linearpredict: Create a Linear Extension of a Distribution
In lestat: A Package for Learning Statistics

Description Usage Arguments Details Value Author(s) See Also Examples

Extends the given probability distribution with new variables which are (multivariate) normally distributed with parameters based on the values of the given probability distribution and values given to the function.

1	linearpredict(object, ...)

object

The probability distribution to be extended. Currently, it should be either a (multivariate) normal distribution, or a (multivariate) normal distribution extended with an extra parameter with either a Gamma or an ExpGamma distribution.

...

A second optional argument may be given, which should then be a matrix with the same number of columns as there are normally distributed variables in the input object. If the matrix has only one row or column, it may be given as a vector. The default is a matrix with one row, consisting of only 1's.

A third optional argument may also be given, which is then the precision matrix of the new normally distributed variables. By default, this matrix is the identity. If the input object contains a Gamma-distributed variable, its value is multiplied with the precision matrix. If the input object contains an ExpGamma-distributed variable y, the value e^{-2y} is multiplied with the precision matrix.

The input is either a (multivariate) variable x with a normal distribution, or a joint distribution consisting of a Gamma- or ExpGamma-distributed variable y, and conditionally on this a (multivariate) normally distributed x. The output is a joint distribution for (z,x) or (z,x,y), where the marginal distribution for x or (x,y) is unchanged, while the conditional distribution for z given x or (x,y) is (multivariate) normal. The expectation and precision for this conditional distribution is Xμ and Pτ, respectively. Here, μ is the expectation of x, while X is the optional second argument. The matrix P is the optional third argument, while τ is either equal to y, when y has a Gamma distribution, or equal to e^{-2y}, when y has an ExpGamma distribution.

A multivariate normal, multivariate Normal-Gamma, or multivariate Normal-ExpGamma distribution, depending on the input.

Petter Mostad <mostad@chalmers.se>

contrast

prior <- normalgamma()
full  <- linearpredict(prior, rep(1, 7))
data  <- simulate(normal(), 7)
posterior <- conditional(full, 1:7, data)
plot(posterior)