Clayton reveals, our current frequentist statistical methods are built on a fundamental error in causal reasoning. This crucial mistake has led to widespread misinterpretations of probability and statistical significance.

Clayton traces the historical development of probability theory, contrasting the subjective Bayesian approach with its frequentist rival. By examining flawed applications of frequentist methods in fields like social science and eugenics, he ultimately makes a case for embracing probability's subjective nature. Only then can it serve as a tool for understanding rather than an oracle of certainty.

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Bernoulli's Bargain: Confusing Observation Accuracy With Inference Accuracy

Bernoulli's "golden theorem," which he named the Large Numbers Law, made a significant impact on the theoretical development of probability. It resolved a long-standing mystery of how the frequency of an event over some number of trials would relate to its assumed probability for each individual trial. Previously known only from intuition, Bernoulli showed mathematically that the two values would necessarily converge to each other as trials increased indefinitely, thus establishing that the likelihood of an event occurring could be estimated by observing its frequency over a long run of trials.

However, as Clayton explains, Bernoulli went a step further with his reasoning than just demonstrating this convergence. He also introduced the concept of moral certainty to quantify how accurate a frequency-based estimate would be, given some finite number of actual observations. His procedure then required a trade-off similar to a project management maxim: "You can have it quickly, well-made, or at low cost. Choose any two." He showed that to estimate a probability within some specified margin of error with a certain degree of moral confidence, one would need to conduct a corresponding number of trials.

As Clayton describes it, the trade-off Bernoulli presented is between the estimate's accuracy, the certainty with which we claim that accuracy, and the sample's size. However, any scientist will know that this choice isn't real. If, for example, we were sampling candies from a bin to see if the machine at the candy factory had been producing the expected ratio of red and green candies, then a large sample would give us a better estimate with greater certainty. That enables us to "pick three" instead of two.

According to Clayton, Bernoulli erred by claiming that after gathering a sufficiently large sample, this data could be used to reverse engineer and deduce the actual probability with the same degree of precision and moral certainty. He made this leap, Clayton argues, by assuming that the idea of closeness was symmetric and could be freely reversed. So, it seemed to Bernoulli, asserting that the observed frequency is very likely to fall within some tolerance of the actual probability was equivalent to the claim that, once observed, the actual probability is very likely within this same tolerance of the observed frequency.

To fully understand why this is a fallacy, we must be careful about which type of probability each of the preceding statements refers to. The former involves how probable an observation (the frequency) is given a hypothesis (the true probability of a single event), whereas the latter is concerned with the probability of a hypothesis (the true probability of a single event) given an observation (the frequency). The first is what we call a sampling probability, and the second is an inferential probability.

Bernoulli's mistake, as Clayton explains it, was not necessarily confusing the two probabilities, though that is sometimes implied by how the wording is phrased. The real mistake was thinking that the probability from sampling was sufficient to produce the inferential probability. As Clayton convincingly demonstrates with an example involving a possible manufacturing error in a bin of candies at a factory, we could have strong prior reasons to believe the true probability is different from the observed frequency, which means that even if we achieved a certain level of moral certainty regarding our estimate's accuracy, it still might be very unlikely to contain the truth.

