Pattern Recognition by Beyerer Jürgen Richter Matthias Nagel Matthias

Author:Beyerer, Jürgen,Richter, Matthias,Nagel, Matthias [Неизв.]
Language: eng
Format: epub
Publisher: De Gruyter
Published: 2018-07-16T21:00:00+00:00

As usual, the data set is D = D1⊎ ⋅ ⋅ ⋅ ⊎ Dc with m ∈ Di ⇔ω(m) = ωi. Taking into account that all quantities in Equation (4.37) are based on the data D, the formula can be extended to

The conceptual difference of the Bayesian view is to actually regard every probability as a conditional probability. Any unconditional distribution is just a convenient utility, if the condition is negligible. This means one actually wants to know the probability that a realized feature m of an random feature m belongs to class ωi, given that the concrete dataset D out of the entirety of D has been observed before. In this sense, P(ω|m) is only an abbreviation for P(ω|m, D) given that P(⋅ , ⋅ , D′) ≈P(⋅ , ⋅ , D||||), if the datasets D′ and D|||| are large enough.

Equation (4.38) can immediately be simplified again, because supervised sampling is assumed. This means that the membership of a sample m in one of the partitions Di is controlled, because its class is known. This has two consequences:

First, though the a priori distribution of the classes P(ω|D) depends on D, one must not use a realization of the random variable D, because the realization is generally not truly sampled but artificially composed. This means that the proportions of the partition D1⊎ ⋅ ⋅ ⋅ ⊎ Dc do not reflect the distribution of the classes. Hence, the assumption is that an a priori distribution P(ω) is known.

Second, one assumes that the class-specific feature distribution does not depend on samples of a different class. This means that

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