> There are two major approaches for assigning probabilities to events: ==objectivist== and ==subjectivist==. The objectivist approach is often used for modelling scientific experiments or well-defined systems. Probabilities are estimated based on observed data or experiments. The subjectivist approach is often used for [[Machine Learning|learning algorithms]] or aggregating beliefs from experts. Both approaches share the same mathematical foundation, expressed formally in the language of [[Measure Theory|measure theory]]. ### Objectivist > **Frequentism** is an objectivist approach to assigning probability that views probability as a relative frequency of success and failure. If you repeat an action many times, like an experiment, how often will you get a positive outcome? Probabilities are estimated based on observed data or experiments. If an event A occurs $k$​ times in $n$ trials, the probability of $A$ is approximated as: $ P(A) = \frac{k}{n} $ In the frequentist approach, probability is interpreted as the long-run relative frequency of an event. As we repeat an experiment many times, the proportion of times an event occurs stabilizes and approaches its true probability. This idea is mathematically justified by the [[Law of Large Numbers (LLN)]], which states that as the number of trials increases, the sample average converges to the true probability. ### Subjectivist > **Bayesian** probability is a subjectivist approach to assigning probability, where probability represents a degree of belief that is updated as new evidence becomes available. You begin with a prior probability, which reflects your initial beliefs, and as you incorporate new evidence (via the likelihood), you update this to obtain a posterior probability. Assigning a probability to an event expresses how likely you think it is, and as you learn new information, your assigned probability adjusts accordingly. $ P(\text{Hypothesis} | \text{Data}) = \text{Data})P(\text{Data} | \text{Hypothesis}) \times P(\text{Hypothesis}) $​