Practical Tips

• Track your daily habits and their outcomes to observe the law of large numbers in action. Start by choosing a simple daily activity, like taking a vitamin or doing 10 minutes of meditation, and record the immediate and long-term effects you notice. Over time, you should see the frequency of certain outcomes (like feeling more energized or less stressed) align with the probabilities you'd expect. This personal data collection can help you understand the impact of consistent habits on your life.
• Apply moral certainty to assess the reliability of information sources. When you encounter new information, rate the source on a moral certainty scale based on factors like the source's track record, expertise, and transparency. This can help you weigh the credibility of news articles, social media posts, or advice from friends. If a source consistently provides accurate information, you might rate it higher on your scale, influencing how much you trust and act on the information it provides.
• When making decisions based on customer feedback, collect data from various touchpoints over an extended period. Instead of relying on feedback from a single event or location, track customer opinions across different services, locations, and times. This could involve setting up feedback kiosks at multiple branches of your business or using comment cards over several months. The broader data collection will give you a more accurate picture of customer satisfaction and areas for improvement.
• Use a fitness tracker to gather your own large sample data on exercise and health outcomes. By consistently wearing a fitness tracker, you can collect extensive data on your physical activity, sleep patterns, heart rate, and more. Over time, analyze this data to identify trends and probabilities related to your health and fitness goals, such as the likelihood of achieving a certain step count leading to weight loss or improved sleep quality.
• Use a random number generator for everyday decisions to challenge your bias. When faced with a choice where you might be influenced by the fallacy, like choosing a checkout line at the grocery store, use a random number generator to pick for you. This can help you see that your perception of the 'fastest' line may not always align with reality.
• Implement a "scenario analysis" approach when faced with complex decisions, like choosing a new car or planning a vacation. Break down the decision into key factors (cost, safety, enjoyment) and assign probabilities to different outcomes based on your research or past experiences. This exercise will help you apply inferential probability to predict the overall satisfaction with your decision, based on the weighted probabilities of each factor.
• Improve your decision-making by conducting informal experiments in everyday situations. For instance, if you're trying to decide which brand of coffee most people prefer, don't just rely on the number of options available; instead, set up a blind taste test with friends or family. Collect and analyze the data to make an informed decision based on the preferences of your sample group, while being mindful that this sample may not represent all coffee drinkers.
• Practice constructing arguments on various topics and then critically assess them for any logical fallacies. You could do this by writing essays or recording yourself speaking. Afterward, review your work to identify any fallacies you may have included, which will help you improve your argumentation skills and avoid these errors in future discussions.

Prosecutor's Fallacy: Mistaking Sampling For Inferential Probability

Clayton also shows that the commonly-cited base rate neglect fallacy and what legal scholars call the prosecutor's fallacy are both essentially the same error: trying to draw conclusions about a hypothesis from its sampling probabilities alone - i.e., Bernoulli's Fallacy.

In the case of base rate neglect, the problem is that the accuracy rates of a test for some condition such a as a disease are usually all stated the wrong way around: "If the patient has the disease, how likely is the test result?" instead of "If we observe this result, how likely is the disease status?" Failing to consider how likely it is that a positive result indicates a true positive rather than a false positive, especially when the condition in question is rare, can lead one to vastly overestimate how probable it is that a diagnosis or a risk assessment is correct. Clayton gives examples of this showing up in medical screening tests for dementia, various cancers, and even drunk driving. The issue, as he points out, is made even more alarming by the fact that rarer conditions also tend to be more severe, which means that, in any risk assessment, we are generally more concerned about false negatives than we are positive results being false. So even if a test has a high sensitivity - a low chance of a false negative - a bias toward minimizing that error can easily lead us to a low specificity, meaning a high chance of a false positive.

The prosecutor's fallacy is logically identical to base rate neglect, but it involves, instead, the probability of some observed data assuming innocence. Clayton tells how Dr. Roy Meadow, a pediatrician, testified in the murder trial for English mother Sally Clark, claiming that the likelihood of two children in a family like hers dying suddenly of natural causes, specifically Sudden Infant Death Syndrome (SIDS), was 1 in 73 million. Her 1999 conviction stemmed from this figure. However, as Clayton points out, the number Meadow calculated—the likelihood of a family experiencing two SIDS deaths—is simply irrelevant to the legal question at hand. If, for example, we know that the average rate of infant double homicide in a community is, for whatever reason, even lower than Meadow's 1 in 73 million figure, then, logically, that data should actually make us more certain of her innocence, not less! What matters is not the absolute improbability of two infants in one family dying of SIDS, but rather how likely this is compared to the alternative—murder.

