spike.slab.glm.prior {BoomSpikeSlab} | R Documentation |

## Zellner Prior for Glm's.

### Description

A Zellner-style spike and slab prior for generalized linear models. It is intended as a base class for LogitZellnerPrior, PoissonZellnerPrior, and potential future extensions.

### Usage

```
SpikeSlabGlmPrior(
predictors,
weight,
mean.on.natural.scale,
expected.model.size,
prior.information.weight,
diagonal.shrinkage,
optional.coefficient.estimate,
max.flips,
prior.inclusion.probabilities)
SpikeSlabGlmPriorDirect(
coefficient.mean,
coefficient.precision,
prior.inclusion.probabilities = NULL,
expected.model.size = NULL,
max.flips = -1)
```

### Arguments

`predictors` |
The design matrix for the regression problem. No missing data is allowed. |

`weight` |
A vector of length |

`mean.on.natural.scale` |
Used to set the prior mean for the intercept. The mean of the response, expressed on the natural scale. This is logit(p-hat) for logits and log(ybar) for Poissons. |

`expected.model.size` |
A positive number less than |

`prior.information.weight` |
A positive scalar. Number of observations worth of weight that should be given to the prior estimate of beta. |

`diagonal.shrinkage` |
The conditionally Gaussian prior for beta (the "slab") starts with a
precision matrix equal to the information in a single observation.
However, this matrix might not be full rank. The matrix can be made
full rank by averaging with its diagonal. |

`optional.coefficient.estimate` |
If desired, an estimate of the regression coefficients can be supplied. In most cases this will be a difficult parameter to specify. If omitted then a prior mean of zero will be used for all coordinates except the intercept, which will be set to mean(y). |

`max.flips` |
The maximum number of variable inclusion indicators the sampler will attempt to sample each iteration. If negative then all indicators will be sampled. |

`prior.inclusion.probabilities` |
A vector giving the prior
probability of inclusion for each variable. If |

`coefficient.mean` |
The prior mean of the coefficients in the maximal model (with all coefficients included). |

`coefficient.precision` |
The prior precision (inverse variance) of the coefficients in the maximal model (with all coefficients included). |

### Details

A Zellner-style spike and slab prior for generalized linear
models. Denote the vector of coefficients by `\beta`

, and the
vector of inclusion indicators by `\gamma`

. These are linked
by the relationship `\beta_i \ne 0`

if ```
\gamma_i =
1
```

and `\beta_i = 0`

if ```
\gamma_i =
0
```

. The prior is

`\beta | \gamma \sim N(b, V)`

`\gamma \sim B(\pi)`

where `\pi`

is the vector of
`prior.inclusion.probabilities`

, and `b`

is the
`optional.coefficient.estimate`

. Conditional on
`\gamma`

, the prior information matrix is

`V^{-1} = \kappa ((1 - \alpha) x^Twx / n + \alpha diag(x^Twx / n))`

The matrix `x^Twx`

is, for suitable choice of the weight vector
`w`

, the total Fisher information available in the data.
Dividing by `n`

gives the average Fisher information in a single
observation, multiplying by `\kappa`

then results in
`\kappa`

units of "average" information. This matrix is
averaged with its diagonal to ensure positive definiteness.

In the formula above, `\kappa`

is
`prior.information.weight`

, `\alpha`

is
`diagonal.shrinkage`

, and `w`

is a diagonal matrix with all
elements set to ```
prior.success.probability * (1 -
prior.success.probability)
```

. The vector `b`

and the matrix
`V^{-1}`

are both implicitly subscripted by `\gamma`

,
meaning that elements, rows, or columsn corresponding to gamma = 0
should be omitted.

The "Direct" version is intended for situations where the predictors are unavailable, or if the user wants more control over the prior precision matrix.

### Value

Returns an object of class `SpikeSlabGlmPrior`

, which is a list
with data elements encoding the selected prior values.

This object is intended for use as a base class for
`LogitZellnerPrior`

and `PoissonZellnerPrior`

.

### Author(s)

Steven L. Scott

### References

Hugh Chipman, Edward I. George, Robert E. McCulloch, M. Clyde, Dean
P. Foster, Robert A. Stine (2001),
"The Practical Implementation of Bayesian Model Selection"
*Lecture Notes-Monograph Series*, Vol. 38, pp. 65-134.
Institute of Mathematical Statistics.

*BoomSpikeSlab*version 1.2.6 Index]