I'm trying to follow the math of Estimating Heston's and Bates’ models parameters using Markov chain Monte Carlo simulation in Journal of Statistical Computation and Simulation, but I'm having trouble in understanding how they find a specific posterior distribution
We assign a prior distribution for $B_t$ of $B_t$ ~ Bernoilli($\lambda_D$)
We then get that the posterior is proportional to $\propto \exp(-\frac 12 [\frac {\Omega+\psi^2}{\Omega} (\frac {Y_t-\bar\mu\Delta t+\frac12 V_{t-1}\Delta t-Z_tB_t}{\sqrt {V_{t-1}\Delta t}})^2-\frac {2\psi}{\Omega}(\frac {V_t-\kappa\theta\Delta t-(1-\kappa\Delta t)V_{t-1}}{\sqrt{V_{t-1}\Delta t}})(\frac {Y_t-\bar\mu\Delta t+\frac 12 V_{t-1}\Delta t-Z_tB_t}{\sqrt{V_{t-1}\Delta t}})+\frac 1\Omega (\frac {V_t-\kappa\theta\Delta t-(1-\kappa\Delta t)V_{t-1}}{\sqrt{V_{t-1}\Delta t}})^2])*(\lambda_D)^{B_t}(1-\lambda_D)^{1-B_t}$
$\propto\exp(-\frac 12 [\frac {\Omega+\psi^2}{\Omega}(\frac {Z_t^2B_t^2-2(Y_t-\bar\mu\Delta t+\frac 12 V_{t-1}\Delta t)Z_tB_t}{V_{t-1}\Delta t})-\frac {2\psi}{\Omega}(\frac {V_t-\kappa\theta\Delta t-(1-\kappa\Delta t)V_{t-1}}{\sqrt{V_{t-1}\Delta t}})(\frac {-Z_tB_t}{\sqrt{V_{t-1}\Delta t}})])*(\frac {\lambda_D}{1-\lambda_D})^{B_t}$
$\propto \exp (-\frac 12 AB_t)*(\frac {\lambda_D}{1-\lambda_D})^{B_t}\propto(\exp(-\frac 12 A)*\frac {\lambda_D}{1-\lambda_D})^{B_t}\propto (\frac {P^*}{1-P^*})^{B_t}$ [1]
So, the posterior distibution of $B_t$ is Bernoillu with probability
$P^*=\frac {\lambda_D\exp(-\frac 12 A)/(1-\lambda_D)}{1+\lambda_D\exp(-\frac 12 A)/(1-\lambda_D)}=\frac 1{(1-\lambda_D)\exp(\frac 12 A)/\lambda_D+1}$
Where
$A=\frac {(\Omega+\psi^2)(Z_t^2-2Z_t(Y_t-\bar\mu\Delta t+\frac 12 V_{t-1}\Delta t))+2\psi(V_t-\kappa\theta\Delta t-(1-\kappa\Delta t)V_{t-1})Z_t}{\Omega V_{t-1}\Delta t}$
So my question is, how do you go from [1] to knowing it's a bernoulli distribution with probability $P^*$