Private Posterior Sampler

Description

Generates samples from the private posterior using a data augmentation framework.

Usage

dapper_sample(
  data_model = NULL,
  sdp = NULL,
  init_par = NULL,
  seed = NULL,
  niter = 2000,
  warmup = floor(niter/2),
  chains = 1
)

Arguments

Details

Generates samples from the private posterior implied by data_model. The data_model input must by an object of class privacy which is created using the new_privacy() constructor. MCMC chains can be run in parallel using furrr::future_map(). See the furrr package documentation for specifics. Long computations can be monitored with the progressr package.

Value

A dpout object which contains: *chain: a draw_matrix object containing niter - warmpup draws from the private posterior. *accept_prob: a (niter - warmup) row matrix containing acceptance probabilities. Each column corresponds to a parameter.

References

Ju, N., Awan, J. A., Gong, R., & Rao, V. A. (2022). Data Augmentation MCMC for Bayesian Inference from Privatized Data. arXiv. doi:10.48550/ARXIV.2206.00710

See Also

new_privacy()

Examples

#simulate confidential data
#privacy mechanism adds gaussian noise to each observation.
set.seed(1)
n <- 100
eps <- 3
y <- rnorm(n, mean = -2, sd = 1)
sdp <- mean(y) + rnorm(1, 0, 1/eps)

post_f <- function(dmat, theta) {
    x <- c(dmat)
    xbar <- mean(x)
    n <- length(x)
    pr_m <- 0
    pr_s2 <- 4
    ps_s2 <- 1/(1/pr_s2 + n)
    ps_m <- ps_s2 * ((1/pr_s2)*pr_m + n * xbar)
    rnorm(1, mean = ps_m, sd = sqrt(ps_s2))
}
latent_f <- function(theta) {
    matrix(rnorm(100, mean = theta, sd = 1), ncol = 1)
}
st_f <- function(xi, sdp, i) {
    xi
}
priv_f <- function(sdp, sx) {
  sum(dnorm(sdp - sx/n, 0, 1/eps, TRUE))
}
dmod <- new_privacy(post_f = post_f,
  latent_f = latent_f,
  priv_f = priv_f,
  st_f = st_f,
  npar = 1)

out <- dapper_sample(dmod,
                    sdp = sdp,
                    init_par = -2,
                    niter = 500)
summary(out)

# for parallel computing we 'plan' a session
# the code below uses 2 CPU cores for parallel computing
library(furrr)
plan(multisession, workers = 2)
out <- dapper_sample(dmod,
                    sdp = sdp,
                    init_par = -2,
                    niter = 500,
                    chains = 2)

# to go back to sequential computing we use
plan(sequential)

Discrete Laplace Distribution

Description

The probability mass function and random number generator for the discrete Laplacian distribution.

Usage

ddlaplace(x, scale = 1, log = FALSE)

rdlaplace(n, scale = 1)

Arguments

Details

Probability mass function

P[X=x] = \dfrac{e^{1/t} - 1}{e^{1/t} + 1} e^{-|x|/t}.

Value

• ddlaplace() returns a numeric vector representing the probability mass function of the discrete Laplace distribution.

• rdlaplace() returns a numeric vector of random samples from the discrete Laplace distribution.

References

Canonne, C. L., Kamath, G., & Steinke, T. (2020). The Discrete Gaussian for Differential Privacy. arXiv. doi:10.48550/ARXIV.2004.00010

Examples

# mass function
ddlaplace(0)

# mass function is vectorized
ddlaplace(0:10, scale = 5)

# generate random samples
rdlaplace(10)

The Discrete Gaussian Distribution

Description

The probability mass function and random number generator for the discrete Gaussian distribution with mean mu and scale parameter sigma.

Usage

ddnorm(x, mu = 0, sigma = 1, log = FALSE)

rdnorm(n, mu = 0, sigma = 1)

Arguments

Details

Probability mass function

P[X = x] = \dfrac{e^{-(x - \mu)^2/2\sigma^2}}{\sum_{y \in \mathbb{Z}} e^{-(x-\mu)^2/2\sigma^2}}.

Value

• ddnorm() returns a numeric vector representing the probability mass function of the discrete Gaussian distribution.

• rdnorm() returns a numeric vector of random samples from the discrete Gaussian distribution.

References

Canonne, C. L., Kamath, G., & Steinke, T. (2020). The Discrete Gaussian for Differential Privacy. arXiv. doi:10.48550/ARXIV.2004.00010

Examples

# mass function
ddnorm(0)

# mass function is also vectorized
ddnorm(0:10, mu = 0, sigma = 5)

# generate random samples
rdnorm(10)

privacy Object Constructor.

Description

Creates a privacy object to be used as input into dapper_sample().

Usage

new_privacy(
  post_f = NULL,
  latent_f = NULL,
  priv_f = NULL,
  st_f = NULL,
  npar = NULL,
  varnames = NULL
)

Arguments

Details

• post_f() is a function which makes draws from the posterior sampler. It has the syntax post_f(dmat, theta). Here dmat is a numeric matrix representing the confidential database and theta is a numeric vector which serves as the initialization point for a one sample draw. The easiest, bug-free way to construct post_f() is to use a conjugate prior. However, this function can also be constructed by wrapping a MCMC sampler generated from other R packages (e.g. rstan, fmcmc, adaptMCMC).

• priv_f() is a function that represents the log of the privacy mechanism density. This function has the form priv_f(sdp, sx) where sdp and sx are both either a numeric vector or matrix. The arguments must appear in the exact stated order with the same variables names as mentioned. Finally, the return value of priv_f() must be a numeric vector of length one.

• st_f() is a function which calculates a summary statistic. It has the syntax st_f(xi, sdp, i) where the three arguments must appear in the stated order. The role of this function is to represent terms in the definition of record additivity. Here i is an integer, while xi is an numeric vector and sdp is a numeric vector or matrix.

• npar is an integer equal to the dimension of theta.

Value

A S3 object of class privacy.

Plot dpout object.

Description

Plot dpout object.

Usage

## S3 method for class 'dpout'
plot(x, ...)

Arguments

Value

trace plots.

Summarise dpout object.

Description

Summarise dpout object.

Usage

## S3 method for class 'dpout'
summary(object, ...)

Arguments

Value

a summary table of MCMC statistics.