How to understand the difference of EM algorithm between PRML and 'Machine learning - a probabilistic perspective' The M step of  EM algorithm in chapter 11.4.7 Machine learning - a probabilistic perspective is denoted:
$$ \theta^{t+1} =\underset{\theta}{argmax}Q(\theta, \theta^{t})=\underset{\theta}{argmax}\sum_{i}E_{q_{i}^{t}}[logp(x_i, z_i|\theta)]$$
$\theta$ is the parameter,$t$ is the step, $Q$ is the expectation, $q_{z_i}$ is a arbitrary distribution over the hidden varaiable $z_i$,$p(x_i,z_i)$ is the distribution of the complete data.
The expectation can be written as:
$$ Q(\theta, \theta^t)=\sum_{i}E_{q_{i}^{t}}[logp(x_i, z_i|\theta)] =\sum_{i}\sum_{z_i}q(z_i^t)logp(x_i, z_i| \theta)  $$
In the E step:
$$q(z_i^t)=p(z_i^t|x_i, \theta^{t})$$
then:
$$Q(\theta, \theta^t)= \sum_{i}\sum_{z_i}p(z_i^t|x_i,\theta^t)logp(x_i, z_i| \theta) \quad \quad(1)$$
The M step in the chapter 9.4 PRML is denoted:
$$ Q(\theta, \theta^{old})=\sum_{Z}q(Z)ln(p(X,Z|\theta))=\sum_{Z}P(Z|X|\theta^{old})lnp(X,Z|\theta) $$
$X$ is all of the observed variables, $Z$ is all of the hidden variables.
In the E step, 
$$ q(Z)=p(Z|X, \theta^{old})=\frac{p(X,Z|\theta^{old})}{\sum_{Z} p(X,Z|\theta^{old})}=\frac{\prod_{n=1}^{N}p(x_n, z_n|\theta^{old})}{\sum_Z\prod_{n=1}^{N}p(x_n, z_n|\theta^{old})}=\prod_{n=1}^{N}p(z_n|x_n, \theta^{old})$$
then:
$$Q(\theta,\theta^{old})=\sum_{Z}\prod_{n=1}^{N}p(z_n|x_n, \theta^{old})\sum_{n=1}^{N}lnp(x_n, z_n|\theta) \quad \quad(2)$$ 
I don't understand why the equation (1) and equation(2) is different?
 A: With respect to Equation 2, \begin{align}Q(\theta,\theta^{old}) 
& = \sum_{Z}  \prod_{n=1}^{N}p(z_n|x_n, \theta^{old})\sum_{n=1}^{N}\log p(x_n, z_n|\theta) 
 \quad \quad(3)\\ 
& = \sum_{z_1} \sum_{z_2} \dots \sum_{z_N} \prod_{n=1}^{N}p(z_n|x_n, \theta^{old}) \sum_{n=1}^{N}\log p(x_n, z_n|\theta) \quad \quad(4)\\ &=  \sum_{z_1} p(z_1|x_1, \theta^{old})\log p(x_1, z_1|\theta) + \sum_{z_2} p(z_2|x_2, \theta^{old})\log p(x_2, z_2|\theta) \dots   \quad \quad(5)  \\ 
& = \sum_{i=1}^{N} \sum_{z_i} p(z_i|x_i)\log p(x_i, z_i|\theta) \\
\\
\end{align}
Here we obtained $(5)$, using the fact that $$\sum_{Z} \prod_{n=1}^{N}p(z_n|x_n, \theta^{old}) \sum_{n=1}^{N} \log p(x_n,z_n| \theta ) = \\ \sum_{z_1} \dots \sum_{z_N} \prod_{n=1}^{N}p(z_n|x_n, \theta^{old}) (\log p(x_1,z_1| \theta ) + \log p(x_2,z_2| \theta ) + \dots) \\
=  \\ \sum_{z_1} \dots \sum_{z_N} \prod_{n=1}^{N}p(z_n|x_n, \theta^{old}) (\log p(x_1,z_1| \theta ) ) + \sum_{z_1} \dots \sum_{z_N} \prod_{n=1}^{N}p(z_n|x_n, \theta^{old}) (\log p(x_2,z_2| \theta ) ) + \dots = \sum_{z_1} p(z_1|x_1, \theta^{old})\log p(x_1, z_1|\theta) + \sum_{z_2} p(z_2|x_2, \theta^{old})\log p(x_2, z_2|\theta)$$
