Conditional Probability  Python and Machine Learning for Integrated Circuits   An Online Book  

Python and Machine Learning for Integrated Circuits http://www.globalsino.com/ICs/  


Chapter/Index: Introduction  A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z  Appendix  
================================================================================= Different from unconditional probability, conditional probability, e.g. P(x_{1}x_{2}), is a concept in probability theory that deals with the probability of an event occurring under a specific condition or given that another event has occurred. That is, it represents the probability of an event x_{1} occurring given that another event x_{2} has occurred. In other words, it's the probability of x_{1} under the condition that x_{2} is true. In machine learning, particularly in probabilistic graphical models or Bayesian statistics, conditional probabilities play a significant role. For example, in Bayesian networks, nodes represent random variables, and the edges between them indicate probabilistic dependencies. The conditional probabilities P(x_{i}parents(x_{i})) specify how each variable depends on its parents in the graph. For instance, P(AB) represents how the probability of one event is affected by the knowledge or occurrence of another event. Conditional probability is denoted as , which reads as "the probability of event occurring given that event has occurred."Mathematically, conditional probability is defined as:  [3818a] Where:
Equivalent Equations of Equation 3818a are: P(AΛB) = P(B)P(AB)  [3818aa] P(AΛB) = P(A)P(BA)  [3818ab] Then, we also can have, P(A)P(BA) = P(B)P(AB)  [3818ac] P(BA) = P(B)P(AB)/P(A)  [3818ad] Equation 3818ad is called Bayes' rule. For a case of P(X3X2), then we have,  [3818b]  [3818c] Then, for a conditional distribution of x_{3} given in a normal distribution, we have, [3818d]  [3818e]  [3818f] This code can be used to compute the mean and covariance matrix of the conditional distribution based on Equations 3818e and 3818f, for instance, in the table below. Table 3818. Computing the mean and covariance matrix of the conditional distribution (code).
Expression 3818d indicates that the conditional distribution of x given is a normal distribution with mean and covariance matrix . Equation 3818e is the mean of the conditional distribution, which appears to be a linear combination involving the mean of x ( ), the covariance between x and x ( ), and the inverse of the covariance of x_{2}(ε_{2,2}^{−1}). Equation 3818f is the covariance matrix of the conditional distribution. It involves the covariance of x_{3}( ), the covariance between and x_{2}(ε_{3,2}), and the inverse of the covariance matrix of x_{2}(ε_{2,2}^{−1}). The expressions resemble those found in the multivariate normal distribution, and the use of these equations would typically assume that the joint distribution of and is multivariate normal.Some examples about conditional probability are:
Conditional probability is a fundamental concept in statistics, machine learning, and many areas of science and engineering. It's widely used for making predictions, making decisions based on data, and understanding the relationships between events. An example of conditional probabilities is: i) Not Cloudy and Cloudy in the Morning: Let A be the event "Not Cloudy in the Morning." Let B be the event "Cloudy in the Morning." The conditional probability of event B (Cloudy in the Morning) given that event A (Not Cloudy in the Morning) has occurred is denoted as P(BA). This represents the probability of it being Cloudy in the Morning given that it's Not Cloudy in the Morning. ii) Predict Rain in the Afternoon: Let C be the event "Rain in the Afternoon." Then, the conditional probability of event C (Rain in the Afternoon) given that events A and B have occurred is denoted as P(CA and B). This represents the probability of Rain in the Afternoon given that it's Not Cloudy and Cloudy in the Morning. Table 3818. Applications of conditional probability.
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