SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\frac {p _ {1} (x)}{\left(p _ {2} (x)\right) ^ {\gamma}} = \frac {1}{\left(\sqrt {2 \pi}\right) ^ {1 - \gamma}} \frac {\sigma_ {2} ^ {\gamma}}{\sigma_ {1}} \exp \left\{x ^ {2} \left(\frac {\gamma \sigma_ {1} ^ {2} - \sigma_ {2} ^ {2}}{2 \sigma_ {1} ^ {2} \sigma_ {2} ^ {2}}\right) + x \left(\frac {\mu_ {1}}{\sigma_ {1} ^ {2}} - \gamma \frac {\mu_ {2}}{\sigma_ {2} ^ {2}}\right) + \gamma \frac {\mu_ {2} ^ {2}}{2 \sigma_ {2} ^ {2}} - \frac {\mu_ {1} ^ {2}}{2 \sigma_ {1} ^ {2}} \right\}. \tag {42}
$$
Condition (40) implies that the quadratic function under the exponent above has a non-negative coefficient for  $x^2$ . Therefore this function either grows unbounded as  $x \to \infty$  (if the coefficients for the quadratic and linear terms are not zero), or constant (if the coefficients for quadratic and linear terms are zero). In either case,  $\int_{\mathbb{R}} p_1(x) / (p_2(x))^{\gamma} dx = \infty$ .
3Informal interpretation: distributions with disjoint supports have empty "intersections" (think of the intersection of sets analogy)
D Proofs and Derivations
D.1 Proof of Proposition 5.1
Our goal is to show that the policy (12) induces the mixture distribution  $p_{M}(x) = \sum_{i = 1}^{m}\omega_{i}p_{i}(x)$ .
Preliminaries. In our proof below we use the notion of "the probability of observing a state  $s \in S$  on a GFlowNet trajectory". Following Bengio et al. [35], we abuse the notation and denote this probability by
$$
p _ {i} (s) \triangleq p _ {i} (\{\tau : s \in \tau \}) = \sum_ {\tau \in \mathcal {T} _ {s _ {0}, s}} \prod_ {t = 0} ^ {| \tau | - 1} p _ {i, F} \left(s _ {t} \mid s _ {t - 1}\right), \tag {43}
$$
where  $\mathcal{T}_{s_0,s}$  is the set of all (sub)trajectories starting at  $s_0$  and ending at  $s$ . The probabilities induced by the policy (12) are denoted by  $p_M(s)$ . Note that  $p_i(s)$  and  $p_M(s)$  should not be interpreted as probability mass functions over the set of states  $S$ . In particular,  $p_i(s_0) = p_M(s_0) = 1$  and sums  $\sum_{s\in S}p_i(s)$ ,  $\sum_{s\in S}p_M(s)$  are not equal to 1 (unless  $S = \{s_0\}$ ). However, the functions  $p_i(\cdot), p_M(\cdot)$  restricted to the set of terminal states  $\mathcal{X}$  give valid probability distributions over  $\mathcal{X}$ :  $\sum_{x\in \mathcal{X}}p_i(x) = \sum_{x\in \mathcal{X}}p_M(x) = 1$ .
By definition  $p_i(\cdot)$  and  $p_M(\cdot)$  satisfy the recurrent relationship
$$
p _ {i} (s) = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {i} \left(s _ {*}\right) p _ {i, F} (s | s _ {*}), \quad p _ {M} (s) = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {M} \left(s _ {*}\right) p _ {M, F} (s | s _ {*}). \tag {44}
$$
The joint distribution of  $y$  and  $\tau$  described in the statement of Proposition 5.1 is  $p(\tau, y = i) = w_i p_i(\tau)$ . This joint distribution over  $y$  and trajectories implies the following expressions for the distributions involving intermediate states  $s$ .
