Bayesian Updating

Bayesian updating means reconsidering how strongly you believe an idea after seeing new information. Compare how well the idea and other explanations would predict what you saw. Your confidence may rise, fall, or stay the same, and a change can be large or small. Pick one belief and write down what evidence would favor another explanation.

As an illustration, imagine confidence as a dial rather than an on/off switch. A hypothesis is an idea you are evaluating. Its prior is your confidence before considering new evidence, meaning the information you observe. Its posterior is your confidence after taking that evidence into account.

The likelihood is the chance of observing the evidence if a particular hypothesis were true. Bayesian updating combines the prior with this likelihood, then divides by the overall chance of the evidence to obtain the posterior. This requires probabilities that follow the rules of probability and evidence whose initial probability is greater than zero. Compare competing explanations: evidence equally likely under an idea and its alternative leaves confidence unchanged. Evidence that strongly favors an alternative can produce a large change, even from a strong prior short of complete certainty. [1]

Consider two hypothetical illustrations. Reliable negative restaurant reviews should lower confidence in a restaurant's quality if those reviews are more likely for a poor restaurant than a good one. Weak sales should lower confidence in strong demand when weak sales are less likely under strong demand than under demand that is not strong, accounting for availability in both predictions. Neither illustration fixes the size of the change without further assumptions.

A positive medical test illustrates why starting information matters. Prevalence means how common the disease is in the relevant population. Sensitivity is the chance of a positive result when someone has the disease; specificity is the chance of a negative result when someone does not. The probability of disease after a positive result depends on all three, so a positive result alone does not establish how high confidence should become. [1]

In a hypothetical science example, suppose confidence in a theory is strong but below certainty. A reliable experimental result that is only a little more likely if the theory is false than true warrants a small reduction in confidence. Assume major and minor problems are the only possibilities. For a separate hypothetical car example, suppose a major problem initially has a 20% chance, and a warning light together with smoke is 40 times as likely under a major problem as under a minor one. Those stipulated probabilities justify a dramatic increase in confidence in a major problem; smoke alone carries no universal numerical meaning.

When updating repeatedly, a posterior can become the next prior, but predictions for the next observation must account for evidence already included. For example, copied restaurant reviews should not be counted as independent reports. Bayesian conditioning—adopting probabilities given evidence accepted as true—is a normative rule, meaning guidance about how confidence should change under stated assumptions. It does not guarantee getting closer to truth: starting probabilities, predictions, or the explanations considered may poorly represent the world.

• Theory: Joyce explains how Bayes’ theorem combines starting probabilities with likelihoods, meaning how probable the observations are under each hypothesis. Evidence can increase, decrease, or leave confidence unchanged. [1]
• Theoretical review drawing on empirical cognitive research: Tenenbaum and colleagues describe models that combine background knowledge with observations to explain aspects of concept learning and causal reasoning. This is a modeling approach, rather than one experiment proving that all human reasoning is Bayesian. [2]
• Empirical findings summarized in the review: Explicit judgments involving numerical probabilities can depart substantially from Bayesian standards. The authors distinguish these tasks from cognitive abilities that their Bayesian models seek to explain. [2]

Updating depends on the starting probabilities and on how well the proposed explanations predict observations. Ordinary conditioning preserves an initial probability of zero or one, so complete certainty can prevent revision through this rule. [1] Tenenbaum and colleagues emphasize that successful generalization depends on suitable constraints and on the true structure remaining within or near the possibilities the learner considers. Their review does not establish that the mind can perform every Bayesian inference. [2] The confidence dial, restaurant, sales, scientist, and car examples here are teaching illustrations with stated assumptions, not empirical findings.