[proofplan] The proof is an ordered chain-rule argument followed by a regrouping according to bidirected districts. Positivity makes each ordered conditional kernel a well-defined ratio of observed prefix marginals. Since the districts partition the observed vertices, the product of all district factors is exactly the product of all ordered chain-rule factors. The identifiability statement follows because every factor in every district product is computed directly from the observed joint law. [/proofplan] [step:Define the ordered conditional kernels from observed prefix marginals] For each $i\in\{1,\dots,n\}$, let $h_i=(v_1,\dots,v_i)$ denote the ordered prefix history in $\mathcal X_1\times\cdots\times\mathcal X_i$. Let $h_0$ denote the empty history. For each $i\in\{1,\dots,n\}$, the prefix marginal mass function or density is the map \begin{align*} p_i:\mathcal X_1\times\cdots\times\mathcal X_i\to[0,\infty) \end{align*} given by \begin{align*} p_i(h_i) := p(v_1,\dots,v_i). \end{align*} For $i=0$, define $p_0$ on the singleton set consisting of the empty history by $p_0(h_0):=1$. For each $i\in\{1,\dots,n\}$, define the ordered conditional kernel \begin{align*} K_i: \{h_i:p_{i-1}(h_{i-1})>0\} \to [0,\infty) \end{align*} by \begin{align*} K_i(h_i) := \frac{p_i(h_i)}{p_{i-1}(h_{i-1})}. \end{align*} By the positivity assumption, every denominator in this definition is positive on the histories under consideration. Thus $K_i(h_i)$ is precisely the displayed ordered conditional quantity $p(v_i\mid v_1,\dots,v_{i-1})$ on its support. [guided] The first task is to make the conditional factors precise. For each $i\in\{1,\dots,n\}$, the symbol $h_i$ denotes the ordered prefix history $(v_1,\dots,v_i)$ in $\mathcal X_1\times\cdots\times\mathcal X_i$, and $h_0$ denotes the empty history. The corresponding observed prefix marginal mass function or density is the map \begin{align*} p_i:\mathcal X_1\times\cdots\times\mathcal X_i\to[0,\infty) \end{align*} given by \begin{align*} p_i(h_i) := p(v_1,\dots,v_i). \end{align*} For the empty prefix we define $p_0$ on the singleton set consisting of the empty history by $p_0(h_0):=1$. Now define, for each $i\in\{1,\dots,n\}$, the map \begin{align*} K_i: \{h_i:p_{i-1}(h_{i-1})>0\} \to [0,\infty) \end{align*} by \begin{align*} K_i(h_i) := \frac{p_i(h_i)}{p_{i-1}(h_{i-1})}. \end{align*} This is the ordered conditional kernel $p(v_i\mid v_1,\dots,v_{i-1})$. The positivity hypothesis is used exactly here: it ensures that the denominator $p_{i-1}(h_{i-1})$ is nonzero whenever the conditional kernel is evaluated. Without this support condition, the ratio would not define a kernel-valued functional of the observed law at that history. [/guided] [/step] [step:Apply the chain rule in the fixed topological order] For every $v$ in the common positive-history support of the ordered kernels, the finite telescoping product gives \begin{align*} \prod_{i=1}^n K_i(h_i) = \prod_{i=1}^n \frac{p_i(h_i)}{p_{i-1}(h_{i-1})}. \end{align*} All intermediate prefix terms cancel, leaving \begin{align*} \prod_{i=1}^n K_i(h_i) = p_n(h_n). \end{align*} Since $p_n(h_n)=p(v)$, we obtain \begin{align*} p(v)=\prod_{i=1}^n p(v_i\mid v_1,\dots,v_{i-1}). \end{align*} [/step] [step:Group the ordered chain-rule factors by bidirected district] For each district $D_j$, define the index set \begin{align*} I_j := \{i\in\{1,\dots,n\}:V_i\in D_j\}. \end{align*} Because $D_1,\dots,D_k$ are the bidirected connected components of $G$, they form a partition of $V$. Therefore the sets $I_1,\dots,I_k$ form a partition of $\{1,\dots,n\}$. By the definition of $Q_{\pi}[D_j]$ and of $K_i$, \begin{align*} Q_{\pi}[D_j](v)=\prod_{i\in I_j} K_i(h_i). \end{align*} Since this is a finite product of nonnegative [real numbers](/page/Real%20Numbers), regrouping factors over the partition $\{I_1,\dots,I_k\}$ gives \begin{align*} \prod_{j=1}^k Q_{\pi}[D_j](v)=\prod_{j=1}^k \prod_{i\in I_j} K_i(h_i). \end{align*} The right-hand side contains each ordered factor $K_i(h_i)$ exactly once, so \begin{align*} \prod_{j=1}^k Q_{\pi}[D_j](v)=\prod_{i=1}^n K_i(h_i). \end{align*} Using the ordered chain-rule factorization from the previous step, \begin{align*} \prod_{j=1}^k Q_{\pi}[D_j](v)=p(v). \end{align*} [/step] [step:Identify each C-component factor from the observed joint law] Fix $j\in\{1,\dots,k\}$. The set $I_j$ is determined by the graph $G$ and the fixed order $\pi$. For each $i\in I_j$, the factor $K_i(h_i)$ is the ratio \begin{align*} K_i(h_i)=\frac{p_i(h_i)}{p_{i-1}(h_{i-1})}, \end{align*} where both $p_i$ and $p_{i-1}$ are prefix marginals of the observed joint law $p$. Hence $K_i$ is identifiable from $p$ on its positive-history support. Taking the finite product over $i\in I_j$ shows that \begin{align*} Q_{\pi}[D_j](v)=\prod_{i\in I_j}\frac{p_i(h_i)}{p_{i-1}(h_{i-1})} \end{align*} is identifiable from the observed joint law $p$ for the fixed order $\pi$. This proves both the Tian-Pearl C-component factorization and the claimed order-dependent identifiability of the factors. The argument does not identify $Q_{\pi}[D_j]$ with the ordinary conditional law of $V_{D_j}$ given $V\setminus D_j$; it identifies the district factor as the specified grouped product of observed ordered conditional kernels. [/step]