5.11 Bayesian approach for parameter estimation

In maximum likelihood estimation, we attempt to find parameters $\{\mathbf{g},\mathbf{s}\}$ that can maximize the marginalized log likelihood $\ell(\mathbf{Y})$.

In Bayesian framework, we aim to determine the posterior distribution of all model parameters ${\mathbf{g},\mathbf{s}},\ldots$: $$\begin{equation} P(\mathbf{g},\mathbf{s},\ldots|\mathbf{Y})=\frac{P(\mathbf{Y}|\mathbf{g},\mathbf{s},\ldots)p(\mathbf{g},\mathbf{s},\ldots)}{P(\mathbf{Y})} \end{equation}$$

Typically, in MCMC programs such as JAGS and nimble, one needs to specify two things:

• priors and
• model likelihood

All parameters have prior distributions

• Item parameters $g$ and $s$ follow $beta(1,1)$, which is a uniform distribution ranging from 0 to 1:

Code

curve(dbeta(x, 1, 1))

• Person’s latent class membership latent.group.index[n] follows a categorical distribution $cat(p)$

• $p$ in the categorical distribution $cat(p)$ follows a dirichlet distribution $dirich(\delta)$

Regarding likelihood, we need to specify the likelihood of observing each item response:

$$\begin{equation} P(Y_{ij}=1|\mathbf{\alpha}_{c})= g_j + (1-s_j-g_j)I(\mathbf{\alpha}_{c}^T\mathbf{q}_j < \mathbf{q}_{j}^T\mathbf{q}_j) \end{equation}$$

Although MCMC is guaranteed to converge to the target distributions of model parameters, one may need a very long MCMC chain to achieve that.