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For the following definitions, two examples will be used. The first is the problem of character recognition given an array of bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative.
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While naive Bayes often fails to produce a good estimate for the correct class probabilities, [16] this may not be a requirement for many applications. For example, the naive Bayes classifier will make the correct MAP decision rule classification so long as the correct class is predicted as more probable than any other class. This is true ...
For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for = values. If no variable's local distribution depends on more than three parent variables, the Bayesian network representation stores at most 10 ⋅ 2 3 = 80 {\displaystyle 10\cdot 2^{3}=80} values.
where the first equality results from the marginalization rule, the second results from Bayes' theorem and the third corresponds to a second application of marginalization. The denominator appears to be a normalization term and can be replaced by a constant . Theoretically, this allows to solve any Bayesian inference problem.
However, this loss function is non-convex and non-smooth, and solving for the optimal solution is an NP-hard combinatorial optimization problem. [4] As a result, it is better to substitute loss function surrogates which are tractable for commonly used learning algorithms, as they have convenient properties such as being convex and smooth.
Bayes' theorem is named after Thomas Bayes (/ b eɪ z /), a minister, statistician, and philosopher. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. His work was published in 1763 as An Essay Towards Solving a Problem in the Doctrine of Chances.
The correct answer can be computed using Bayes' theorem. The event G is that the student observed is a girl, and the event T is that the student observed is wearing trousers. To compute the posterior probability P ( G | T ) {\displaystyle P(G|T)} , we first need to know: