Inverse probability

 

Inverse probability[edit]

In probability theory and applications, Bayes' rule relates the odds of event ${\displaystyle {�}_1}$ to event ${\displaystyle {�}_2}$, before (prior to) and after (posterior to) conditioning on another event ${\displaystyle �}$. The odds on ${\displaystyle {�}_1}$ to event ${\displaystyle {�}_2}$ is simply the ratio of the probabilities of the two events. When arbitrarily many events ${\displaystyle �}$ are of interest, not just two, the rule can be rephrased as posterior is proportional to prior times likelihood, ${\displaystyle �(�|�)\propto �(�)�(�|�)}$ where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as ${\displaystyle �}$ varies, for fixed or given ${\displaystyle �}$ (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). See Inverse probability and Bayes' rule.

Summary of probabilities[edit]

Summary of probabilities

Event	Probability
A	${\displaystyle �(�)\in [0,1]}$
not A	${\displaystyle �({�}^{\complement })=1-�(�)}$
A or B
A and B
A given B	${\displaystyle �(�\mid �)=\frac{�(�\cap �)}{�(�)}=\frac{�(�|�)�(�)}{�(�)}}$

Relation to randomness and probability in quantum mechanics[edit]

Main article: Randomness

See also: Quantum fluctuation § Interpretations

In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions were known (Laplace's demon) (but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In the case of a roulette wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty (though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled – as Thomas A. Bass' Newtonian Casino revealed). This also assumes knowledge of inertia and friction of the wheel, weight, smoothness, and roundness of the ball, variations in hand speed during the turning, and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in the kinetic theory of gases, where the system, while deterministic in principle, is so complex (with the number of molecules typically the order of magnitude of the Avogadro constant 6.02×10²³) that only a statistical description of its properties is feasible.

Probability theory is required to describe quantum phenomena.^[34] A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics. The objective wave function evolves deterministically but, according to the Copenhagen interpretation, it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously remarked in a letter to Max Born: "I am convinced that God does not play dice".^[35] Like Einstein, Erwin Schrödinger, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality.^[36] In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes