# Mean Estimation on the Diagonal of Product Manifolds
## 3 Diffusion Mean Estimation
This is illustrated in Figure 2._  
The idea of conditioning diffusion bridge processes on the diagonal of a product manifold originates from the facts established in Examples 1-3. We sample the mean by sampling from the conditional distribution \(Y_{1}|Y_{1}=\cdots=Y_{n}\) from Example 2 using a guided proposal scheme on the product manifolds \(M^{n}\) and on each step of the sampling projecting to the diagonal as in Example 3. Turning now to the manifold situation, we replace the normal distributions with mean \(x_{i}\in\mathbb{R}^{d}\) and variance \(T/w_{i}\) with Brownian motions started at \(x_{i}\in M\) and evaluated at time \(T/w_{i}\). Please note that the Brownian motion density, the heat kernel, is symmetric in its coordinates: \(p_{t}(x,y)=p_{t}(y,x)\). We will work with multiple process and indicate with superscript the density with respect to a particular process, e.g., \(p_{T}^{X}\). Note also that change of the evaluation time \(T\) is equal to scaling the variance, i.e., \(p_{aT}^{X}(x,y)=p_{T}^{Xa}(x,y)\) where \(X^{\alpha}\) is a Brownian motion with variance of the increments scaled by \(\alpha>0\). This gives the following theorem, first stated in [7] with sketch proof:  
**Theorem 1**. : _Let \(X_{t}=(X_{1,t}^{w_{1}^{-1}},\ldots,X_{n,t}^{w_{n}^{-1}})\) consist of \(n\) independent Brownian motions on \(M\) with variance \(w_{i}^{-1}\) and \(X_{i,0}=x_{i}\), and let \(\mathbb{P}^{*}\) the law of the conditioned process \(Y_{t}=X_{t}|X_{T}\in N\), \(N=\mathrm{diag}(M^{n})\). Let \(v\) be the random variable \(Y_{1,T}\). Then \(v\) has density \(p_{v}^{Y}(y)\propto\prod_{i=1}^{n}p_{T/w_{i}}(x_{i};y)\) and \(v=Y_{i,T}\) for all \(i\) a.s. (almost surely)._  
Figure 2: The mean estimator viewed as a projection onto the diagonal of a product manifold. Conditioning on the closest point in the diagonal yields a density on the diagonal depending on the time to arrival \(T>0\). As \(T\) tends to zero the density convergence to the Dirac-delta distribution (grey), whereas as \(T\) increases the variance of the distribution increases (rouge). **Proof. **\(p_{T}^{X}((x_{1},\ldots,x_{n}),(y,\ldots,y))=\prod_{i=1}^{n}p_{T}^{X_{i}^{n-1}}(x_{i},y)\) because the processes \(X_{i,t}\) are independent. By symmetry of the Brownian motion and the time rescaling property, \(p_{T}^{X_{i}^{n-1}}(x_{i},y)\)\(=p_{T/w^{i}}(y,x_{i})\). For elements \((y,\ldots,y)\in\text{diag}(M^{n})\) and \(x\in M^{n}\), \(p_{v}(y)=p_{T}^{X}(x,y)\propto p_{T}^{X}(x,y)\). As a result of the conditioning, \(v=Y_{1,T}=\cdots=Y_{n,T}\). In combination, this establishes the result. Consequently, the set of modes of \(p_{v}\) equal the set of the maximizers for the likelihood \(L(y;x_{1},\ldots,x_{n})=\prod_{i=1}^{n}p_{T/w_{i}}(x_{i};y)\) and thus the weighted diffusion mean. This result is the basis for the sampling scheme. Intuitively, if the distribution of \(v\) is relatively well behaved (e.g., close to normal), a sample from \(v\) will be close to a weighted diffusion mean with high probability. In practice, however, we cannot sample \(Y_{t}\) directly. Instead, we will below use guided proposal schemes resulting in processes \(\tilde{Y}_{t}\) with law \(\bar{\mathbb{P}}\) that we can actually sample and that under certain assumptions, will be absolutely continuous with respect to \(Y_{t}\) with explicitly computable likelihood ratio so that \(\mathbb{P}^{*}=\frac{\varphi(\tilde{Y}_{T})}{\mathbb{E}^{\mathbb{P}}[\varphi( \tilde{Y}_{T})]}\bar{\mathbb{P}}\). **Corollary 1**. : _Let \(\bar{\mathbb{P}}\) be the law of \(\tilde{Y}_{t}\) and \(\varphi\) be the corresponding correction factor of the guiding scheme. Let \(\bar{\varphi}\) be the random variable \(\tilde{Y}_{1,T}\) with law \(\frac{\varphi(\tilde{Y}_{T})}{\mathbb{E}^{\mathbb{P}}[\varphi(\tilde{Y}_{T})]}\bar {\mathbb{P}}\). Then \(\bar{\varphi}\) has density \(p_{\bar{\varphi}}(y)\propto\prod_{i=1}^{n}p_{T/w_{i}}(x_{i};y)\)._  
We now proceed to actually construct the guided sampling schemes.