Context

• The term "prosecutor's fallacy" gained prominence in the late 20th century as DNA evidence became more common in courtrooms, highlighting the need for accurate interpretation of statistical data in legal settings.
• Understanding these fallacies requires knowledge of conditional probability, which is the probability of an event occurring given that another event has already occurred. Misinterpretation often arises from not properly applying Bayes' Theorem, which helps calculate these probabilities.
• This mathematical formula is used to update the probability of a hypothesis based on new evidence. It highlights the importance of considering both the base rate and the likelihood of the evidence given the hypothesis.
• Sensitivity is the ability of a test to correctly identify those with the condition (true positive rate), while specificity is the ability to correctly identify those without the condition (true negative rate). Both are important for evaluating test accuracy.
• In legal contexts, the consequences of false negatives and false positives can differ significantly. For instance, in criminal cases, a false negative might mean a guilty person goes free, while a false positive could lead to wrongful conviction. Balancing these risks is crucial in both medical and legal decision-making.
• This cognitive bias occurs when people ignore the general prevalence of an event (the base rate) in favor of specific information. For example, if a medical test is 95% accurate, people might ignore the fact that the disease is very rare, leading to incorrect conclusions about the likelihood of having the disease after a positive test result.
• Sally Clark's conviction was eventually overturned in 2003 after it was revealed that key medical evidence had been withheld from the defense, further illustrating the complexities and pitfalls of relying on statistical evidence in court.
• Meadow's calculation focused on the rarity of SIDS without considering the broader context of statistical evidence. In legal settings, it's crucial to compare the probability of different scenarios, such as natural death versus homicide, rather than focusing solely on the rarity of one event.
• In criminal cases, the standard is "beyond a reasonable doubt." This requires a comprehensive evaluation of all evidence, including statistical probabilities, to ensure that conclusions are not drawn from isolated data points.

Multiple Comparisons Problem: Cherry-Picking Hypotheses in Analysis

Clayton argues that the ability of researchers to sort through possible relationships within a data set to find that "one that works," as Bem and others managed to do, is what leads to the problem of multiple comparisons and an increased likelihood of false positive findings. He also demonstrates that a Bayesian probability framework can handle this situation naturally by requiring that more specific hypotheses—those chosen after the fact to fit the available data—must start with a lower probability. This corresponds to raising the bar for significance, which is the exact opposite of how orthodox statistics operates.

Context

• Daryl Bem is a psychologist known for his controversial experiments on psi phenomena, such as precognition. His studies were criticized for methodological issues, including the use of multiple comparisons, which can lead to false positives.
• This term refers to the practice of analyzing data in multiple ways to find significant results. Bayesian methods penalize this by requiring stronger evidence for hypotheses that emerge from such practices.
• In Bayesian analysis, hypotheses that are formulated after examining the data (post hoc) are assigned lower prior probabilities. This means that more evidence is needed to reach the same level of confidence, effectively raising the threshold for significance.

P-Values' Incompleteness: Deciding "Whether" Vs. "How Strongly"

This section examines arguably the greatest sin of significance testing and its use of p-values: the mistaken belief that it's meaningful to say either that "nothing is there" or that "something is there" based solely on how likely the observed data is under an assumption the "nothing" is true. Clayton reviews the lengthy record of criticism p-values have endured, primarily regarding their frequent misinterpretation, and explains why the efforts to restore them will not resolve the fundamental problem. He also describes the "crud factor," the frustration of anyone seeking to analyze large data sets: the unavoidable small correlations that exist between most variables that will, when the dataset is sufficiently large, inevitably be statistically significant even though they may be of no real scientific interest.

Crud Factor: Insignificant Correlations in Big Datasets

The crud factor, Clayton explains, is the tendency for larger and larger statistical analyses to necessarily uncover tiny correlations between basically everything that would be significant by statistical standards if analyzed according to the p-value test, even if these correlations weren't the result of the experimenters trying hypotheses until one proved effective. For example, a study of 57,000 adolescents found significant relationships between birth order and an amusing collection of responses to survey questions about religious views, culinary interest, occupational plans, and membership in farm youth clubs. These differences genuinely existed in the findings, but were likely of no practical significance to the research in question. Clayton points out that with enough data, any real effect, of which there are always some, will become visible, though this doesn't always imply that the result supports the scientific hypothesis it was supposed to.

In a way, crud factor is the worst possible outcome of the Bernoulli fallacy—i.e., of ignoring prior probabilities. The hypothesis that there is precisely no correlation is highly improbable, but frequentist procedures force us to make some judgment of accepting it or rejecting it. This is what led to the situation of medical researchers seeming to disagree amongst themselves about whether certain anti-inflammatory drugs cause heart problems or not. The explanation is that in every instance the null hypothesis—exactly zero effect on heart attacks—was undoubtedly untrue, but only some of the studies had enough power to reject it, and those that did had likely found effects that were small or idiosyncratic to their group of participants, which made them unscientific.