$$
p (y = i) = \omega_ {i}, \tag {45}
$$
$$
p (\tau | y = i) = p _ {i} (\tau) = \prod_ {t = 0} ^ {| \tau | - 1} p _ {i, F} \left(s _ {t} \mid s _ {t - 1}\right), \tag {46}
$$
$$
p (\tau) = \sum_ {i = 1} ^ {m} p (\tau | y = i) p (y = i) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} (\tau), \tag {47}
$$
$$
p (s \mid y = i) = p _ {i} (s), \tag {48}
$$
$$
p (s) = \sum_ {i = 1} ^ {m} p (s \mid y = i) p (y = i) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} (s). \tag {49}
$$
Proof. Using the notation introduced above, we can formally state our goal. We need to show that  $p_{M}(x)$  induced by  $p_{M,F}$  gives the mixture distribution
$$
p _ {M} (x) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} (x). \tag {50}
$$
We prove a more general equation for all states  $s \in S$
$$
p _ {M} (s) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} (s) \tag {51}
$$
by induction over the DAG  $(S, \mathcal{A})$ .
Base case. Consider the initial state  $s_0 \in S$ . By definition  $p_i(s_0) = p_M(s_0) = 1$  which implies
$$
p _ {M} \left(s _ {0}\right) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} \left(s _ {0}\right). \tag {52}
$$
Inductive step. Consider a state  $s$  such that (51) holds for all predecessor states  $s_*: (s_* \to s) \in \mathcal{A}$ . For such a state we have
$$
\begin{array}{l} p _ {M} (s) = \sum_ {s _ {*}: (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {M} \left(s _ {*}\right) p _ {M, F} (s \mid s _ {*}) \quad \{u s e d (4 4) \} (53) \\ = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {M} \left(s _ {*}\right)\left(\sum_ {i = 1} ^ {m} p \left(y = i \mid s _ {*}\right) p _ {i, F} \left(s \mid s _ {*}\right)\right) \quad \{\text {u s e d d e f i n i t i o n o f} p _ {M, F} \} (54) \\ = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} \frac {p _ {M} \left(s _ {*}\right)}{p \left(s _ {*}\right)} \left(\sum_ {i = 1} ^ {m} p \left(s _ {*} \mid y = i\right) p \left(y = i\right) p _ {i, F} (s \mid s _ {*})\right) \quad \{\text {u s e d B a y e s ’ t h e o r e m} \} (55) \\ = \sum_ {s _ {*} \cdot (s _ {*} \rightarrow s) \in \mathcal {A}} \frac {p _ {M} \left(s _ {*}\right)}{\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} \left(s _ {*}\right)} \left(\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} \left(s _ {*}\right) p _ {i, F} (s \mid s _ {*})\right) \quad \{u s e d (4 5), (4 8), (4 9) \} (56) \\ = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} \left(\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} \left(s _ {*}\right) p _ {i, F} (s \mid s _ {*})\right) \quad \{\text {u s e d i n d u c t i o n h y p o t h e s i s} \} (57) \\ = \sum_ {i = 1} ^ {m} \omega_ {i} \left(\sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {i} \left(s _ {*}\right) p _ {i, F} (s \mid s _ {*})\right) \quad \{\text {c h a n g e d s u m m a t i o n o r d e r} \} (58) \\ = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i} (s), \quad \{\text {u s e d} (4 4) \} (59) \\ \end{array}
$$
which proves (51) for  $s$ .
D.2 Proof of Proposition 5.2
Claim 1. Our goal is to prove the relationship (13) for all non-terminal states  $s \in S \setminus \mathcal{X}$ . To prove this relationship, we invoke several important properties of Markovian probability flows on DAGs [35].
