How do we reach decisions to act on something? One way to answer this question is the action driven by a decision bringing positive feedback. The feedback is often accompanied by monetary or other forms of rewards; it thereby motivates us to make such decisions in the future.

In the terminology of psychological experiments, the feedback could conceive as been shown in the form of data, which are collected from human and/or animal subjects. In other words, the theories (our prior beliefs) behind every decision that entails an action (prediction) and we then collect data to check how close the prediction fit the data. How? Often the closer our prediction matches the resultant data, the more rewards we might receive. Hence when there are some mismatches between the predictions and the data, we would likely modify our theories/beliefs. They then become posterior belief. This intuitive idea of human decision-making is described by the well-known Bayes’ theorem (Bayes, Price, & Canton, 1763):

\[\begin{align*} & P(\theta | y) = \frac{P(y | \theta) P(\theta)}{P(y)} \end{align*}\]

1. y represents data. For example, a serial of response times in seconds, c(0.533, 0.494, 0.494, …);
2. θ represents a set of parameters. That is, it is a parameter vector;
3. P(θ) represents our prior belief in the form of a probability distribution, which is fully accounted for the parameter vector. This is often dubbed the prior distribution.
4. P(y | θ) represents the mechanism accounting for the data. This is often dubbed (data’s) likelihood function.
5. P(θ | y) represents posterior belief, which similar to the prior belief, is often dubbed the posterior distribution.