# SuSiE: Sum of Single Effects Model

##### Posted on Jan 22, 2024

writing the sparse vector of regression coefficients as a sum of “single-effect” vectors, each with one non-zero element

a new fitting procedure: iterative Bayesian stepwise selection (IBSS)

a Bayesian analogue of stepwise selection methods

instead of selecting a single variable at each step, IBSS computes a distribution on variables that captures uncertainty in which variable to select

$y = Xb + e$

variables $j$ with non-zero effects as “effect variables”

Assume variables 1 and 4 are two effect variables, but each completely corrleated with another non-effect variable, say $x_1 = x_2$ and $x_3 = x_4$

we may conclude that there are (at least) two effect variables, and that

$(b_1\neq 0 \text{ or } b_2 \neq 0)\text{ and } (b_3\neq 0\text{ or } b_4\neq 0)$

The goal: is to provide methods that directly produce this kind of inferential statement.

two approaches:

• select groups of variables
• Bayesian approach

the marginal posterior inclusion probability (PIP) of each variable

$PIP_j = \Pr(b_j\neq 0\mid X, y)$

level $\rho$ credible set: a subset of variables that has probability $\rho$ or greater of containing at least one effect variable. Equivalently, the probability that all variables in the credible set have zero regression coefficients is $1-\rho$ or less.

primary aim: report as many credible sets as the data support, each with as few variables as possible second goal: prioritize the variables within each credible set, assigning each a probability that reflects the strength of the evidence for that variable being an effect variable.

### Posterior under single-effect regression model

• $\alpha$ is the vector of PIPs
• from $\alpha$, one can compute a level $\rho$ credible set, $CS(\alpha, \rho)$

## Numerical Comparisons

b specified by two parameters:

• S: the number of effect variables
• $\phi$: the proportion of variance in $y$ explained by $X$

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