By Proposition 16 of Bengio et al. [35] for the given GFlowNet forward policy  $p_F(\cdot|\cdot)$  there exists a unique backward policy  $p_B(\cdot|\cdot)$  such that the probability of any complete trajectory  $\tau = (s_0 \to \ldots \to s_{|\tau|} = x)$  in DAG  $(S, \mathcal{A})$  can be expressed as
$$
p (\tau) = p (x) \prod_ {t = 1} ^ {| \tau |} p _ {B} \left(s _ {t - 1} \mid s _ {t}\right), \tag {60}
$$
and the probability of observing a state  $s \in S$  on a trajectory can be expressed as
$$
p (s) = \sum_ {x \in \mathcal {X}} p (x) \sum_ {\tau \in \mathcal {T} _ {s, x}} \prod_ {t = 1} ^ {| \tau |} p _ {B} \left(s _ {t - 1} \mid s _ {t}\right), \tag {61}
$$
where  $\mathcal{T}_{s,x}$  is the set of all (sub)trajectories starting at  $s$  and ending at  $x$ . Moreover,  $p_F(\cdot |\cdot)$  and  $p_B(\cdot |\cdot)$  are related through the "detailed balance condition" [35, Proposition 21]
$$
p (s) p _ {F} \left(s ^ {\prime} \mid s\right) = p \left(s ^ {\prime}\right) p _ {B} \left(s \mid s ^ {\prime}\right), \quad \forall (s \rightarrow s ^ {\prime}) \in \mathcal {A}. \tag {62}
$$
By the statement of Proposition 5.2, in the probabilistic model  $p(x,y)$ , the marginal distribution  $p(x)$  is realized by the GFlowNet forward policy  $p_F(\cdot |\cdot)$  and  $y$  is independent of intermediate states  $s$ . The joint distribution  $p(s,y)$  is given by
$$
\begin{array}{l} p (s, y) = \sum_ {x \in \mathcal {X}} p (x, y) \sum_ {\tau \in \mathcal {T} _ {s, x}} \prod_ {t = 1} ^ {| \tau |} p _ {B} \left(s _ {t - 1} \mid s _ {t}\right) (63) \\ = \sum_ {x \in \mathcal {X}} p (x, y) \sum_ {s ^ {\prime}: (s \rightarrow s ^ {\prime}) \in \mathcal {A}} p _ {B} \left(s \mid s ^ {\prime}\right) \sum_ {\tau \in \mathcal {T} _ {s ^ {\prime}, x}} \prod_ {t = 1} ^ {| \tau |} p _ {B} \left(s _ {t - 1} \mid s _ {t}\right) (64) \\ \end{array}
$$
$$
\begin{array}{l} = \sum_ {s ^ {\prime}: (s \rightarrow s ^ {\prime}) \in \mathcal {A}} p _ {B} (s | s ^ {\prime}) \underbrace {\sum_ {x \in \mathcal {X}} p (x , y) \sum_ {\tau \in \mathcal {T} _ {s ^ {\prime} , x}} \prod_ {t = 1} ^ {| \tau |} p _ {B} \left(s _ {t - 1} \mid s _ {t}\right)} _ {p \left(s ^ {\prime}, y\right)} (65) \\ = \sum_ {s ^ {\prime}: (s \rightarrow s ^ {\prime}) \in \mathcal {A}} p _ {B} \left(s \mid s ^ {\prime}\right) p \left(s ^ {\prime}, y\right). (66) \\ \end{array}
$$
Expressing the conditional probability  $p(y|s)$  through the joint  $p(s,y)$  we obtain
$$
\begin{array}{l} p (y \mid s) = \frac {p (s , y)}{p (s)} \quad \{\text {u s e d} \} (67) \\ = \frac {1}{p (s)} \sum_ {s ^ {\prime}: (s \rightarrow s ^ {\prime}) \in \mathcal {A}} p _ {B} \left(s \mid s ^ {\prime}\right) p \left(s ^ {\prime}, y\right) \quad \{u s e d (6 6) \} (68) \\ = \sum_ {s ^ {\prime}: (s \rightarrow s ^ {\prime}) \in \mathcal {A}} p _ {B} \left(s \mid s ^ {\prime}\right) \frac {p \left(s ^ {\prime}\right)}{p (s)} p \left(y \mid s ^ {\prime}\right) \quad \left\{\text {d e c o m p o s e d} p \left(s ^ {\prime}, y\right) = p \left(s ^ {\prime}\right) p \left(y \mid s ^ {\prime}\right)\right\} (69) \\ = \sum_ {s ^ {\prime}: (s \rightarrow s ^ {\prime}) \in \mathcal {A}} p _ {F} \left(s ^ {\prime} \mid s\right) p \left(y \mid s ^ {\prime}\right), \quad \{\text {u s e d} (6 2) \} (70) \\ \end{array}
$$
which proves (13).
