Consider a Wiener process with zero drift, infintesimal variance $\sigma^2$, and an unknown starting value $\nu$. That is, \begin{align} Y_t \sim \mathcal{N}(\nu, t\sigma^2). \end{align}

Now, suppose that we have a Gaussian likelihood for each $Y_1, \dotsc, Y_t$: $X_1, \dotsc, X_t$ having means $\mu_i$ and precisions $\lambda_i$.

Question 1: If we know $\sigma$ and have $X_{i\le t}$, what is the posterior on $Y_t$?

Intuitively, I think we modify each $X_{t-i}$ by adding $i\sigma^2$ to its variance, and then the combined likelihood on $Y_t$ is Gaussian with mean and precision: \begin{align} \mu^\star &= \frac{\sum_{i\le t}\mu_i\lambda_i}{\lambda^\star} \\\\ \lambda^\star &= \sum_{i\le t}\lambda_i. \end{align}

Question 2: If we don't know $\sigma$, but have $X_{i\le t}$, what is the posterior on $Y_t$ and $\sigma$?

It seems that we would want to estimate $\sigma$ by setting a gamma prior on $\sigma^{-2}$. (That's what we would do in the special case that each of the $X_i$ has zero variance.) I'm having trouble proceeding from here. (Doesn't this make the combined likelihood on $Y_t$ student's t-distributed?)

Consider a Wiener process with zero drift, infintesimal variance $\sigma^2$, and an unknown starting value $\nu$. That is, \begin{align} Y_t \sim \mathcal{N}(\nu, t\sigma^2). \end{align}

Now, suppose that we don't observe the $Y_i$ directly, but rather have a corresponding Gaussian likelihood for each $Y_i$: $X_i$ having mean $\mu_i$ and precision $\lambda_i$.

Question 1: If we know $\sigma$ and have $X_{i\le t}$, what is the posterior on $Y_t$?

Intuitively, I think we modify each $X_{t-i}$ by adding $i\sigma^2$ to its variance, and then the combined likelihood on $Y_t$ is Gaussian with mean and precision: \begin{align} \mu^\star &= \frac{\sum_{i\le t}\mu_i\lambda_i}{\lambda^\star} \\\\ \lambda^\star &= \sum_{i\le t}\lambda_i. \end{align}

Question 2: If we don't know $\sigma$, but have $X_{i\le t}$, what is the posterior on $Y_t$ and $\sigma$?

It seems that we would want to estimate $\sigma$ by setting a gamma prior on $\sigma^{-2}$. (That's what we would do in the special case that each of the $X_i$ has zero variance.) I'm having trouble proceeding from here. (Doesn't this make the combined likelihood on $Y_t$ student's t-distributed?)