Hierarchical Exponential Family Model

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
)

Arguments

n: An integer scalar. The size of the posterior sample required.
model: A character string. Abbreviated name for the response-population distribution combination. For a hierarchical normal model see hanova1 (hierarchical one-way analysis of variance (ANOVA)).
data: A numeric matrix. The format depends on model. See Details.
...: Optional further arguments to be passed to ru.
prior: The log-prior for the parameters of the hyperprior distribution. If the user wishes to specify their own prior then prior must be an object returned from a call to set_user_prior. Otherwise, prior is a character scalar giving the name of the required in-built prior. If prior is not supplied then a default prior is used. See Details.
hpars: A numeric vector. Used to set parameters (if any) in an in-built prior.
param: A character scalar. If param = "trans" (the default) then the marginal posterior of hyperparameter vector \(\phi\) is reparameterized in a way designed to improve the efficiency of sampling from this posterior. If param = "original" the original parameterization is used. The former tends to make the optimizations involved in the ratio-of-uniforms algorithm more stable and to increase the probability of acceptance, but at the expense of slower function evaluations.
init: A numeric vector of length 2. Optional initial estimates for the search for the mode of the posterior density of the hyperparameter vector \(\phi\).
nrep: A numeric scalar. If nrep is not NULL then nrep gives the number of replications of the original dataset simulated from the posterior predictive distribution. Each replication is based on one of the samples from the posterior distribution. Therefore, nrep must not be greater than n. In that event nrep is set equal to n.

Value

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.

Details

Conditional on population-specific 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 ratio-of-uniforms parameterization param.

Beta-binomial: 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 2-column 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).

Gamma-Poisson: 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 2-column 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 \(\phi1 = log(\alpha/\beta), \phi2 = log(\beta).\)

References

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/

Examples

############################ Beta-binomial #################################

# ------------------------- Rat tumor data ------------------------------- #

# Default prior, sampling on (rotated) (log(mean), log(alpha + beta)) scale
rat_res <- hef(model = "beta_binom", data = rat)
# \donttest{
# Hyperparameters alpha and beta
plot(rat_res)

# Parameterization used for sampling
plot(rat_res, ru_scale = TRUE)

# }
summary(rat_res)
#> ru bounding box:  
#>                box       vals1       vals2 conv
#> a        1.0000000  0.00000000  0.00000000    0
#> b1minus -0.2382163 -0.40313465 -0.03906169    0
#> b2minus -0.2174510  0.05447431 -0.35297538    0
#> b1plus   0.2231876  0.36718395 -0.06551365    0
#> b2plus   0.2512577  0.05665707  0.44459818    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.5194805
#> 
#> sample summary 
#>      alpha              beta       
#>  Min.   : 0.8536   Min.   : 5.064  
#>  1st Qu.: 1.7943   1st Qu.:10.800  
#>  Median : 2.2262   Median :13.646  
#>  Mean   : 2.4495   Mean   :14.678  
#>  3rd Qu.: 2.8735   3rd Qu.:17.155  
#>  Max.   :13.2346   Max.   :76.543  

# Choose rats with extreme sample probabilities
pops <- c(which.min(rat[, 1] / rat[, 2]), which.max(rat[, 1] / rat[, 2]))
# Population-specific posterior samples: separate plots
plot(rat_res, params = "pop", plot_type = "both", which_pop = pops)

# Population-specific 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")
# \donttest{
plot(rat_res)

plot(rat_res, ru_scale = TRUE)

# }
summary(rat_res)
#> ru bounding box:  
#>                box       vals1      vals2 conv
#> a        1.0000000  0.00000000  0.0000000    0
#> b1minus -1.0464012 -1.85116847  0.8473716    0
#> b2minus -0.7515215  0.06414713 -1.0929862    0
#> b1plus   1.2453108  2.53614928  1.3267447    0
#> b2plus   1.6246189  0.52814754  3.8051940    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.4761905
#> 
#> sample summary 
#>      alpha            beta      
#>  Min.   :0.883   Min.   : 5.32  
#>  1st Qu.:1.761   1st Qu.:10.54  
#>  Median :2.220   Median :13.36  
#>  Mean   :2.388   Mean   :14.27  
#>  3rd Qu.:2.799   3rd Qu.:16.83  
#>  Max.   :7.281   Max.   :44.10  

# 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 ratio-of-uniforms 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)
# \donttest{
plot(rat_res, ru_scale = TRUE)

# }
# This is the estimated location of the posterior mode
rat_res$f_mode
#> [1] -1.785783  2.741549

# User-defined 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)
# \donttest{
plot(rat_res)

# }
summary(rat_res)
#> ru bounding box:  
#>                box       vals1       vals2 conv
#> a        1.0000000  0.00000000  0.00000000    0
#> b1minus -0.2425978 -0.41087097 -0.04439263    0
#> b2minus -0.2145118  0.05190271 -0.34376799    0
#> b1plus   0.2280150  0.37607515 -0.07085000    0
#> b2plus   0.2730012  0.06004980  0.51162996    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.5208333
#> 
#> sample summary 
#>      alpha             beta       
#>  Min.   : 1.034   Min.   : 7.039  
#>  1st Qu.: 2.280   1st Qu.:13.740  
#>  Median : 2.930   Median :17.441  
#>  Mean   : 3.182   Mean   :19.082  
#>  3rd Qu.: 3.746   3rd Qu.:22.565  
#>  Max.   :10.487   Max.   :57.031  

############################ Gamma-Poisson #################################

# ------------------------ Pump failure data ------------------------------ #

pump_res <- hef(model = "gamma_pois", data = pump)
# Hyperparameters alpha and beta
# \donttest{
plot(pump_res)

# }
# Parameterization used for sampling
plot(pump_res, ru_scale = TRUE)

summary(pump_res)
#> ru bounding box:  
#>                box       vals1       vals2 conv
#> a        1.0000000  0.00000000  0.00000000    0
#> b1minus -0.5174980 -0.91869101 -0.06060116    0
#> b2minus -0.5150835  0.15757254 -0.92429417    0
#> b1plus   0.4124640  0.65433383 -0.11046433    0
#> b2plus   0.4224941  0.08788857  0.67847965    0
#> 
#> estimated probability of acceptance:  
#> [1] 0.5136107
#> 
#> sample summary 
#>      alpha             beta       
#>  Min.   :0.2271   Min.   :0.2641  
#>  1st Qu.:0.8036   1st Qu.:1.2776  
#>  Median :1.0741   Median :1.9060  
#>  Mean   :1.1493   Mean   :2.1830  
#>  3rd Qu.:1.4325   3rd Qu.:2.7548  
#>  Max.   :4.8299   Max.   :8.7179  

# 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)