Claim 2. Our goal is to show that the classifier-guided policy (14) induces the conditional distribution  $p(y|x)$ .
We know (Section D.1) that the state probabilities induced by the marginal GFlowNet policy  $p_F(\cdot|\cdot)$  satisfy the recurrence
$$
p (s) = \sum_ {s _ {*}: (s _ {*} \rightarrow s) \in \mathcal {A}} p \left(s _ {*}\right) p _ {F} (s \mid s _ {*}). \tag {71}
$$
Let  $p_{y}(\cdot)$  denote the state probabilities induced by the classifier-guided policy (14). These probabilities by definition (Section D.1) satisfy the recurrence
$$
p _ {y} (s) = \sum_ {s _ {*}: (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {y} \left(s _ {*}\right) p _ {F} (s \mid s _ {*}, y). \tag {72}
$$
We show that
$$
p _ {y} (s) = p (s \mid y), \tag {73}
$$
by induction over DAG  $(S, \mathcal{A})$ .
Base case. Consider the initial state  $s_0$ . By definition  $p_y(s_0) = 1$ . At the same time  $p(s_0|y) = p(\{\tau : s_0 \in \tau\}|y) = 1$ . Therefore  $p_y(s_0) = p(s_0|y)$ .
Inductive step. Consider a state  $s$  such that (73) holds for all predecessor states  $s_*: (s_* \to s) \in \mathcal{A}$ . For such a state we have
$$
\begin{array}{l} p _ {y} (s) = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {y} \left(s _ {*}\right) p _ {F} (s \mid s _ {*}, y) \quad \{u s e d (7 2) \} (74) \\ = \sum_ {s _ {*}: (s _ {*} \rightarrow s) \in \mathcal {A}} p _ {y} \left(s _ {*}\right) p _ {F} (s \mid s _ {*}) \frac {p (y \mid s)}{p (y \mid s _ {*})} \quad \{\text {u s e d} (1 4) \} (75) \\ = \sum_ {s _ {*}: (s _ {*} \rightarrow s) \in \mathcal {A}} p \left(s _ {*} \mid y\right) p _ {F} (s \mid s _ {*}) \frac {p (y \mid s)}{p (y \mid s _ {*})} \quad \{\text {u s e d i n d u c t i o n h y p o t h e s i s} \} (76) \\ = \sum_ {s _ {*} : (s _ {*} \rightarrow s) \in \mathcal {A}} \frac {p (y | s _ {*}) p (s _ {*})}{p (y)} p _ {F} (s | s _ {*}) \frac {p (y | s)}{p (y | s _ {*})} \quad \left\{\text {u s e d B a y e s ’ t h e o r e m} \right\} (77) \\ = \frac {p (y \mid s)}{p (y)} \sum_ {s _ {*}: (s _ {*} \rightarrow s) \in \mathcal {A}} p \left(s _ {*}\right) p _ {F} (s \mid s _ {*}) \quad \{\text {r e a r r a n g e d t e r m s} \} (78) \\ \end{array}
$$
$$
\begin{array}{l} = \frac {p (y \mid s)}{p (y)} p (s) \quad \{\text {u s e d} (7 1) \} (79) \\ = p (s \mid y), \quad \{\text {u s e d B a y e s} \} (80) \\ \end{array}
$$
which proves (73) for state  $s$ .
