# Mean Estimation on the Diagonal of Product Manifolds
## 2 Background
### Diffusion Bridges  
The proposed sampling scheme for the (weighted) diffusion mean builds on simulation methods for conditioned diffusion processes, diffusion bridges. Here, we outline ways to simulate conditioned diffusion processes numerically in both the Euclidean and manifold context. Euclidean Diffusion Bridges  
Let \((\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P})\) be a filtered probability space, and \(X\) a \(d\)-dimensional Euclidean diffusion \([0,T]\) satisfying the stochastic differential equation (SDE)  
\[dX_{t}=b_{t}(X_{t})dt+\sigma_{t}(X_{t})dW_{t},\quad X_{0}=x, \tag{6}\]  
where \(W\) is a \(d\)-dimensional Brownian motion. Let \(v\in\mathbb{R}^{d}\) be a fixed point. Conditioning \(X\) on reaching \(v\) at a fixed time \(T>0\) gives the bridge process \(X|X_{T}=v\). Denoting this process \(Y\), Doob's \(h\)-transform shows that \(Y\) is a solution of the SDE (see e.g., [16])  
\[ \begin{split} dY_{t}&=\tilde{b}_{t}(Y_{t})dt+\sigma _{t}(Y_{t})d\tilde{W}_{t},\quad Y_{0}=x\\ \tilde{b}_{t}(y)&=b_{t}(y)+a_{t}(y)\
abla_{y}\log p _{T-t}(y,v),\end{split}  \tag{7}\]  
where \(p_{t}(\cdot,\cdot)\) denotes the transition density of the diffusion \(X\), \(a=\sigma\sigma^{T}\), and where \(\tilde{W}\) is a \(d\)-dimensional Brownian motion under a changed probability law. From a numerical viewpoint, in most cases, the transition density is intractable and therefore direct simulation of (7) is not possible. If we instead consider a Girsanov transformation of measures to obtain the system (see, e.g., [17] Theorem 1)  
\[ \begin{split} d\gamma_{t}&=\tilde{b}_{t}(Y_{t})dt+ \sigma_{t}(Y_{t})d\tilde{W}_{t},\quad Y_{0}=x\\ \tilde{b}_{t}(y)&=b_{t}(y)+\sigma_{t}(y)h(t,y),\end{split}  \tag{8}\]  
the corresponding change of measure is given by  
\[\left.\frac{d\mathbb{P}^{h}}{d\mathbb{P}}\right|_{\mathcal{F}_{t}}=e^{\int_{0 }^{t}h(s,X_{s})^{T}dW_{t}-\frac{1}{2}\int_{0}^{t}\|h(s,X_{s})\|^{2}ds}. \tag{9}\]  
From (7), it is evident that \(h(t,x)=\sigma^{T}\
abla_{x}\log p_{T-t}(x,v)\) gives the diffusion bridge. However, different choices of the function \(h\) can yield processes which are absolutely continuous regarding the actual bridges, but which can be simulated directly. [PERSON] and [PERSON] [17] suggested to use \(h(t,x)=\sigma_{t}^{-1}(x)\
abla_{x}\log q_{T-t}(x,v)\), where \(q\) denotes the transition density of a standard Brownian motion with mean \(v\), i.e., \(q_{t}(x,v)=(2\pi t)^{-d/2}\exp(-\|x-v\|^{2}/2t)\). They furthermore proposed a method that would disregard the drift term \(b\), i.e., \(h(t,x))=\sigma_{t}^{-1}(x)\
abla_{x}\log q_{T-t}(x,v)-\sigma_{t}^{-1}(x)b_{t} (x)\). Under certain regularity assumptions on \(b\) and \(\sigma\), the resulting processes converge to the target in the sense that \(\lim_{t\to T}Y_{t}=v\) a.s. In addition, for bounded continuous functions \(f\), the conditional expectation is given by  
\[\mathbb{E}[f(X)|X_{T}=v]=\mathbb{C}\mathbb{E}[f(Y)\varphi(y)], \tag{10}\]  
where \(\varphi\) is a functional of the whole path \(Y\) on \([0,T]\) that can be computed directly. From the construction of the \(h\)-function, it can be seen that the missing drift term is accounted for in the correction factor \(\varphi\). The simulation approach of [17] can be improved by the simulation scheme introduced by [PERSON] et al. [18]. Here, an \(h\)-function defined by \(h(t,x)=\
abla_{x}\log\hat{p}_{T-t}(x,v)\) is suggested, where \(\hat{p}\) denotes the transition density of an auxiliary diffusion process with known transition densities. The auxiliary process can for example be linear because closed-form solutions of transition densities for linear processes are available. Under the appropriate measure \(\mathbb{P}^{h}\), the guided proposal process is a solution to  
\[dY_{t}=b_{t}(Y_{t})dt+a_{t}(Y_{t})\
abla_{x}\log\hat{p}_{T-t}(x,v)|_{x=Y_{t}} dt+\sigma_{t}(Y_{t})dW_{t}. \tag{11}\]  
Note the factor \(a(t,y)\) in the drift in (7) which is also present in (11) but not with the scheme proposed by [17]. Moreover, the choice of a linear process grants freedom to model.