Context

• In large datasets, data mining can lead to overfitting, where models capture noise instead of the underlying pattern, resulting in findings that don't generalize beyond the sample data.
• The replication crisis in science refers to the difficulty in reproducing the results of many studies. This issue is partly due to the reliance on statistical significance without considering the robustness and reproducibility of findings.
• The p-value measures the probability of observing data as extreme as the observed data, assuming the null hypothesis is true. It doesn't measure the size or importance of an effect, leading to potential misinterpretation when small effects are deemed significant.
• Statistical significance does not always equate to practical significance. Ignoring prior probabilities can result in overemphasizing findings that are statistically significant but have little real-world relevance.
• When multiple hypotheses are tested simultaneously, the chance of finding at least one statistically significant result due to random chance increases. This can lead to false positives and is a common issue in large datasets.
• These are effects that are peculiar to a specific study or sample and may not generalize to other populations. They can arise due to unique characteristics of the sample or specific conditions under which the study was conducted.
• In complex systems, such as biological or social systems, it is rare for two variables to have absolutely no relationship. Even small, seemingly insignificant factors can create some level of correlation, making the idea of "no correlation" highly unlikely.
• Journals often prefer to publish studies with significant findings, which can lead to a skewed understanding of research areas, as studies with null results are less likely to be published.

Misconceptions About P-Values: Criticism and the Asa's Response

In reaction to the widespread replication crisis and recognition of its cause in the ubiquitous misuse of p-values, the ASA issued a set of clarifying principles to guide research practice and publication standards. Clayton explains the ASA's position regarding p-values essentially boils down to this: they're acceptable if their meaning is correctly interpreted. However, he argues this is a nonsensical instruction because the approaches to inferring with p-values were initially taught precisely because of their inherent potential for misinterpretations. They seemed to yield the correct results for simple survey sampling questions concerning a population mean because for those problems, the Bayesian and frequentist inferences overlapped. This was because the prior information about the likely parameter value was weak, so it could be safely ignored without consequence, and the alternative hypotheses were simple negations of the null, which made the choice of what to reject obvious. But none of this can be expected to generalize outside those special cases. So even with a thorough understanding of what p-values actually do and do not mean, we can still expect the same errors to continue happening because those errors are built into the tests' logical structure.

Practical Tips

• Create a p-value cheat sheet for everyday decisions to practice interpreting statistical significance. For instance, when reading reports or studies that mention p-values, use your cheat sheet to determine what the p-value implies about the strength of the evidence. This could include a simple scale that relates p-values to levels of evidence, such as "p < 0.01: strong evidence against the null hypothesis, p < 0.05: moderate evidence, p > 0.05: weak or no evidence."

Other Perspectives

• Weak prior information may not always be safe to ignore, as it can still influence the outcome of an analysis, especially in cases where data is limited or noisy.
• In many cases, alternative hypotheses are complex and involve predictions about the magnitude or nature of the effect, which cannot be captured by a simple negation of the null.
• The logical structure of p-value tests is sound when the tests are applied in appropriate contexts and with a clear understanding of their assumptions and limitations.

Optional Stopping: Ignoring Data Due to Potential Outcomes

As Clayton describes it, a specific pathology of NHST is needing to consider other, more extreme figures of the statistic we might have obtained, but that were not actually observed, to determine whether some observed outcome of a test statistic like the sample mean is significant. In other words, the method depends on being able to identify tail regions of an imagined sampling distribution that necessarily contain values of the statistic that aren't available to us—implying that we would ignore our data if those values, instead of the ones we did actually get, had been observed.

Clayton illustrates this with a hypothetical lab experiment where an assistant named Alex is tasked with conducting multiple trials that each, individually, could result in either G for good or B for bad. The research hypothesis is that, according to the new experimental protocol, the likelihood of successful (good) results should exceed the baseline even probability of seeing a good or bad outcome. The null assumption is, therefore, that both outcomes have equal likelihood. Alex conducts six trials, noting the results in a GGGGGB sequence. The problem is that, based on the traditional method, this could be interpreted in multiple ways, depending on what information we have about Alex's planned experiment.

If, for example, Alex had intended to end the experiment regardless of results following six trials, then we would analyze this data using the binomial distribution for how many times a successful result occurred in the six trials. But if, instead, Alex's plan had been to keep conducting trials until the first bad result was recorded—perhaps because each trial was expensive and destroying a vital lab component—then we would use a negative binomial distribution to calculate how likely the sequence is. These, however, yield differing likelihoods. Which interpretation, then, should the scientists use when evaluating the data? In the traditional view, this depends on whether Alex would have been allowed to continue the experiment beyond one failure based on other possible outcomes. What if, say, a backup set of the equipment was available, but Alex was unaware of it? Should that be considered as having the possibility to continue beyond one bad result? Or suppose the building's smoke alarm went off, but was triggered by the smoke emitted from Alex's sixth trial. If a similar fire had occurred after, say, only one experiment, would Alex have continued and ignored the alarm?