D.3 Proof of Theorem 5.3
By Proposition 5.1 we have that the policy
$$
p _ {M, F} \left(s ^ {\prime} \mid s\right) = \sum_ {i = 1} ^ {m} p _ {i, F} \left(s ^ {\prime} \mid s\right) \widetilde {p} (y = i \mid s), \tag {81}
$$
generates the mixture distribution  $p_{M}(x) = \frac{1}{m}\sum_{i = 1}^{m}p_{i}(x)$ .
In the probabilistic model  $\widetilde{p}(x, y_1, \ldots, y_n)$  the marginal distribution  $\widetilde{p}(x) = p_M(x)$  is realized by the mixture policy  $p_{M,F}$ . Therefore,  $\widetilde{p}(x, y_1, \ldots, y_n)$  satisfies the conditions of Proposition 5.2 which states that the conditional distribution  $\widetilde{p}(x|y_1, \ldots, y_n)$  is realized by the classifier-guided policy
$$
p _ {F} \left(s ^ {\prime} \mid s, y _ {1}, \dots , y _ {n}\right) = p _ {M, F} \left(s ^ {\prime} \mid s\right) \frac {\widetilde {p} \left(y _ {1} , \dots , y _ {n} \mid s ^ {\prime}\right)}{\widetilde {p} \left(y _ {1} , \dots , y _ {n} \mid s\right)} = \frac {\widetilde {p} \left(y _ {1} , \dots y _ {n} \mid s ^ {\prime}\right)}{\widetilde {p} \left(y _ {1} , \dots y _ {n} \mid s\right)} \sum_ {i = 1} ^ {m} p _ {i, F} \left(s ^ {\prime} \mid s\right) \widetilde {p} (y = i \mid s). \tag {82}
$$
D.4 Detailed Derivation of Classifier Training Objective
This section provides a more detailed step-by-step derivation of the non-terminal state classifier training objective (18).
Step 1. Our goal is to train a classifier  $\widetilde{Q}(y_1, \ldots, y_n | s)$ . This classifier can be obtained as the optimal solution of
$$
\min  _ {\phi} \mathbb {E} _ {\hat {\tau}, \hat {y} _ {1}, \dots , \hat {y} _ {n} \sim \widetilde {p} (\tau , y _ {1}, \dots , y _ {n})} \left[ \ell (\hat {\tau}, \hat {y} _ {1}, \dots , \hat {y} _ {n}; \phi) \right], \tag {83}
$$
where  $\ell(\cdot)$  is defined in equation (17). An unbiased estimate of the loss (and its gradient) can be obtained by sampling  $(\widehat{\tau}, \widehat{y}_1, \dots, \widehat{y}_n)$  and evaluating (17) directly. However sampling tuples  $(\tau, y_1, \dots, y_n)$  is not straightforward. The following steps describe our proposed approach to the estimation of expectation in (83).
Step 2. The expectation in (83) can be expressed as
$$
\mathbb {E} _ {\widehat {\tau}, \widehat {y} _ {1} \sim \widetilde {p} (\tau , y _ {1})} \left[ \sum_ {\widehat {y} _ {2} = 1} ^ {m} \dots \sum_ {\widehat {y} _ {n} = 1} ^ {m} \left(\prod_ {i = 2} ^ {n} \widetilde {p} \left(y _ {i} = \widehat {y} _ {i} \mid x = \widehat {x}\right)\right) \ell (\widehat {\tau}, \widehat {y} _ {1}, \dots , \widehat {y} _ {n}; \phi) \right], \tag {84}
$$
where we re-wrote the expectation over  $(y_{2},\ldots ,y_{n})|\tau$  as the explicit sum of the form  $\mathbb{E}_{q(z)}[g(z)] = \sum_{z\in \mathcal{Z}}q(z)g(z)$ . The expectation over  $(\tau ,y_{1})$  can be estimated by sampling pairs  $(\widehat{\tau},\widehat{y}_1)$  as described in the paragraph after equation (17): 1)  $\widehat{y}_1\sim \widetilde{p} (y_1)$  and 2)  $\widehat{\tau}\sim p_{\widehat{y}_1}(\tau)$ . The only missing part is the probabilities  $\widetilde{p} (y_i = \widehat{y}_i|x = \widehat{x})$  which are not directly available.
