# Mean Estimation on the Diagonal of Product Manifolds
## 3 Diffusion Mean Estimation  
The standard setup for diffusion mean estimation described in the literature (e.g., [13]) is as follows: Given a set of observations \(x_{1},\ldots,x_{n}\in M\), for each observation \(x_{i}\), sample a guided bridge process approximating the bridge \(X_{i,t}|X_{i,T}=x_{i}\) with starting point \(x_{0}\). The expectation over the correction factors can be computed from the samples, and the transition density can be evaluated using (5). An iterative maximum likelihood approach using gradient descent to update \(x_{0}\) yielding an approximation of the diffusion mean in the final value of \(x_{0}\). The computation of the diffusion mean, in the sense just described, is, similarly to the Frechet mean, computationally expensive. We here explore the idea first put forth in [7]: We turn the situation around to simulate \(n\) independent Brownian motions starting at each of \(x_{1},\ldots,x_{n}\), and we condition the \(n\) processes to coincide at time \(T\). We will show that the value \(x_{1,T}=\cdots=x_{n,T}\) is an estimator of the diffusion mean. By introducing weights in the conditioning, we can similarly estimate the weighted diffusion mean. The construction can concisely be described as a single Brownian motion on the \(n\)-times product manifold \(M^{n}\) conditioned to hit the diagonal \(\text{diag}(M^{n})=\{(x,\ldots,x)|x\in M\}\subset M^{n}\). To shorten notation, we denote the diagonal submanifold \(N\) below. We start with examples with \(M\) Euclidean to motivate the construction. **Example 1**. : _Consider the two-dimensional Euclidean multivariate normal distribution_  
\[ \begin{pmatrix}X\\ Y\end{pmatrix} \sim N\bigg{(} \begin{pmatrix}\mu_{1}\\ \mu_{2}\end{pmatrix} , \begin{pmatrix}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{pmatrix} \bigg{)}.\]  
_The conditional distribution of \(X\) given \(Y=y\) follows a univariate normal distribution_  
\[X|Y=y\sim N\Big{(}\mu_{1}+\sigma_{12}\sigma_{22}^{-1}(y-\mu_{2}),\sigma_{11}- \sigma_{12}\sigma_{22}^{-1}\sigma_{21}\Big{)}.\]  
_This can be seen from the fact that if \(X\sim N(\mu,\Sigma)\) then for any linear transformation \(AX+b\sim N(b+A\mu,A\Sigma A^{T})\). Defining the random variable \(Z=X-\sigma_{12}\sigma_{22}^{-1}Y\), the result applied to \((Z,X)\) gives \(Z\sim N\Big{(}\mu_{1}-\sigma_{12}\sigma_{22}^{-1}\mu_{2},\sigma_{11}-\sigma_{12 }\sigma_{22}^{-1}\sigma_{21}\Big{)}\). The conclusion then follows from \(X=Z+\sigma_{12}\sigma_{22}^{-1}Y\). Please note that \(X\) and \(Y\) are independent if and only if \(\sigma_{12}=\sigma_{21}=0\) and the conditioned random variable is in this case identical in law to \(X\)._Let now \(x_{1},\ldots,x_{n}\in M\) be observations and let \(x=(x_{1},\ldots,x_{n})\in M^{n}\) be an element of the \(n\)-product manifold \(M\times\cdots\times M\) with the product Riemannian metric. We again first consider the case \(M=\mathbb{R}^{d}\):  
**Example 2**. : _Let \(Y_{i}\sim N\Big{(}x_{i},\frac{T}{w_{i}}I_{d}\Big{)}\) be independent random variables. The conditional distribution \(Y_{1}|Y_{1}=\cdots=Y_{n}\) is normal \(N\Big{(}\frac{\sum_{i}w_{i}x_{i}}{\sum_{i}w_{i}}\frac{T}{\sum_{i}w_{i}}\Big{)}\). This can be seen inductively: The conditioned random variable \(Y_{1}|Y_{1}=Y_{2}\) is identical to \(Y_{1}|Y_{1}-Y_{2}=0\). Now let \(X=Y_{1}\) and \(Y=Y_{1}-Y_{2}\) and refer to Example 1. To conclude, assume \(Z_{n}:=Y_{1}|Y_{1}=\cdots,=Y_{n-1}\) follows the desired normal distribution. Then \(Z_{n}|Z_{n}=Y_{n}\) is normally distributed with the desired parameters and \(Z_{n}|Z_{n}=Y_{n}\) is identical to \(Y_{1}|Y_{1}=\cdots=Y_{n}\)._  
The following example establishes the weighted average as a projection onto the diagonal. **Example 3**. : _Let \(x\) be a point in \((\mathbb{R}^{d})^{n}\) and let \(P\) be the orthogonal projection to the diagonal of \((\mathbb{R}^{d})^{n}\) such that \(Px=\Big{(}\frac{1}{nd}\sum_{i=1}^{nd}x_{i}\ldots\frac{1}{nd}\sum_{i=1}^{nd}x_{ i}\Big{)}^{T}\). We see that the projection yields \(n\) copies of the arithmetic mean of the coordinates.