Package 'NeuralEstimators' reference manual

Title:	Likelihood-Free Parameter Estimation using Neural Networks
Description:	An 'R' interface to the 'Julia' package 'NeuralEstimators.jl'. The package facilitates the user-friendly development of neural point estimators, which are neural networks that map data to a point summary of the posterior distribution. These estimators are likelihood-free and amortised, in the sense that, after an initial setup cost, inference from observed data can be made in a fraction of the time required by conventional approaches; see Sainsbury-Dale, Zammit-Mangion, and Huser (2024) <doi:10.1080/00031305.2023.2249522> for further details and an accessible introduction. The package also enables the construction of neural networks that approximate the likelihood-to-evidence ratio in an amortised manner, allowing one to perform inference based on the likelihood function or the entire posterior distribution; see Zammit-Mangion, Sainsbury-Dale, and Huser (2024, Sec. 5.2) <doi:10.48550/arXiv.2404.12484>, and the references therein. The package accommodates any model for which simulation is feasible by allowing the user to implicitly define their model through simulated data.
Authors:	Matthew Sainsbury-Dale [aut, cre]
Maintainer:	Matthew Sainsbury-Dale <[email protected]>
License:	GPL (>= 2)
Version:	0.1.1
Built:	2024-11-04 03:31:19 UTC
Source:	CRAN

NeuralEstimators: Likelihood-Free Parameter Estimation using Neural Networks

Description

An 'R' interface to the 'Julia' package 'NeuralEstimators.jl'. The package facilitates the user-friendly development of neural point estimators, which are neural networks that map data to a point summary of the posterior distribution. These estimators are likelihood-free and amortised, in the sense that, after an initial setup cost, inference from observed data can be made in a fraction of the time required by conventional approaches; see Sainsbury-Dale, Zammit-Mangion, and Huser (2024) doi:10.1080/00031305.2023.2249522 for further details and an accessible introduction. The package also enables the construction of neural networks that approximate the likelihood-to-evidence ratio in an amortised manner, allowing one to perform inference based on the likelihood function or the entire posterior distribution; see Zammit-Mangion, Sainsbury-Dale, and Huser (2024, Sec. 5.2) doi:10.48550/arXiv.2404.12484, and the references therein. The package accommodates any model for which simulation is feasible by allowing the user to implicitly define their model through simulated data.

Author(s)

Maintainer: Matthew Sainsbury-Dale [email protected]

assess a neural estimator

Description

assess a neural estimator

Usage

assess(
  estimators,
  parameters,
  Z,
  estimator_names = NULL,
  parameter_names = NULL,
  use_gpu = TRUE,
  verbose = TRUE
)
assess(
  estimators,
  parameters,
  Z,
  estimator_names = NULL,
  parameter_names = NULL,
  use_gpu = TRUE,
  verbose = TRUE
)

Arguments

`estimators`	a list of (neural) estimators
`parameters`	true parameters, stored as a pxK matrix, where p is the number of parameters in the statistical model and K is the number of sampled parameter vectors
`Z`	data simulated conditionally on the `parameters`. If `Z` contains more data sets than parameter vectors, the parameter matrix will be recycled by horizontal concatenation.
`estimator_names`	list of names of the estimators (sensible defaults provided)
`parameter_names`	list of names of the parameters (sensible defaults provided)
`use_gpu`	a boolean indicating whether to use the GPU if it is available (default true)
`verbose`	a boolean indicating whether information should be printed to the console

Value

a list of two data frames: runtimes, contains the total time taken for each estimator, while estimates is a long-form data frame with columns:

"estimator"; the name of the estimator
"parameter"; the name of the parameter
"truth"; the true value of the parameter
"estimate"; the estimated value of the parameter
"m"; the sample size (number of iid replicates)
"k"; the index of the parameter vector in the test set
"j"; the index of the data set

computes a Monte Carlo approximation of an estimator's bias

Description

computes a Monte Carlo approximation of an estimator's bias

Usage

bias(assessment, ...)
bias(assessment, ...)

Arguments

`assessment`	an object returned by `assess()` (or the `estimates` data frame of this object)
`...`	optional arguments inherited from `risk` (excluding the argument `loss`)

Value

a dataframe giving the estimated bias

bootstrap

Description

Compute bootstrap estimates from a neural estimator

Usage

bootstrap(estimator, Z, B = 400, blocks = NULL, use_gpu = TRUE)
bootstrap(estimator, Z, B = 400, blocks = NULL, use_gpu = TRUE)

Arguments

`estimator`	a neural estimator
`Z`	either a list of data sets simulated conditionally on the fitted parameters (parametric bootstrap); or a single observed data set containing independent replicates, which will be sampled with replacement `B` (non-parametric bootstrap)
`B`	number of non-parametric bootstrap estimates (default 400)
`blocks`	integer vector specifying the blocks in non-parameteric bootstrap (default `NULL`). For example, with 5 replicates, the first two corresponding to block 1 and the remaining three corresponding to block 2, `blocks` should be `c(1,1,2,2,2)`. The bootstrap sampling algorithm aims to produce bootstrap data sets that are of a similar size to `Z`, but this can only be achieved exactly if all blocks are equal in length.
`use_gpu`	a boolean indicating whether to use the GPU if it is available (default `TRUE`)

Value

p × B matrix, where p is the number of parameters in the model and B is the number of bootstrap samples

encodedata

Description

for data Z with missing (NA) entries, returns an augmented data set (U, W) where W encodes the missingness pattern as an indicator vector and U is the original data Z with missing entries replaced by a fixed constant c.