Step 3. Our proposal is to approximate these probabilities as  $\widetilde{p}(y_1 = j|x = \widehat{x}) \approx w_j(\widehat{x};\phi) = \widetilde{Q}_{\phi}(y_1 = j|x = \widehat{x})$ . The idea here is that the terminal state classifier  $\widetilde{Q}_{\phi}(y_1|x)$ , when trained to optimality, produces outputs exactly equal to the probabilities  $\widetilde{p}(y_1|x)$ , and the more the classifier is trained the better is the approximation of the probabilities.
Step 4. Steps 1-3, give a procedure where the computation of the non-terminal state classification loss requires access to the terminal state classifier. As we described in the paragraph preceding equation (18), we propose to train non-terminal and terminal classifiers simultaneously and introduce "target network" parameters. The weights  $w$  are computed by the target network  $\tilde{Q}_{\overline{\phi}}$ .
Combining all the steps above, we arrive at objective (18) which we use to estimate the expectation in (83).
Note that equation (18) involves summation over  $\widehat{y}_2, \ldots, \widehat{y}_n$  with  $m^{n-1}$  terms in the sum. If values of  $n$  and  $m$  are small, the sum can be evaluated directly. In general, one could trade off estimation accuracy for improved speed by replacing the summation with Monte Carlo estimation. In this case, the values  $\widehat{y}_k$  are sampled from the categorical distributions  $Q_{\overline{\phi}}(y|x)$ . Note that labels can be sampled in parallel since  $y_i$  are independent given  $x$ .
D.5 Assumptions and Proof of Proposition A.1
This subsection provides a formal statement of the assumptions and a more detailed formulation of Proposition A.1.
The assumptions, the formulation of the result, and the proof below closely follow those of Theorem 1 of Peluchetti [53]. Theorem 1 in [53] generalizes the result of Brigo [57] (Corollary 1.3), which derives the SDE for mixtures of 1D diffusion processes.
We found an error in the statement and the proof of Theorem 1 (Appendix A.2 of [53]). The error makes the result of [53] for  $D$ -dimensional diffusion processes disagree with the result of [57] for 1-dimensional diffusion processes.
Here we provide a corrected version of Theorem 1 of [53] in a modified notation and a simplified setting (mixture of finite rather than infinite number of diffusion processes). Most of the content is directly adapted from [53].
Notation:
- for a vector-valued  $f: \mathbb{R}^D \to \mathbb{R}^D$ , the divergence of  $f$  is denoted as  $\nabla \cdot (f(x)) = \sum_{d=1}^{D} \frac{\partial}{\partial x_d} f_d(x)$ ,
- for a scalar-values  $a: \mathbb{R}^D \to \mathbb{R}$ , the divergence of the gradient of  $a$  (the Laplace operator) is denoted by  $\Delta(a(x)) = \nabla \cdot (\nabla a(x)) = \sum_{d=1}^{D} \frac{\partial^2}{\partial x_d^2} a(x)$ .
Assumption 1 (SDE solution). A given  $D$ -dimensional  $\mathrm{SDE}(f,g)$ :
$$
d x _ {t} = f _ {t} \left(x _ {t}\right) d t + g _ {t} d w _ {t}, \tag {85}
$$
with associated initial distribution  $p_0(x)$  and integration interval  $[0, T]$  admits a unique strong solution on  $[0, T]$ .
