0
$\begingroup$

At some point, in Bishop's book 'Pattern recognition and Machine Learning', (p.75) he is talking about multinomial distributions in a classification context, introducing a suitable probability distribution $p(\bf x | \mu)$:enter image description here

with given constraints for $\bf x$ and $\bf \mu$.

What I don't understand is why the distribution is normalized, i.e. equality 2.27. How does he achieve that?

$\endgroup$
1
$\begingroup$

Note that the possible values of $\ \mathbf{x}\ $ are $\ \mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_K\ $, where $\ \mathbf{e}_j\ $ is the unit vector of $\ \mathbb{R}^K\ $ whose $\ j^\text{th}\ $ entry is $1$ and all of whose other entries are $0$. Thus, \begin{align} \sum_{\mathbf{x}}\prod_{k=1}^K\mu_k^{x_k}&=\sum_{j=1}^K\prod_{k=1}^K\mu_k^{\left(\mathbf{e}_j\right)_k}\\ &= \sum_{j=1}^K \prod_{k=1}^K\mu_k^{\delta_{jk}}\\ &= \sum_{j=1}^K\mu_j\\ &=1\ . \end{align}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.