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Specify prior hyperparameters for EM algorithm

Source: R/specify_priors.R
specify_priors.Rd

A function that allows the user to specify the prior hyperparameters for the EM algorithm in a structure accepted by JANE.

Usage

specify_priors(
  D = 2,
  K = 2,
  model,
  family = "bernoulli",
  noise_weights = FALSE,
  n_interior_knots = NULL,
  a,
  b,
  c,
  G,
  nu,
  e,
  f,
  h,
  l,
  e_2,
  f_2,
  m_1,
  o_1,
  m_2,
  o_2
)

Arguments

D

An integer specifying the dimension of the latent positions (default is 2).

K

An integer specifying the total number of clusters (default is 2).

model

A character string specifying the model:

  • 'NDH': undirected network with no degree heterogeneity (or connection strength heterogeneity if working with weighted network)

  • 'RS': undirected network with degree heterogeneity (and connection strength heterogeneity if working with weighted network)

  • 'RSR': directed network with degree heterogeneity (and connection strength heterogeneity if working with weighted network)

family

A character string specifying the distribution of the edge weights.

  • 'bernoulli': for unweighted networks; utilizes a Bernoulli distribution with a logit link (default)

  • 'lognormal': for weighted networks with positive, non-zero, continuous edge weights; utilizes a log-normal distribution with an identity link

  • 'poisson': for weighted networks with edge weights representing non-zero counts; utilizes a zero-truncated Poisson distribution with a log link

noise_weights

A logical; if TRUE then a Hurdle model is used to account for noise weights, if FALSE simply utilizes the supplied network (converted to an unweighted binary network if a weighted network is supplied, i.e., (A > 0.0)*1.0) and fits a latent space cluster model (default is FALSE).

n_interior_knots

An integer specifying the number of interior knots used in fitting a natural cubic spline for degree heterogeneity (and connection strength heterogeneity if working with weighted network) models (i.e., 'RS' and 'RSR' only; default is NULL).

a

A numeric vector of length \(D\) specifying the mean of the multivariate normal prior on \(\mu_k\) for \(k = 1,\ldots,K\), where \(\mu_k\) represents the mean of the multivariate normal distribution for the latent positions of the \(k^{th}\) cluster.

b

A positive numeric scalar specifying the scaling factor on the precision of the multivariate normal prior on \(\mu_k\) for \(k = 1,\ldots,K\), where \(\mu_k\) represents the mean of the multivariate normal distribution for the latent positions of the \(k^{th}\) cluster.

c

A numeric scalar \(\ge\) \(D\) specifying the degrees of freedom of the Wishart prior on \(\Omega_k\) for \(k = 1,\ldots,K\), where \(\Omega_k\) represents the precision of the multivariate normal distribution for the latent positions of the \(k^{th}\) cluster.

G

A numeric \(D \times D\) matrix specifying the inverse of the scale matrix of the Wishart prior on \(\Omega_k\) for \(k = 1,\ldots,K\), where \(\Omega_k\) represents the precision of the multivariate normal distribution for the latent positions of the \(k^{th}\) cluster.

nu

A positive numeric vector of length \(K\) specifying the concentration parameters of the Dirichlet prior on \(p\), where \(p\) represents the mixture weights of the finite multivariate normal mixture distribution for the latent positions.

e

A numeric vector of length 1 + (model =='RS')*(n_interior_knots + 1) + (model =='RSR')*2*(n_interior_knots + 1) specifying the mean of the multivariate normal prior on \(\beta_{LR}\), where \(\beta_{LR}\) represents the coefficients of the logistic regression model.

f

A numeric p.s.d square matrix of dimension 1 + (model =='RS')*(n_interior_knots + 1) + (model =='RSR')*2*(n_interior_knots + 1) specifying the precision of the multivariate normal prior on \(\beta_{LR}\), where \(\beta_{LR}\) represents the coefficients of the logistic regression model.

h

A positive numeric scalar specifying the first shape parameter for the Beta prior on \(q\), where \(q\) is the proportion of non-edges in the "true" underlying network converted to noise edges. Only relevant when noise_weights = TRUE.

l

A positive numeric scalar specifying the second shape parameter for the Beta prior on \(q\), where \(q\) is the proportion of non-edges in the "true" underlying network converted to noise edges. Only relevant when noise_weights = TRUE.

e_2

A numeric vector of length 1 + (model =='RS')*(n_interior_knots + 1) + (model =='RSR')*2*(n_interior_knots + 1) specifying the mean of the multivariate normal prior on \(\beta_{GLM}\), where \(\beta_{GLM}\) represents the coefficients of the zero-truncated Poisson or log-normal GLM. Only relevant when noise_weights = TRUE & family != 'bernoulli'.

f_2

A numeric p.s.d square matrix of dimension 1 + (model =='RS')*(n_interior_knots + 1) + (model =='RSR')*2*(n_interior_knots + 1) specifying the precision of the multivariate normal prior on \(\beta_{GLM}\), where \(\beta_{GLM}\) represents the coefficients of the zero-truncated Poisson or log-normal GLM. Only relevant when noise_weights = TRUE & family != 'bernoulli'.

m_1

A positive numeric scalar specifying the shape parameter for the Gamma prior on \(\tau^2_{weights}\), where \(\tau^2_{weights}\) is the precision (on the log scale) of the log-normal weight distribution. Note, this value is scaled by 0.5, see 'Details'. Only relevant when noise_weights = TRUE & family = 'lognormal'.

o_1

A positive numeric scalar specifying the rate parameter for the Gamma prior on \(\tau^2_{weights}\), where \(\tau^2_{weights}\) is the precision (on the log scale) of the log-normal weight distribution. Note, this value is scaled by 0.5, see 'Details'. Only relevant when noise_weights = TRUE & family = 'lognormal'.

m_2

A positive numeric scalar specifying the shape parameter for the Gamma prior on \(\tau^2_{noise \ weights}\), where \(\tau^2_{noise \ weights}\) is the precision (on the log scale) of the log-normal noise weight distribution. Note, this value is scaled by 0.5, see 'Details'. Only relevant when noise_weights = TRUE & family = 'lognormal'.

o_2

A positive numeric scalar specifying the rate parameter for the Gamma prior on \(\tau^2_{noise \ weights}\), where \(\tau^2_{noise \ weights}\) is the precision (on the log scale) of the log-normal noise weight distribution. Note, this value is scaled by 0.5, see 'Details'. Only relevant when noise_weights = TRUE & family = 'lognormal'.