Assumption 2 (SDE density). A given  $D$ -dimensional  $\mathrm{SDE}(f,g)$  with associated initial distribution  $p_0(x)$  and integration interval  $[0,T]$  admits a marginal density on  $(0,T)$  with respect to the  $D$ -dimensional Lebesgue measure that uniquely satisfies the Fokker-Plank (Kolmogorov-forward) partial differential equation (PDE):
$$
\frac {\partial p _ {t} (x)}{\partial t} = - \nabla \cdot \left(f _ {t} (x) p _ {t} (x)\right) + \frac {1}{2} \Delta \left(g _ {t} ^ {2} p _ {t} (x)\right). \tag {86}
$$
Assumption 3 (positivity). For a given stochastic process, all finite-dimensional densities, conditional or not, are strictly positive.
Theorem D.1 (Diffusion mixture representation). Consider the family of  $D$ -dimensional SDEs on  $t \in [0, T]$  indexed by  $i \in \{1, \dots, m\}$ ,
$$
d x _ {i, t} = f _ {i, t} \left(x _ {i, t}\right) d t + g _ {i, t} d w _ {i, t}, \quad x _ {i, 0} \sim p _ {i, 0}, \tag {87}
$$
where the initial distributions  $p_{i,0}$  and the Wiener processes  $w_{i,t}$  are all independent. Let  $p_{i,t}, t \in (0,T)$  denote the marginal density of  $x_{i,t}$ . For mixing weights  $\{\omega_i\}_{i=1}^m$ ,  $\omega_i \geq 0$ ,  $\sum_{i=1}^m \omega_i = 1$ , define the mixture marginal density  $p_{M,t}$  for  $t \in (0,T)$  and the mixture initial distribution  $p_{M,0}$  by
$$
p _ {M, t} (x) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {i, t} (x) \quad p _ {M, 0} (x) = \sum_ {i = 1} ^ {m} \omega_ {i} p _ {M, 0} (x). \tag {88}
$$
Consider the  $D$ -dimensional SDE on  $t \in [0, T]$  defined by
$$
f _ {M, t} (x) = \frac {\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i , t} (x) f _ {i , t} (x)}{p _ {M , t} (x)}, \quad g _ {M, t} (x) = \sqrt {\frac {\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i , t} (x) g _ {i , t} ^ {2}}{p _ {M , t} (x)}}, \tag {89}
$$
$$
d x _ {t} = f _ {M, t} \left(x _ {t}\right) d t + g _ {M, t} \left(x _ {t}\right) d w _ {t}, \quad x _ {M, 0} \sim p _ {M, 0}. \tag {90}
$$
It is assumed that all diffusion processes  $x_{i,t}$  and the diffusion process  $x_{M,t}$  satisfy the regularity Assumptions 1, 2, and 3. Then the marginal distribution of the diffusion  $x_{M,t}$  is  $p_{M,t}$ .
Proof. For  $0 < t < T$  we have that
$$
\begin{array}{l} \frac {\partial p _ {M , t} (x)}{\partial t} = \frac {\partial}{\partial t} \left(\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i, t} (x)\right) (91) \\ = \sum_ {i = 1} ^ {m} \omega_ {i} \frac {\partial p _ {i , t} (x)}{\partial t} (92) \\ = \sum_ {i = 1} ^ {m} \omega_ {i} \left(- \nabla \cdot \left(f _ {i, t} (x) p _ {i, t} (x)\right) + \frac {1}{2} \Delta \left(g _ {i, t} ^ {2} p _ {i, t} (x)\right)\right) (93) \\ = \sum_ {i = 1} ^ {m} \omega_ {i} \left(- \nabla \cdot \left(\frac {p _ {i , t} (x) f _ {i , t} (x)}{p _ {M , t} (x)} p _ {M, t} (x)\right) + \frac {1}{2} \Delta \left(\frac {p _ {i , t} (x) g _ {i , t} ^ {2}}{p _ {M , t} (x)} p _ {M, t} (x)\right)\right) (94) \\ = - \nabla \cdot \left(\sum_ {i = 1} ^ {m} \frac {\omega_ {i} p _ {i , t} (x) f _ {i , t} (x)}{p _ {M , t} (x)} p _ {M, t} (x)\right) + \frac {1}{2} \Delta \left(\sum_ {i = 1} ^ {m} \frac {\omega_ {i} p _ {i , t} (x) g _ {i , t} ^ {2}}{p _ {M , t} (x)} p _ {M, t} (x)\right) (95) \\ = - \nabla \cdot \left(f _ {M, t} (x) p _ {M, t} (x)\right) + \frac {1}{2} \Delta \left(g _ {M, t} ^ {2} p _ {M, t} (x)\right). (96) \\ \end{array}
$$
The second is an exchange of the order of summation and differentiation, the third line is the application of the Fokker-Planck PDEs for processes  $x_{i,t}$ , the fourth line is a rewriting in terms of  $p_{M,t}$ , the fifth line is another exchange of the order of summation and differentiation. The result follows by noticing that  $p_{M,t}(x)$  satisfies the Fokker-Planck equation of SDE( $f_M, g_M$ ).
