By James V Stone

Discovered by means of an 18th century mathematician and preacher, Bayes' rule is a cornerstone of recent chance concept. during this richly illustrated ebook, various available examples is used to teach how Bayes' rule is really a usual outcome of logic reasoning. Bayes' rule is then derived utilizing intuitive graphical representations of chance, and Bayesian research is utilized to parameter estimation. As an reduction to realizing, on-line machine code (in MatLab, Python and R) reproduces key numerical effects and diagrams. the educational kind of writing, mixed with a entire word list, makes this an amazing primer for rookies who desire to get to grips with the elemental ideas of Bayesian analysis.

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**Extra info for Bayes’ Rule: A Tutorial Introduction to Bayesian Analysis**

**Example text**

3. Once all those bias markings have been removed from the coins, we choose one coin, flip it, and observe that it lands heads up. 9 given that a head Xh was observed? 75. 9 ) that the coin chosen has a bias of 69,9 and that coins chosen from our container landed heads up is the area a, whereas the average probability p(xh) that a coin lands heads up is the area (a + b) (see the sum rule above). 19) which is a posterior or inverse probability (see Chapter 1). 37 2 Bayes9 Rule in Pictures T h e P r o d u c t R u le U sing G e om etric Area: In order to find the posterior probability using Bayes’rule (instead of the joint probability, as above), we will need the product rule.

4. 11, do you see a hill or a crater? Now turn the page upside-down. : Is this a hill or a crater? Try turning the book upside-down. (Barringer crater, with permission, United States Geological Survey). to a crater). This illusion almost certainly depends on the fact that your visual system assumes that the scene is lit from above. 11 as a hill, and the inverted version as a crater (which it is, in reality). In terms of Bayes’rule, the image data are equally consistent with a hill and a crater, where each interpretation corresponds to a different maximum likelihood value.

9 ,Xh), each of which has a probability represented by the area of one quadrilateral. The area of all areas sum to one (not drawn to scale). 4. P rob a b ility A s G e o m e tric A rea Here, we show how joint probabilities behave just like areas, which allows the sum and product rules to be illustrated geometrically. This general idea is then extended to obtain expressions for conditional probabilities (eg likelihoods), which behave like ratios of areas. 3 represent the proportion of times each combination is observed.