it's well known that the scale mixture of normal distributions is equivalent to a Student t model, that is $$ t_{(v)}(x|\mu,\sigma^2)=\int_0^\infty N(x|\mu,\sigma^2/\lambda)\times G(\lambda|v/2,v/2)d\lambda. $$
a full Bayesian analysis is completed by assigning prior distributions for $\mu$ and $\sigma^2$, say $$ \mu\sim N(\mu_0,\sigma_0^2)\mbox{ and } \sigma^2\sim G(\alpha,\beta), $$ where $\mu_0,\sigma_0^2,\alpha,\beta, v$ known.
Considering a random sample $(x_1,...x_n)$, modeled according to the following models:
Model I: $(x_1,...x_n)|\mu,\sigma^2\sim t_{(v)}(\mu,\sigma^2)$
$\mu\sim N(\mu_0,\sigma_0^2)\mbox{ and } \sigma^2\sim G(\alpha,\beta)$
Model II: $(x_1,...x_n)|\mu,\sigma^2,\lambda\sim N(\mu,\sigma^2/\lambda)$
$\lambda\sim G(\lambda|v/2,v/2)$
$\mu\sim N(\mu_0,\sigma_0^2)\mbox{ and } \sigma^2\sim G(\alpha,\beta)$
When I simulate from the posterior distribution from each model, the posterior estimates for $\mu$ and $\sigma^2$ are not the same in the two models. I was expecting to obtain equivalent posterior inferences from the two models. Am I missing something here? thanks.