Proof of Proposition A.1. Below, we show that the result of Proposition A.1 follows from Theorem D.1.
First, we rewrite  $f_{M,t}(x_t)$  and  $g_{M,t}(x_t)$  in (89) in terms of the classifier probabilities (21):
$$
f _ {M, t} \left(x _ {t}\right) = \frac {\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i , t} \left(x _ {t}\right) f _ {i , t} \left(x _ {t}\right)}{p _ {M , t} \left(x _ {t}\right)} = \sum_ {i = 1} ^ {m} p (y = i \mid x _ {t}) f _ {i, t} \left(x _ {t}\right), \tag {97}
$$
$$
g _ {M, t} \left(x _ {t}\right) = \sqrt {\frac {\sum_ {i = 1} ^ {m} \omega_ {i} p _ {i , t} \left(x _ {t}\right) g _ {i , t} ^ {2}}{p _ {M , t} \left(x _ {t}\right)}} = \sqrt {\sum_ {i = 1} ^ {m} p (y = i \mid x _ {t}) g _ {i , t} ^ {2}}. \tag {98}
$$
With these expressions, we apply the result of Theorem D.1 to the base forward processes  $dx_{i,t} = f_{i,t}(x_{i,t})dt + g_{i,t}dw_{i,t}$  and obtain the mixture forward process in equation (19).
From the forward process, we derive the backward process following Song et al. [34]. Using the result of Anderson [38], the backward process for (19) is given by
$$
d x _ {t} = \left[ f _ {M, t} \left(x _ {t}\right) - \nabla_ {x _ {t}} \left(g _ {M, t} ^ {2} \left(x _ {t}\right)\right) - g _ {M, t} ^ {2} \left(x _ {t}\right) \nabla_ {x _ {t}} \log p _ {M, t} \left(x _ {t}\right) \right] d t + g _ {M, t} \left(x _ {t}\right) d \bar {w} _ {t}. \tag {99}
$$
Note that the term  $\nabla_{x_t}(g_{M,t}^2 (x_t))$  is due to the fact that the diffusion coefficient  $g_{M,t}(x_t)$  in (19) is a function of  $x$  (cf., equation (16), Appendix A in [34]). This term can be transformed as follows
$$
\begin{array}{l} \nabla_ {x _ {t}} \left(g _ {M, t} ^ {2} \left(x _ {t}\right)\right) = \nabla_ {x _ {t}} \left(\sum_ {i = 1} ^ {m} p (y = i \mid x _ {t}) g _ {i, t} ^ {2}\right) (100) \\ = \sum_ {i = 1} ^ {m} g _ {i, t} ^ {2} \nabla_ {x _ {t}} p (y = i | x _ {t}) (101) \\ = \sum_ {i = 1} ^ {m} p (y = i | x _ {t}) g _ {i, t} ^ {2} \nabla_ {x _ {t}} \log p (y = i | x _ {t}) (102) \\ \end{array}
$$