hef {bang}  R Documentation 
Produces random samples from the posterior distribution of the parameters of certain hierarchical exponential family models.
hef(
n = 1000,
model = c("beta_binom", "gamma_pois"),
data,
...,
prior = "default",
hpars = NULL,
param = c("trans", "original"),
init = NULL,
nrep = NULL
)
n 
An integer scalar. The size of the posterior sample required. 
model 
A character string. Abbreviated name for the
responsepopulation distribution combination.
For a hierarchical normal model see 
data 
A numeric matrix. The format depends on 
... 
Optional further arguments to be passed to

prior 
The logprior for the parameters of the hyperprior
distribution. If the user wishes to specify their own prior then

hpars 
A numeric vector. Used to set parameters (if any) in an inbuilt prior. 
param 
A character scalar.
If 
init 
A numeric vector of length 2. Optional initial estimates
for the search for the mode of the posterior density of the
hyperparameter vector 
nrep 
A numeric scalar. If 
Conditional on populationspecific parameter vectors
\theta
1, ..., \theta
J
the observed response data y
1, ..., y
J within each
population are modelled as random samples from a distribution in an
exponential family. The population parameters \theta
1, ...,
\theta
J
are modelled as random samples from a common
population distribution, chosen to be conditionally conjugate
to the response distribution, with hyperparameter vector
\phi
. Conditionally on
\theta
1, ..., \theta
J
, y
1, ..., y
J
are independent of each other and are independent of \phi
.
A hyperprior is placed on \phi
. The user can either
choose parameter values of a default hyperprior or specify their own
hyperprior using set_user_prior
.
The ru
function in the rust
package is used to draw a random sample
from the marginal posterior of the hyperparameter vector \phi
.
Then, conditional on these values, population parameters are sampled
directly from the conditional posterior density of
\theta
1, ..., \theta
J
given \phi
and the data.
We outline each model
, specify the format of the
data
, give the default (log)priors (up to an additive constant)
and detail the choices of ratioofuniforms parameterization
param
.
Betabinomial: For j = 1, ..., J
,
Yj  pj
are i.i.d binomial(nj, pj)
,
where pj
is the probability of success in group j
and nj
is the number of trials in group j
.
pj
are i.i.d. beta(\alpha, \beta)
, so
and \phi = (\alpha, \beta)
.
data
is a 2column matrix: the numbers of successes in column 1
and the corresponding numbers of trials in column 2.
Priors:
prior = "bda"
(the default):
log \pi(\alpha, \beta) =  2.5 log(\alpha + \beta),
\alpha > 0, \beta > 0.
[See Section 5.3 of Gelman et al. (2014).]
prior = "gamma"
: independent gamma priors on \alpha
and \beta
, i.e.
log \pi(\alpha, \beta) =
(s1  1)log\alpha  r1 \alpha +
(s2  1)log\beta  r2 \beta, \alpha > 0, \beta > 0.
where the respective shape (s1
, s2
) and rate
(r1
, r2
) parameters are specified using
hpars
= (s1, r1, s2, r2)
. The default setting is
hpars = c(1, 0.01, 1, 0.01).
Parameterizations for sampling:
param = "original"
is (\alpha, \beta
),
param = "trans"
(the default) is
\phi1 = logit(\alpha/(\alpha+\beta)) = log(\alpha/\beta),
\phi2 = log(\alpha+\beta)
.
See Section 5.3 of Gelman et al. (2014).
GammaPoisson: For j = 1, ..., J
,
Yj  \lambda
j are i.i.d Poisson(e
j\lambda
j),
where
ej
is the exposure in group j
, based on the
total length of observation time and/or size of the population at
risk of the event of interest and \lambda
j is the mean number
of events per unit of exposure.
\lambda
j are i.i.d. gamma(\alpha, \beta)
, so
\phi = (\alpha, \beta)
.
data
is a 2column matrix: the counts yj
of the numbers of
events in column 1 and the corresponding exposures ej
in column 2.
