# How can we make bernoulli samples when implementing MCMC for SSVS?

I'm implementing Stochastic Search Variable Selection(SSVS) for logistic regression using R. I'm using Spike and Slab prior with indicators $$\lambda_{j}$$. Here's my model structure:
$$Y ∼ Bernoulli(\mu(X))$$
$$log {μ(X)\over1−μ(X)} = X\beta$$
$$\beta_{j} \overset{ind}{∼}\lambda_{j}N(0,\sigma_1^2)+(1−λj)N(0,\sigma_2^2) \ for \ j=1,···,10$$

$$\lambda_{j} \overset{ind}{∼} Bernoulli(1/2)$$
$$\sigma_1^2 ∼ InverseGamma(1, 20), \ \sigma_2^2 ∼ Gamma(1, 20)$$

And I will make $$T$$ MCMC samples of $$\beta_j, \sigma_1^2, \sigma_2^2, \lambda_j$$. When $${1\over T}\sum_{i=1}^T \lambda_{j,i} > 0.5$$, predictor $$X_j$$ is selected.

But how can I sample $$\lambda$$? It's Bernoulli random variable, so its value is either 0 or 1. But for Metropolis - Hastings proposal, I need $$q(\lambda'|\lambda)$$, which is correlated with last sample. How can I correlate proposal of $$\lambda'$$ with current $$\lambda$$? Any help should be appreciated.

You have correctly identified that specifying $$\lambda_i \sim \text{Bernoulli}(1/2)$$ means that there's really no parameter and no updating here, i.e. there will always be 50% probability for the first mixture component and 50% for the second one.
One solution is to have an extra parameter e.g. like this: $$\lambda_i \sim \text{Bernoulli}(\pi)$$ with some prior (e.g. $$\text{Beta}(0.5, 0.5)$$ or something else) for $$\pi$$.
Note that applying the criterion $$\frac{1}{T} \sum_{i=1}^T \lambda_{i,j} > 0.5$$ is pretty arbitrary and the coefficients you estimate would be very different if you re-fit after selecting those covariates you select on this basis. It's a problem of post-selection inference, which tends to be problematic, if done in such a way that the variable selection is ignored.
By the way, you have a number of options in R (and other programming languages), where you would not have to code up your own sampler. E.g. the brms R package (that uses Stan in the background) implements has its own spike-and-slab prior implementation that has fat-tails via t-distributions (the horseshoe or regularized horseshoe prior - see the documentation). The link also offers further links to some of the papers on this that describe why this particular prior may be expected to have some good properties.