Assume $Y_1$, $Y_2$, $\ldots$ ,$Y_n$ are random variables over a regular lattice indexed by $i= 1,2,\ldots,n$ where $Y_i\in\{1,2,...,K\}$. Let the probability of a particular configuration $\textbf{y}= (y_1,y_2,...,y_n)$ be given by
$$\mathsf P(\textbf{Y}=\textbf{y}) =C\cdot\text{exp}\left(\sum_{i=1}^n\alpha_{i,y_i}+\frac{1}{2}\beta\sum_{i=1}^n\sum_{j\in N(i)}1(y_i=y_j)\right)$$
where $C$ is the normalizing constant, $N(i)$ is the set of neighbor points of $i$ and $1(.)$ is the indicator function. This model is known as Potts model and is popular in image analysis. Show that
$$\mathsf P\left(Y_i=k\mid Y_j=y_j, j\neq i\right)=\frac{\text{exp}\left(\alpha_{i,k}+\beta\sum_{j\in N(i)}1(y_j=k)\right)}{\sum_l\text{exp}\left(\alpha_{i,l}+\beta\sum_{j\in N(i)}1(y_j=l)\right)}$$
I first have some notational questions.
- Does the bolded $\textbf{Y}$ just mean that it's a vector?
- Does $\mathsf P\left(Y_i=k\mid Y_j=y_j, j\neq i\right)$ mean we know what a specific value for $Y_j$ is or does it mean we know what $Y_j$ is $\textbf{for all}$ $j\neq i$?
My try:
I discovered Brook's Lemma which says that if we let $\textbf{y}_0=(y_{10},\ldots y_{n0})$ be any fixed point in the support of $p(.)$ then
$$p(y_1,\ldots, y_n)=\frac{p(y_1\mid y_2,\ldots, y_n)}{p(y_{10}\mid y_2,\ldots y_n)}\cdots \frac{p(y_n\mid y_{10},\ldots, y_{n-1,0})}{p(y_{n0}\mid y_{10},\ldots, y_{n-1,0})}\cdot p(y_{10},\ldots, y_{n0})$$
which seems useful in that it relates the joint distribution in terms of its conditioned individual components. If my second notational question is correct, then I believe any one of the denominators in this expression gives the desired probability $\mathsf P\left(Y_i=k\mid Y_j=y_j, j\neq i\right)$ but it's not clear to me how to proceed or even if making use of this lemma is a viable approach. Any hints to get me going in the right direction would be greatly appreciated.
As a side question, how would one interpret the parameters $\alpha_{i,y_i}$ and $\beta$ in the joint distribution?
Note: I wasn't sure what to title this question so feel free to change it.