notes:bayesian_classification

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notes:bayesian_classification [2013/03/15 13:12] andy [Combining words] |
notes:bayesian_classification [2013/03/15 14:13] andy |
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Please forgive the slightly loose use of notation, there are a few too many dimensions over which to iterate for clarity. | Please forgive the slightly loose use of notation, there are a few too many dimensions over which to iterate for clarity. | ||

- | One slight simplification to note is that as $P(C_i)$ is presumably determined by dividing a number of trained messages by the total number of messages trained, this means that the total number of messages trained can be cancelled out between the numerator and denominator and the raw number of messages in each category used instead. | + | One slight simplification to note results from the fact that $P(C_i)$ is presumably determined by dividing a number of trained messages by the total number of messages trained. Let $N_{C_i}$ indicate the number of messages trained in category $C_i$, $N$ indicate the number of messages trained overall and $N_{C_i}(W_a)$ indicate the number of messages containing token $W_a$ that were trained in category $C_i$. Thus the equation above becomes: |

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+ | \begin{equation*} P(C_i|W_a \cap W_b \cap ... \cap W_z) = \frac{\frac{1}{N}N_{C_i}\prod\limits_{j=a}^z{\frac{N_{C_i}(W_j)}{N_{C_i}}}}{\frac{1}{N}\sum\limits_{k=1}^n{N_{C_k}\prod\limits_{j=a}^z{\frac{N_{C_k}(W_j)}{N_{C_k}}}}} \end{equation*} | ||

+ | \begin{equation} \Rightarrow P(C_i|W_a \cap W_b \cap ... \cap W_z) = \frac{\prod\limits_{j=a}^z{N_{C_i}(W_j)}}{N_{C_i}^{x-1}\sum\limits_{k=1}^n{\frac{1}{N_{C_k}^{x-1}}\prod\limits_{j=a}^z{N_{C_k}(W_j)}}} \end{equation} | ||

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+ | Where $x$ is the total number of words. This version may help avoid underflow, but may instead be susceptible to overflow due to the exponentiation involved. As a result, it may be preferable to move the divisions back inside the iterations. | ||

==== Two-category case ==== | ==== Two-category case ==== | ||

notes/bayesian_classification.txt · Last modified: 2013/03/15 14:13 by andy