As these thought experiments demonstrate, the results produced by a significance test can be subject to change, depending on our assumptions about the outcomes we didn't get. This sensitivity to counterfactual assumptions makes NHST so incoherent. In contrast, the Bayesian procedure doesn't care about the other outcomes we might have obtained because it focuses solely on the ones that were actually observed. Different stopping rules don't affect this type of reasoning since the observed data must necessarily include any aspect that led to the experiment's conclusion.

Practical Tips

• Engage with friends or family in discussions about hypothetical extremes in everyday situations. For instance, when planning a family vacation, discuss not only the expected costs and activities but also what could happen if costs were significantly higher or if unexpected events occurred. This will not only prepare you for a wider range of possibilities but also train you to routinely consider more extreme scenarios, which is a key aspect of NHST (Null Hypothesis Significance Testing) thinking.
• Apply the concept of tail regions to everyday decision-making by setting up "if-then" scenarios. For example, if you're trying to decide whether to bring an umbrella when there's a forecast of rain, consider the "tail" as the unlikely event of rain. If you're in the tail region (low chance of rain), then you might decide not to bring an umbrella. This simple strategy helps you understand the idea of probability and risk assessment in your daily life.
• Create a "chance box" for everyday uncertainties. Write down different possible outcomes of a situation you're uncertain about on slips of paper and place them in a box. When faced with that situation, draw a slip to decide the outcome, acting as if each possibility has an equal chance. For example, if you're unsure about what hobby to pick up, write down options like painting, learning an instrument, or coding, and let the chance box decide for you.
• Start a "context club" with friends or colleagues where you discuss recent news or studies and deliberately focus on the context in which the information was gathered. This could be a monthly meetup where each person brings an article or a piece of data and explains the context behind it. By doing this, you'll train yourself to automatically consider the experimental plan or context when interpreting new information.
• Experiment with different stopping rules in your daily habits to see which ones lead to better outcomes. For instance, when trying to establish a new exercise routine, set a rule that you'll stop your workout after 30 minutes or when you've completed three sets of each exercise, whichever comes first. This approach can help you avoid overexertion and maintain consistency.
• You can test the impact of different outcomes by creating a "What If" journal. Start by identifying a decision you're facing, then write down several possible outcomes for each option. Over the next weeks, revisit your journal and note any new outcomes that occur, reflecting on how they compare to your initial predictions. This practice can sharpen your decision-making skills by making you more aware of the range of possible results and how they might influence your choices.
• Engage in discussions with friends or colleagues about the implications of counterfactual thinking in daily life. During a casual conversation, bring up a recent news event or a personal experience and explore different "what if" scenarios. Discussing how things could have been different can shed light on the importance of understanding and recognizing the influence of counterfactual assumptions on our perceptions and decisions.

The "Sure-Thing" Hypothesis: Accepting Claims From Sample Probability

Clayton uses the scenario of a cube die with a strange predetermined pattern to show the dangers of accepting a hypothesis purely based on how probable it makes the observed data. For instance, take the hypothesis "This die has a inner mechanism that makes it generate the precise results we just witnessed." No matter how unlikely the "fair die" idea is, the significance test will not allow us to reject it because the observed data is certain according to this alternative. This is because we can always imagine an alternative hypothesis of this kind ("the data had to come out that way), which demonstrates the same issue of multiple comparisons we saw with Daryl Bem's ESP experiment.

Practical Tips

• Use the scientific method in your kitchen. If you think a certain ingredient affects the flavor of a dish, conduct blind taste tests with friends or family. Prepare the dish with and without the ingredient and see if the tasters can identify a difference. This can help you understand the actual impact of individual ingredients on the overall taste.
• Use the idea of alternative hypotheses to foster open-minded discussions in your social circles. When debating a topic with friends or family, encourage each person to present a different perspective or explanation for the issue at hand. This not only broadens the conversation but also promotes a culture of understanding that there can be many valid viewpoints, which can lead to more nuanced and comprehensive discussions.

