Probabilistic reasoning and Bayesian belief updating

Laplace

”Probability theory is nothing but common sense reduced to calculation.”

Probabilistic reasoning uses subjective probabilities—the interpretation of probabilities as degrees of belief—which can be computed using Bayes’ rule, a mathematical theory of learning that specifies how to combine prior knowledge with information provided by new data.

Bayes’ rule for updating beliefs

Suppose that a learner assigns prior probabilities $P(h)$ to each hypothesis $h\in \mathcal{H}$. Given new observed data $d$, we can calculate posterior probabilities assigned to each hypothesis by

$$P(h|d)=\frac{P(d|h)P(h)}{P(d)}=\frac{P(d|h)P(h)}{{\sum }_{h^{\prime }\in \mathcal{H}}P(d|h^{\prime })P(h^{\prime })},$$

where the expansion in the final denominator follows from the marginalization principle $P(d)={\sum }_{h}P(d,h)$ and the chain rule $P(d,h)=P(d|h)P(h)$. This is often expressed as $P(h|d)\propto P(d|h)P(h)$ to emphasize the denominator’s role as a normalizing constant.

From a cognitive science perspective, Bayes’ rule describes how a rational agent should approach the problem of induction. Bayes’ rule encodes two facts about how our beliefs change in response to new evidence: if we believe an event has a low probability, then the probably is still low in spite of reliable evidence; and if new evidence is unreliable, then our beliefs will change very little.

Related notes:

• Frequentist, subjectivist, and primitivist interpretations of a probabilistic locution
• Conditional probability and Bayes’ rule
• Bayesian models of cognition
• Pragmatic Bayesian modeling

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Bayesian likelihood

The term $P(d|h)$ is the likelihood, which give the probability of observing $d$ if