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has been summed over the index
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probability for a given cell. In addition there are product multinomial models that involve fixed row or column totals. Although we may be interested in one of the multinomial models, it is easier to fit the Poisson model. Such a model is called a surrogate Poisson model and, with correctly selected model terms, will give the same estimates (Birch, 1963) the multinomial model. Where one of the factors has two levels, a binomial model may be fitted, giving the same parameter estimates (see Venables and Ripley (1994, $7.3) for a n example). This can be extended to a response factor with more than two levels by using an MLP with appropriate activation and penalty functions. In such a case an MLP with no hidden layers is fitting a multinomial model without a surrogate Poisson model4. Say that for the response factor we have three levels and, for a particular cell, we have the following counts, yz, y3), for the three levels. Then the target vector is ( y ~ / y .y ~ / y .y3/y.) and the penalty function is weighted by y.. , , In other words, the targets are observed probabilities and the MLP models these probabilities. A hierarchy of nested models may be fitted, from the saturated model (with a separate term for each cell) to the intercept model and the final model may be selected via the AIC criterion. We provide a brief discussion of the AIC criterion and further references in Section 5.3.5, (p. 61).
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4See Ripley (1994b) for a n Spackage ( multinom ) t h a t allows such a hierarchy of contingency table models t o be fitted single-layer MLP models.
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A DERIVATION OF THE SOFTMAX ACTIVATION FUNCTION
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A DERIVATION OF THE SOFTMAX ACTIVATION FUNCTION
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The softmax activation function
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ensures the condition of equation (4.8) is met and allows the use of the cross-entropy penalty function. However, by modeling P ( r ( C )we can give a better justification for the use of the softmax activation function. For a classification scheme using the sampling paradigm, for a two-class problem, P(C1Iz) may be modeled
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+ exp { - log [#3]- log [
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(4.9)
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Now, if we are making some distributional assumptions about P(xlC,,), it is a standard procedure to base a test of C1 versus on the likelihood ratio,
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For computational reasons, we take minus the log of the likelihood and minimize this with respect to the parameters of the distribution P(zlCq).
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= - log(LR)
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If, for example, we assume that the classes have Gaussian distributions with a common covariance matrix C, and means p1 and pz
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ACTIVATION AND PENALTY FUNCTIONS
subsumes the constant terms. Hence in this case the posterior probabilities can a logistic function of a linear combination of the variables. In this be written treatment the logistic function only arises a convenient mathematical step. Jordan (1995) comments that (4.9) will only be useful if the log likelihood has some convenient and tractable form, it does in the example above. However, if we start with multiple classes and assume that P(XIC,) is a distribution from the exponential family of distributions parameterized by (O,, $), we can derive the softmax activation function directly. Note that the distributions are assumed to have a common scale 4.
Note that {a($)}-lO;lX - { a ( $ ) } - l b ( O q l ) +log{P(C,,)} is a linear combination of the variables with an offset or bias term and that (4.10) is the softmax activation function. This shows that modeling the posterior a softmax function is invariant to a family of classification problems where the distributions are drawn from the same exponential family with equal scale parameters. The logistic activation function is then recovered a special case of softmax.