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The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. [1]
The choice of the prior distribution is used to impose restrictions on , e.g. , with the beta distribution as a common choice due to (i) being defined between 0 and 1, (ii) being able to produce a variety of shapes, and (iii) yielding a posterior distribution of the standard form if combined with the likelihood function ().
An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for ...
The prior probability may also quantify prior knowledge or information about . P ( B ∣ A ) {\displaystyle P(B\mid A)} is the likelihood function , which can be interpreted as the probability of the evidence B {\displaystyle B} given that A {\displaystyle A} is true.
Bayesian-specific workflow comprises three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function ; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used to start a new research cycle.
Posterior = Likelihood × Prior ÷ ... this process can be interpreted as "support from independent evidence adds", and the log-likelihood is the "weight of evidence".
Bayesian probability (/ ˈ b eɪ z i ə n / BAY-zee-ən or / ˈ b eɪ ʒ ən / BAY-zhən) [1] is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation [2] representing a state of knowledge [3] or as quantification of a personal belief.
where (|) denotes the posterior, (|) the likelihood, () the prior, and () the evidence (also referred to as the marginal likelihood or the prior predictive probability of the data). Note that the denominator p ( D ) {\displaystyle p(D)} is normalizing the total probability of the posterior density p ( θ | D ) {\displaystyle p(\theta |D)} to ...