Bayesian Approach: Unified Framework for Probabilistic Inference

Having thoroughly dismantled the frequentist school, Clayton concludes with a prescription for how inferences in statistics should be done. He presents the Bayesian approach as a natural extension of logical reasoning to cases of uncertainty and shows its strength in handling even the most pathological examples from among those considered earlier.

Probability as a Logical Extension of Deduction to Uncertainty

Clayton explains that Jaynes's probability theory is based on the concept that probability is simply a generalization of deductive reasoning to cases where we have less-than-perfect information. The sum rule and the product rule—which respectively describe how a proposition's probability relates to its negation, and how a joint proposition's probability relates to its individual component parts—can be seen as analogous to logical axioms that allow us to combine propositions consistently. Bayes' theorem then serves to update our beliefs regarding a hypothesis, based on new evidence and those basic logical operations. In this interpretation, then, probability concerns information and plausibility, and frequencies are simply elements of the inferential mechanism rather than a definition.

Embracing the Subjective Nature of Preliminary Knowledge in Inference

A highly contentious aspect of Bayesian analysis is the function of prior information, which, as Clayton has shown many times, is necessarily a part of any logically complete probabilistic inference. In the Bayesian framework, prior information isn't about an experimenter's feelings or biases toward a particular result, but rather about their assumptions and the knowledge that informs their view of the world. Recognizing this subjectivity is crucial for allowing rational disagreement between observers presented with identical information. From this perspective, subjectivity doesn't imply relativism; it simply acknowledges that different individuals might have different degrees of certainty about how plausible a given claim is based on their unique experiences and what they believe regarding how the world works.

Practical Tips

• Develop a "Bayesian belief network" for a complex decision, like choosing a career path, by mapping out how different pieces of prior knowledge (like job market statistics, educational requirements, and personal skills) influence the likelihood of various outcomes. This visual tool can help you see the logical structure of your beliefs and how they interconnect.
• You can enhance decision-making by starting a "Rational Disagreement Journal" where you record instances where you and another person have different opinions despite having the same information. This practice will help you identify patterns in how you interpret information differently from others, leading to a better understanding of your own subjective biases. For example, after a meeting where data is presented, note down your conclusions and compare them with a colleague's. Discuss the reasons behind your differing views to uncover subjective influences.

Reclaiming Probabilistic Analysis as a Tool for Understanding, Not an Oracle

Clayton emphasizes that even when grounded in a consistent system of logic, probabilistic conclusions don't provide perfect or final answers. Instead, they offer a structured way to form conclusions in uncertain situations and incorporate new information into our beliefs. Frequentist methods, by attempting to eliminate prior information and present inferences as deriving solely from data, give an illusion of certainty that is demonstrably false. As he suggests, embracing the necessary subjectivity of Bayesian methods is the only way to begin to reclaim probability as a tool for understanding, rather than as a mysterious oracle issuing pronouncements of "truth" we are obliged to accept.

Practical Tips

• Embrace uncertainty in decision-making by using a "confidence scale" when faced with choices. Instead of seeking a definitive answer, rate your confidence in each option on a scale from 1 to 10. This acknowledges the probabilistic nature of outcomes and helps you become comfortable with uncertainty. For example, if you're deciding between two job offers, assign a confidence level to each based on factors like company stability, personal growth opportunities, and work-life balance.
• Play board games or card games that involve chance with friends or family. As you play, discuss the odds of certain plays or outcomes occurring. This social activity can demystify probability by showing it in action in a fun, relatable context.

Benefits of Bayesian Methods: Managing Complexity, Uncertainty, and "Nuisance Parameters"

Clayton presents Bayesianism as a unified framework for probabilistic inference, capable of handling problems of arbitrary complexity, incorporating prior knowledge, and explicitly accounting for the uncertainty in both data and model parameters. He points out the power of Bayesian approaches lies in their comprehensive treatment of probabilities, where anything not known with certainty can be given a probability distribution, and any question that can be logically phrased in probabilities can (at least in principle) be answered. Furthermore, by explicitly incorporating all available information into the inference, this approach provides a natural safeguard against overfitting or exaggerating the significance of small effects.

