Bayesian Finite Mixture Modelling

MATLAB code for Bayesian Computation of Finite Mixture Models.

SUMMARY

Treatment Response Models (Finite Mixture of 2SLS Models)

Continuous Treatment

$\begin{aligned} D_i &= \gamma_{0,\tilde{c}_i} + \mathbf{Z}_i\mathbf{\gamma}_{z,\tilde{c}_i} + v_i\\ Y_i &= \beta_{0,\tilde{c}_i} + \mathbf{X}_i\mathbf{\beta}_{x,\tilde{c}_i} + \delta_{\tilde{c}_i} D_i + \epsilon_i\\ \end{aligned}$

where

$\begin{aligned} \begin{bmatrix} v_i \\ u_i \end{bmatrix} \bigm\vert \mathbf{Z},\mathbf{X} &\overset{ind}{\sim} \mathcal{N}\left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_{v, \tilde{c}_i}^2 & \sigma_{vu,\tilde{c}_i}\\ \sigma_{vu, \tilde{c}_i} & \sigma_{u,\tilde{c}_i}^2\\ \end{bmatrix} \right) \end{aligned} \equiv \mathcal{N}\left( \mathbf{0}, \mathbf{\Sigma}_{\tilde{c}_i} \right)$

and $\text{Pr}(\tilde{c}_i = g) = \pi_g, \quad \text{for } g = 1,\ldots,G \text{ and } \sum_{g=1}^G \pi_g = 1$.

Binary Treatment

$\begin{aligned} D^*_i &= \gamma_{0,\tilde{c}_i} + \mathbf{Z}_i\mathbf{\gamma}_{z,\tilde{c}_i} + v_i\\ D_i &= \mathbb{1}\{D^*_i \geq 0\}\\ Y_i &= \beta_{0,\tilde{c}_i} + \mathbf{X}_i\mathbf{\beta}_{x,\tilde{c}_i} + \tau_{\tilde{c}_i} D_i + \epsilon_i\\ \end{aligned}$

where

$\begin{aligned} \begin{bmatrix} v_i \\ u_i \end{bmatrix} \bigm\vert \mathbf{Z},\mathbf{X} &\overset{ind}{\sim} \mathcal{N}\left( \begin{bmatrix} 0 \\ 0 \end{bmatrix}, \begin{bmatrix} \sigma_{v, \tilde{c}_i}^2 & \sigma_{vu,\tilde{c}_i}\\ \sigma_{vu, \tilde{c}_i} & \sigma_{u,\tilde{c}_i}^2\\ \end{bmatrix} \right) \end{aligned} \equiv \mathcal{N}\left( \mathbf{0}, \mathbf{\Sigma}_{\tilde{c}_i} \right)$

and $\text{Pr}(\tilde{c}_i = g) = \pi_g, \quad \text{for } g = 1,\ldots,G \text{ and } \sum_{g=1}^G \pi_g = 1$.

USAGE

• Gibbs samplers: MCMCsamplers

• Permutation algorithms to handle label-switching: permutation

• Simulations: SyntheticMonteCarlo