I am working on a time series model involving Kalman filters and smoothing to estimate state variables Y_i. The part of the model is structured as follows:

Y_1, …, Y_n are iid.

Y_i ~ MN(M, Σ, V)

Σ ~ IW(ν, Ξ).

The matrix V is known and constant.

As part of a Gibbs sampling process to estimate Y and Σ, I need to compute the posterior distribution p(Σ | Y_1, …, Y_n).

While I have managed to derive p(Y_1, …, Y_n | Σ) using the smoothing equations from the Kalman filter, I am struggling to find p(Σ | Y_1, …, Y_n).

My Understanding and Attempts:

Conjugacy: I am aware that there is a conjugacy between the Inverse Wishart distribution and the multivariate normal distribution, which suggests that the posterior of Σ should also follow an Inverse Wishart distribution.
Marginal Distribution of Y: I understand that the marginal p(Y) in the joint distribution (Y, Σ) is a Matrix-T(M,V,ν,Ξ), but this hasn’t led me to a solution for p(Σ|Y).

Specific Question:

How can I compute p(Σ | Y_1, …, Y_n)? Any pointers or resources would be really helpful.