5.12 Bayesian approach for parameter estimation

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

In Bayesian framework, we aim to determine the posterior distribution of all model parameters $g,s,\dots$: $$P(g,s,\dots |Y)=\frac{P(Y|g,s,\dots )p(g,s,\dots )}{P\left(Y\right)}$$

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 $cat\left(p\right)$

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

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

$$P(Y_{ij}=1|{\alpha }_c)=g_j+(1-s_j-g_j)I({\alpha }_c^T{q}_j<q_j^T{q}_j)$$

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