Overcoming Fear of Priors: Knowledge Integration and Assumption Testing

Clayton acknowledges the discomfort many researchers have felt regarding the use of prior probability assignments, but he argues their concerns are largely unfounded. For many problems, simply recognizing that all probability statements are conditioned on some background information is enough to reveal the appropriate prior probability for the parameter or hypothesis of interest. In other cases, it might turn out that the exact shape of the prior isn't essential to the conclusions because the data alone is sufficient to produce a reasonably precise posterior distribution. For situations where we do have a choice between multiple plausible prior distributions, various hypotheses can be explicitly tested by simulating data from those distributions and examining their consequences for our inferences.

For those still committed to some idea of objectivity, Clayton suggests using a weak prior to express a conservative skepticism about the likely value of the parameter or the credibility of the research hypothesis. Crowd-sourcing prior likelihoods or preregistering them before data collection, for example, could provide further safeguards against biases. However, as he notes, the purported objectivity of frequentist techniques is a facade. The fact that these techniques do not explicitly make use of prior information simply means that the prior assumptions are hidden within the selection of tests or estimators. By requiring researchers to explicitly declare their prior beliefs, Bayesian approaches actually make statistical analysis more transparent and allow for more meaningful comparisons between studies with different theoretical frameworks.

Practical Tips

• Engage in community science projects where data collection is a key component. Look for local or online citizen science initiatives that require data collection, such as observing wildlife, tracking weather patterns, or participating in health studies. By contributing data and observing how it's used to draw conclusions, you'll get a hands-on understanding of the power of data-driven analysis.
• Create a simple spreadsheet to test financial decisions with variable inputs. Use a tool like Microsoft Excel or Google Sheets to create a model where you can adjust factors like interest rates, inflation, and salary growth to see how these changes might affect your long-term savings or mortgage payments.
• Organize a virtual "likelihood party" with friends or colleagues where you discuss upcoming events or decisions and estimate their outcomes. Each participant can share their predictions and the reasoning behind them. This collective forecasting exercise can reveal different perspectives and help balance individual biases with group insights.
• Start a book club with a twist where each member reads a different book on the same topic and shares their insights. This encourages the application of Bayesian thinking as you'll compare and contrast the conclusions drawn from different theoretical perspectives. For instance, if the topic is climate change, members might read books focusing on economic, scientific, and sociopolitical frameworks, then discuss how these different angles provide a more comprehensive understanding of the issue.

The Power of Post-Analysis Probability Distributions: Full Knowledge Over Binary Decision

Clayton argues that Bayesian reasoning's advantage is its lack of definitive conclusions. Instead, it provides a full probability distribution showing a range of possible values for the parameter or the hypothesis in question, weighted according to all the information available to us. That distribution then serves as the foundation for making further predictions, designing new experiments, or updating our beliefs based on new data. This comprehensive approach contrasts sharply with frequentist methods, which simplify the inferential process into a binary choice regarding dismissing a hypothesis. Clayton compares this to choosing a destination on a map based solely on whether a road sign says "WRONG WAY" without knowing anything about where the road actually goes. By providing a full probability distribution, instead of just a yes or no conclusion about a specific null hypothesis, Bayesian inferences allow us to see for ourselves exactly what the data has told us.

Context

• By considering the entire distribution, decision-makers can weigh the probabilities of different outcomes, leading to more informed and flexible decision-making processes.
• The distribution incorporates prior knowledge or beliefs about the parameter before observing the data. This prior can be based on previous studies, expert opinion, or theoretical considerations.
• While frequentist methods use confidence intervals to estimate parameters, these intervals are often misinterpreted. They represent a range of values that would contain the true parameter a certain percentage of the time if the experiment were repeated, not the probability of the parameter being within that range.
• Bayesian reasoning is inherently iterative, meaning individuals can continuously update their understanding as new data becomes available. This dynamic process supports ongoing learning and adaptation.

Escaping Statistics Traps: Unbiased Estimators & Sufficient Statistics

Clayton explains that by making explicit all the informational content of a statistical analysis, particularly through incorporating prior probabilities, the Bayesian method avoids some of the common statistical traps of frequentism, such as the obsession with finding an estimator that is unbiased without consideration for its other properties, or with seeking out a sufficient statistic in the data without regard to whether such a statistic even exists at all.

