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I am considering the following non-linear state space model:

$X_t=\frac{X_{t-1}}{2}+25\frac{X_{t-1}}{1+X_{t-1}^2}+8\cos{1.2t}+\epsilon_t, \epsilon_t\sim N(0,\sigma_x^2 ) $

$Y_t=\frac{X_t^2}{20}+\eta_t,\eta_t\sim N(0,\sigma_y^2)$

I want to make inference on $\theta=(\sigma_x^2,\sigma_y^2)$ and the unobserved states $X_{1:T}$ in this model using Particle Gibbs. In order to this I need to derive the conditional posterior for $\theta$, $p(\theta\mid y_{1:T},x_{1:T})$. Following Andrieu et al.(2010), I assign the following priors

$\sigma_x^2\sim IG(a,b), \sigma_y^2\sim IG(a,b)$, where $IG$ denotes the inverse Gamma distribution. In order to derive the conditional posterior, I consider the following approach:

$p(\theta\mid y_{1:T},x_{1:T})=\frac{p(\theta,y_{1:T},x_{1:T})}{p(y_{1:T},x_{1:T})} \propto p(\theta, y_{1:T},x_{1:T})=p(y_{1:T}\mid x_{1:T},\theta)p(x_{1:T}\mid \theta)p(\theta)=\prod_{i=1}^{T}p(y_i\mid x_i,\theta)p(x_1)\prod_{t=2}^{T}p(x_t\mid x_{t-1},\theta)p(\theta).$

Due to the model dynamics it might be true that $p(y_i\mid x_i,\theta)$ and $p(x_t\mid x_{t-1},\theta)$ are Gaussian. Can anybody tell me whether the approach outlined above is right? Do I just need to plug in all the densities, and try to figure out what the posterior will be in this specific case?

Thank you!

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Your approach is basically right. As a function of $\theta$, you should find that the functions $p(y_t | x_t,\theta)$ and $p(x_t | x_{t-1}, \theta)$ have the form of inverse Gamma distributions. Thus when you multiply these together for all $t$ with the prior you should get an inverse Gamma distribution for the conditional posterior.

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