Priors:
prior = "gamma"
(the default): independent gamma priors
on \alpha
and \beta
, i.e.
log \pi(\alpha, \beta) =
(s1  1)log\alpha  r1 \alpha +
(s2  1)log\beta  r2 \beta, \alpha > 0, \beta > 0.
where the respective shape (s1
, s2
) and rate
(r1
, r2
) parameters are specified using
hpars
= (s1, r1, s2, r2)
. The default setting is
hpars = c(1, 0.01, 1, 0.01).
Parameterizations for sampling:
param = "original"
is (\alpha, \beta
),
param = "trans"
(the default) is
\phi1 = log(\alpha/\beta), \phi2 = log(\beta).
An object (list) of class "hef"
, which has the same
structure as an object of class "ru" returned from ru
.
In particular, the columns of the n
row matrix sim_vals
contain the simulated values of \phi
.
In addition this list contains the arguments model
, data
and prior
detailed above, an n
by J
matrix
theta_sim_vals
: column j
contains the simulated values of
\theta
j
and call
: the matched call to hef
.
If nrep
is not NULL
then this list also contains
data_rep
, a numerical matrix with nrep
columns.
Each column contains a replication of the first column of the original
data data[, 1]
, simulated from the posterior predictive
distribution.
Gelman, A., Carlin, J. B., Stern, H. S. Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014) Bayesian Data Analysis. Chapman & Hall / CRC. http://www.stat.columbia.edu/~gelman/book/
The ru
function in the rust
package for details of the arguments that can be passed to ru
via
hef
.
hanova1
for hierarchical oneway analysis of
variance (ANOVA).
set_user_prior
to set a userdefined prior.
############################ Betabinomial #################################
#  Rat tumor data  #
# Default prior, sampling on (rotated) (log(mean), log(alpha + beta)) scale
rat_res < hef(model = "beta_binom", data = rat)
# Hyperparameters alpha and beta
plot(rat_res)
# Parameterization used for sampling
plot(rat_res, ru_scale = TRUE)
summary(rat_res)
# Choose rats with extreme sample probabilities
pops < c(which.min(rat[, 1] / rat[, 2]), which.max(rat[, 1] / rat[, 2]))
# Populationspecific posterior samples: separate plots
plot(rat_res, params = "pop", plot_type = "both", which_pop = pops)
# Populationspecific posterior samples: one plot
plot(rat_res, params = "pop", plot_type = "dens", which_pop = pops,
one_plot = TRUE, add_legend = TRUE)
# Default prior, sampling on (rotated) (alpha, beta) scale
rat_res < hef(model = "beta_binom", data = rat, param = "original")
plot(rat_res)
plot(rat_res, ru_scale = TRUE)
summary(rat_res)
# To produce a plot akin to Figure 5.3 of Gelman et al. (2014) we
# (a) Use the same prior for (alpha, beta)
# (b) Don't use axis rotation (rotate = FALSE)
# (c) Plot on the scale used for ratioofuniforms sampling (ru_scale = TRUE)
# (d) Note that the mode is relocated to (0, 0) in the plot
rat_res < hef(model = "beta_binom", data = rat, rotate = FALSE)
plot(rat_res, ru_scale = TRUE)
# This is the estimated location of the posterior mode
rat_res$f_mode
# Userdefined prior, passing parameters
# (equivalent to prior = "gamma" with hpars = c(1, 0.01, 1, 0.01))
user_prior < function(x, hpars) {
return(dexp(x[1], hpars[1], log = TRUE) + dexp(x[2], hpars[2], log = TRUE))
}
user_prior_fn < set_user_prior(user_prior, hpars = c(0.01, 0.01))
rat_res < hef(model = "beta_binom", data = rat, prior = user_prior_fn)
plot(rat_res)
summary(rat_res)
############################ GammaPoisson #################################
#  Pump failure data  #
pump_res < hef(model = "gamma_pois", data = pump)
# Hyperparameters alpha and beta
plot(pump_res)
# Parameterization used for sampling
plot(pump_res, ru_scale = TRUE)
summary(pump_res)
# Choose pumps with extreme sample rates
pops < c(which.min(pump[, 1] / pump[, 2]), which.max(pump[, 1] / pump[, 2]))
plot(pump_res, params = "pop", plot_type = "dens", which_pop = pops)