Value

A list of S3 class "JANE.priors" representing prior hyperparameters for the EM algorithm, in a structure accepted by JANE.

Details

Prior on \(\boldsymbol{\mu}_k\) and \(\boldsymbol{\Omega}_k\) (note: the same prior is used for \(k = 1,\ldots,K\)) :

$$\boldsymbol{\Omega}_k \sim Wishart(c, \boldsymbol{G}^{-1})$$ $$\boldsymbol{\mu}_k | \boldsymbol{\Omega}_k \sim MVN(\boldsymbol{a}, (b\boldsymbol{\Omega}_k)^{-1})$$

Prior on \(\boldsymbol{p}\):

For the current implementation we require that all elements of the nu vector be \(\ge 1\) to prevent against negative mixture weights for empty clusters. $$\boldsymbol{p} \sim Dirichlet(\nu_1 ,\ldots,\nu_K)$$

Prior on \(\boldsymbol{\beta}_{LR}\): $$\boldsymbol{\beta}_{LR} \sim MVN(\boldsymbol{e}, \boldsymbol{F}^{-1})$$

Prior on \(q\): $$q \sim Beta(h, l)$$

Zero-truncated Poisson

Prior on \(\boldsymbol{\beta}_{GLM}\): $$\boldsymbol{\beta}_{GLM} \sim MVN(\boldsymbol{e}_{2}, \boldsymbol{F}_{2}^{-1})$$

Log-normal

Prior on \(\tau^2_{weights}\): $$\tau^2_{weights} \sim Gamma(\frac{m_1}{2}, \frac{o_1}{2})$$

Prior on \(\boldsymbol{\beta}_{GLM}\): $$\boldsymbol{\beta}_{GLM}|\tau^2_{weights} \sim MVN(\boldsymbol{e}_{2}, (\tau^2_{weights}\boldsymbol{F}_{2})^{-1})$$

Prior on \(\tau^2_{noise \ weights}\): $$\tau^2_{noise \ weights} \sim Gamma(\frac{m_2}{2}, \frac{o_2}{2})$$

Unevaluated calls can be supplied as values for specific hyperparameters. This is particularly useful when running JANE for multiple combinations of K and D. See 'examples' section below for implementation examples.

Examples

# \donttest{
# Simulate network
mus <- matrix(c(-1,-1,1,-1,1,1), 
              nrow = 3,
              ncol = 2, 
              byrow = TRUE)
omegas <- array(c(diag(rep(7,2)),
                  diag(rep(7,2)), 
                  diag(rep(7,2))), 
                  dim = c(2,2,3))
p <- rep(1/3, 3)
beta0 <- 1.0
sim_data <- JANE::sim_A(N = 100L, 
                        model = "RS",
                        mus = mus, 
                        omegas = omegas, 
                        p = p, 
                        params_LR = list(beta0 = beta0), 
                        remove_isolates = TRUE)
                        
                        
# Specify prior hyperparameters
D <- 3L
K <- 5L
n_interior_knots <- 5L

a <- rep(1, D)
b <- 3
c <- 4
G <- 10*diag(D)
nu <- rep(2, K)
e <- rep(0.5, 1 + (n_interior_knots + 1))
f <- diag(c(0.1, rep(0.5, n_interior_knots + 1)))

my_prior_hyperparameters <- specify_priors(D = D,
                                           K = K,
                                           model = "RS",
                                           n_interior_knots = n_interior_knots,
                                           a = a,
                                           b = b,
                                           c = c,
                                           G = G,
                                           nu = nu,
                                           e = e,
                                           f = f)
                                           
# Run JANE on simulated data using supplied prior hyperparameters
res <- JANE::JANE(A = sim_data$A,
                  D = D,
                  K = K,
                  initialization = "GNN",
                  model = "RS",
                  case_control = FALSE,
                  DA_type = "none",
                  control = list(priors = my_prior_hyperparameters))

# Specify prior hyperparameters as unevaluated calls
n_interior_knots <- 5L
e <- rep(0.5, 1 + (n_interior_knots + 1))
f <- diag(c(0.1, rep(0.5, n_interior_knots + 1)))

my_prior_hyperparameters <- specify_priors(model = "RS",
                                           n_interior_knots = n_interior_knots,
                                           a = quote(rep(1, D)),
                                           b = b,
                                           c = quote(D + 1),
                                           G = quote(10*diag(D)),
                                           nu = quote(rep(2, K)),
                                           e = e,
                                           f = f)
                                           
# # Run JANE on simulated data using supplied prior hyperparameters (NOT RUN)
# future::plan(future::multisession, workers = 5)
# res <- JANE::JANE(A = sim_data$A,
#                    D = 2:5,
#                    K = 2:10,
#                    initialization = "GNN",
#                    model = "RS",
#                    case_control = FALSE,
#                    DA_type = "none",
#                    control = list(priors = my_prior_hyperparameters))
# future::plan(future::sequential)
                
                                                         
# }