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Basics of probability and statistics
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
Conditional probability is the probability of occurrence of one event with some relationship to (conditioning to) one or more other events. For two events A and B with P(B)>0, the conditional probability of A and B, P(A|B) is defined as P(A|B)=P(A∩B)P(B),P(B)>0
Basic Approaches of Artificial Intelligence and Machine Learning in Thermal Image Processing
Published in U. Snekhalatha, K. Palani Thanaraj, Kurt Ammer, Artificial Intelligence-Based Infrared Thermal Image Processing and Its Applications, 2023
U. Snekhalatha, K. Palani Thanaraj, Kurt Ammer
This algorithm is based on the Bayes theorem which is based on the concept of conditional probability. Conditional probability is the estimation of the probability of one event, given that another one has already occurred. The Naïve Bayes is the fastest algorithm when compared to other algorithm techniques and is also relatively simple and easy to implement. This algorithm needs a small amount of training data to evaluate the necessary parameters. The Bayes algorithm is a supervised learning algorithm. It is possible to apply it in both binary and multiclass classifications. The Bayes algorithm can be mathematically represented as follows: Pa|y=py|apa/Py where p(y|a) = Likelihood, p(a) = Class prior probability, P(a|y) = Posterior probability, and P(y) = Predictor probability.
Getting started with supply chain analytics
Published in Peter W. Robertson, Supply Chain Analytics, 2020
At a very basic level, probability assigns the likelihood of an outcome (an ‘event’) between the numerical values of 0 and 1, where 0 signifies the event will not happen and 1 signifies the event will always happen. Additionally, all possible results of an event must add up to 1. Table 3.4 provides an example where the frequencies of differing customer demand levels over a period of one year (365 days) are shown. As can be seen, the individual instance probabilities are all between 0 and 1, thus: 0≤Pevent≤1
Bayesian estimate of the elastic modulus of concrete box girders from dynamic identification: a statistical framework for the A24 motorway in Italy
Published in Structure and Infrastructure Engineering, 2021
Angelo Aloisio, Dag Pasquale Pasca, Rocco Alaggio, Massimo Fragiacomo
Bayes’ theorem describes the probability of an event, based on prior knowledge of conditions possibly related to the event (Aloisio, Battista, Alaggio, Antonacci, & Fragiacomo, 2020; Gelman et al., 2013). The probability of having the EM below a given value indicated as updated to the experimental evidence from dynamic tests can be written as: where is the posterior probability, that is, the probability of observing if the expected first natural frequency f is below the measured one is the likelihood distribution, that is, the probability of observing natural frequencies f below is the prior distribution, that is, the probability of observing E below is the marginal likelihood.
Towards the cognitive and psychological perspectives of crowd behaviour: a vision-based analysis
Published in Connection Science, 2021
Elizabeth B. Varghese, Sabu M. Thampi
Bayesian approaches are based on the prominent theorem in probability known as the Bayes' theorem which states that the probability of the occurrence of an event depends on the conditional probability of the prior knowledge of the event. For a given hypothesis H, and event E, the Bayes' theorem is stated as: where and are conditional probabilities. The theorem is analogous to human mind that consists of conditional probabilities and statistical correlations that can predict an event based on statistical representations and probability calculations (Jiang et al., 2017). The Bayesian theorem was utilised in analysing crowd behaviour in the field of computer vision and is discussed below.
A note on independence in probability
Published in International Journal of Mathematical Education in Science and Technology, 2020
Intuitively, two events are considered to be independent of each other if the occurrence of one does not affect the probability of the other one occurring. Formally, two events A and B are said to be independent if and only if (Grimmett & Stirzaker, 2001; Miller, 2001). For the Venn diagram in Figure 1 this equality becomes , which may be rearranged to give the following quadratic equation in r: From this we obtain noting that in order for r to be a real number, we require q and s such that