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A continuous spike-and-slab prior mixes a narrow continuous probability distribution around zero with a much wider component. It permits practical effect selection while retaining a continuous posterior density. Its narrow component is not the exact-zero mass of a point-null mixture prior. A latent component indicator aids computation; a wide-component draw can still have small magnitude, so posterior effect thresholds and component membership answer different questions.

Integrating the positive component's normal distribution prior gives its prior predictive distribution:The negative component gives $Y_i\mid -\sim N(0,1)$. Let $f_1(y)={\varphi }_{1+V}(y)$ and $f_0(y)={\varphi }_1(y)$ denote those centred normal densities. The point-null mixture prior assigns component odds $q/(1-q)$, so the posterior odds of the positive component areIts posterior probability is $O_i/(1+O_i)$. Evidence depends on $y_i^2$, so large effects of either sign favour the positive component. This is the marginal distribution before observing $y_i$, not the posterior predictive law for a second observation conditional on it.

Introduce independent component indicators $z_i\mid q\sim \mathrm{Bernoulli}(q)$, and auxiliary slab effects $u_i\sim N(0,V)$, independent of the indicators. Set ${\theta }_i=z_i{u}_i$. Then inactive genes have exactly zero effect, while active genes have the stipulated normal distribution. With $V>0$ supplied as known data and elicited positive beta shapes, one suitable WinBUGS model isThe normal arguments are precisions, not variances. Supply all observed y values, monitor q, theta and optionally z, and assess Markov chain Monte Carlo convergence diagnostics. The proper auxiliary prior remains defined for inactive genes, where $u_i$ is not informed by the likelihood. This represents the exact point-null mixture prior, rather than replacing its point mass by a narrow continuous spike. Conditional on the indicators, $q\mid z$ has beta distribution $\mathrm{Beta}(a_q+{\sum }_i{z}_i,b_q+N-{\sum }_i{z}_i)$, providing a useful check on the update.