I am self learning Variational Inference. Currently I am reading the chapter 21 book from Murphy 1 and trying to understand the Ising model (21.3.2). The Ising model here is used as denoising technique to turn a gray valued image into a binary image.
The model is described as follows: assuming a latent variable $x_i \in \{-1, +1\}$, the joint model form is $p(\textbf{x}, \textbf{y}) = p(\textbf{x})p(\textbf{y}|\textbf{x})$, where $\textbf{y}$ denotes the gray values of the image and $\textbf{x}$ denotes the binary values of the corresponding pixels. The prior is $$p(\textbf{x}) \propto \exp(E_0(\textbf{x})),$$ $$E_0(\textbf{x}) = -\sum_{i=1}^{D}\sum_{j\in Nb(i)}W_{i,j}x_ix_j,$$ where $W$ is a weighting matrix (usually positive), and $Nb(i)$ denotes the first-degree hidden neighbors of pixel $i$.
Then, the author goes on and defines the likelihood as $$p(\textbf{y}|\textbf{x}) = \prod_{i}p(y_i|x_i) = \exp(\sum_{i}-L_i(x_i)).$$ Finally, the posterior is presented as $$p(\textbf{x}|\textbf{y}) \propto \exp(-E(\textbf{x})),$$ $$E(\textbf{x}) = E_0(\textbf{x}) - \sum_{i}L(x_i).$$
I do not follow what $L(\cdot)$ here represents, how the author arrived to the likelihood in that form, and how the posterior is determined since it should be intractable.