In Bayesian inference, nothing suffices as a statistic other than the probability of all the data taken together, and we choose an estimator only when required to report a single summary value, which typically we do not. Those who demand the use of estimators without bias, for instance, are effectively throwing away data given the fact that the bias corrected estimate will necessarily have a higher variance. And those who fret over whether their estimator accounts for every piece of valuable information will automatically be directed toward a sufficient statistic if such a statistic can be found. According to Clayton, the takeaway is that the Bayesian approach, which consistently uses all the information in a manner aligned with the axioms of probability (derived as consequences of logical reasoning about plausibilities), never finds itself in a statistical dead end, whereas the frequentist methods often do.

Practical Tips

• Enhance your critical thinking by playing "Bayesian Detective" in everyday situations, where you actively seek out prior information and adjust your conclusions as new data comes in. When reading news articles, start by stating your prior belief about the topic, then with each piece of new information, adjust your belief accordingly, and see where you stand at the end of the article.
• Improve your understanding of statistics in the real world by engaging in "Statistical Scavenger Hunts" where you look for examples of statistics in the media or reports and critically assess whether they seem to be sufficient for the claims made. Take notes on your findings and discuss them with peers to develop a keener sense for when statistics are being used effectively and when they might be misleading or unnecessary.
• You can visualize the impact of bias correction on variance using simple data simulations in a spreadsheet. Create two columns of random data representing two datasets—one with bias correction applied and one without. Use built-in functions to calculate the variance of each dataset. This hands-on activity will help you see how bias correction affects variance without needing advanced statistical software.
• Develop a habit of reflective journaling to distill essential insights from daily experiences. At the end of each day, write down the most significant events and what you learned from them. Over time, you'll begin to notice patterns and can identify which experiences hold the most valuable lessons, effectively creating your own "sufficient statistics" for personal growth.

Beyond Objectivity: Embracing Science as Human Endeavor and Likely Insights

According to Clayton, a key component of escaping Bernoulli's Fallacy, and for rectifying the damage done by its continued influence in modern science, is to recognize that drawing inferences from statistics isn't an objective process and will necessarily never produce completely final answers. Instead, any inference we draw—whether a medical drug increases the chance of heart problems, a genetic marker is linked to the incidence rate of a disease, a suspect is guilty of a crime, a signal is present in our data, one variable correlates with another, etc.—will be conditioned on whatever assumptions we are willing to bring to the table. Using probability terms, the quality of these inferences comes not from their objectivity, but from their validity—that is, from whether we have abided by the consistent probabilistic reasoning rules provided by Bayes' theorem and its axioms.

Experimenters' Regress and Trustworthy Authorities

Clayton describes how in his work, sociologist Harry Collins identified something he termed "experimenter's regress," which refers to a logical circularity between theory and experimental data. This can be viewed as an analogue to Hume's induction dilemma: by what right can we trust our observations to inform our beliefs unless we already know those observations to be accurate? If we demand absolute objectivity, it would seem we are forever caught in a catch-22 where no experiment can ever truly establish a surprising or counterintuitive claim due to the possibility of some systematic error lurking within the experiment itself. But in science, as in daily life, we accept these judgments of authority all the time without requiring a complete logical proof, because experience has taught us they are reliable.

The only way out, therefore, is to recognize that scientific information is probabilistic and that all understanding is ultimately subjective. This requires a degree of humility from researchers who are claiming their theories are probable truths grounded in evidence rather than pronouncements of certainty, but science is never truly independent of human perception anyway.

Context

• The concept suggests that scientific progress is not purely objective but is influenced by social factors, including trust in the scientific community and the reputations of researchers, which can affect how results are interpreted and accepted.
• While science strives for objectivity, complete objectivity is unattainable because human perception and interpretation are involved at every stage, from designing experiments to interpreting data.
• Humans have cognitive limitations that prevent them from processing all available information. Relying on trusted authorities helps manage these limitations by delegating complex analyses to those with specialized knowledge.
• All scientific measurements have some degree of uncertainty, which is often quantified and reported. This uncertainty reflects the limitations of instruments and methods, reinforcing the probabilistic nature of scientific findings.
• Philosophers like Karl Popper have argued that scientific theories should be falsifiable, meaning they can be tested and potentially proven wrong. This approach emphasizes the tentative nature of scientific knowledge.
• Observations in science are often influenced by the theoretical framework that scientists use, meaning that what is observed is partly shaped by existing theories and expectations.