The indicator vector W is stored in the second-to-last dimension of Z, which should be singleton. If the second-to-last dimension is not singleton, then two singleton dimensions will be added to the array, and W will be stored in the new second-to-last dimension.

Usage

encodedata(Z, c = 0)
encodedata(Z, c = 0)

Arguments

`Z`	data containing `NA` entries
`c`	fixed constant with which to replace `NA` entries

Value

Augmented data set (U, W). If Z is provided as a list, the return type will be a JuliaProxy object; these objects can be indexed in the usual manner (e.g., using [[), or converted to an R object using juliaGet() (note however that juliaGet() can be slow for large data sets).

Examples

## Not run: 
library("NeuralEstimators")
Z <- matrix(c(1, 2, NA, NA, 5, 6, 7, NA, 9), nrow = 3)
encodedata(Z)
encodedata(list(Z, Z))
## End(Not run)
## Not run: 
library("NeuralEstimators")
Z <- matrix(c(1, 2, NA, NA, 5, 6, 7, NA, 9), nrow = 3)
encodedata(Z)
encodedata(list(Z, Z))
## End(Not run)

estimate

Description

estimate parameters from observed data using a neural estimator

Usage

estimate(estimator, Z, theta = NULL, use_gpu = TRUE)
estimate(estimator, Z, theta = NULL, use_gpu = TRUE)

Arguments

`estimator`	a neural estimator
`Z`	data; format should be amenable to the architecture of `estimator`
`theta`	parameter vectors (only for neural estimators that take both the data and parameters as input, e.g., neural ratio estimators)
`use_gpu`	a boolean indicating whether to use the GPU if it is available (default true)

Value

a matrix of parameter estimates (i.e., estimator applied to Z)

Examples

## Not run: 
library("NeuralEstimators")
library("JuliaConnectoR")

## Observed data: 100 replicates of a univariate random variable
Z = matrix(rnorm(100), nrow = 1)

## Construct an (un-trained) point estimator
estimator <- initialise_estimator(1, architecture = "MLP")

## Apply the estimator
estimate(estimator, Z)
## End(Not run)
## Not run: 
library("NeuralEstimators")
library("JuliaConnectoR")

## Observed data: 100 replicates of a univariate random variable
Z = matrix(rnorm(100), nrow = 1)

## Construct an (un-trained) point estimator
estimator <- initialise_estimator(1, architecture = "MLP")

## Apply the estimator
estimate(estimator, Z)
## End(Not run)

Initialise a neural estimator

Description

Helper function for initialising a neural estimator.

The estimator is couched in the DeepSets framework so that it can be applied to data with an arbitrary number of independent replicates (including the special case of a single replicate).

Usage

initialise_estimator(
  p,
  architecture,
  d = 1,
  estimator_type = "point",
  depth = 3,
  width = 32,
  activation = "relu",
  activation_output = "identity",
  variance_stabiliser = NULL,
  kernel_size = NULL,
  weight_by_distance = TRUE,
  probs = c(0.025, 0.975)
)
initialise_estimator(
  p,
  architecture,
  d = 1,
  estimator_type = "point",
  depth = 3,
  width = 32,
  activation = "relu",
  activation_output = "identity",
  variance_stabiliser = NULL,
  kernel_size = NULL,
  weight_by_distance = TRUE,
  probs = c(0.025, 0.975)
)

Arguments

`p`	number of unknown parameters in the statistical model
`architecture`	a string: for unstructured data, one may use a fully-connected MLP ("MLP"); for data collected over a grid, a convolutional neural network ("CNN"); and for graphical or irregular spatial data, a graphical neural network ("GNN").
`d`	for unstructured multivariate data (i.e., when `architecture = "MLP"`), the dimension of the data (e.g., `d = 3` for trivariate data); otherwise, if `architecture` is `"CNN"` or `"GNN"`, the argument `d` controls the number of input channels (e.g., `d = 1` for univariate spatial processes).
`estimator_type`	the type of estimator; either "point" or "interval".
`depth`	the number of hidden layers. Either a single integer or an integer vector of length two specifying the depth of inner (summary) and outer (inference) network of the DeepSets framework. Since there is an input and an output layer, the total number of layers is `sum(depth) + 2`.
`width`	a single integer or an integer vector of length `sum(depth)` specifying the width (or number of convolutional filters/channels) in each layer.
`activation`	the (non-linear) activation function of each hidden layer. Accepts a string of Julia code (default `"relu"`).
`activation_output`	the activation function of the output layer layer. Accepts a string of Julia code (default `"identity"`).
`variance_stabiliser`	a function that will be applied directly to the input, usually to stabilise the variance.: a string ('log' for the natural logarithm, or 'cbrt' for the cube-root function), or a string of Julia code that will be converted to a Julia function using `juliaEval()`.
`kernel_size`	(applicable only to CNNs) a list of length `depth[1]` containing lists of integers of length D, where D is the dimension of the convolution (e.g., D = 2 for two-dimensional convolution).
`weight_by_distance`	(applicable only to GNNs) flag indicating whether the estimator will weight by spatial distance; if true (default), a `WeightedGraphConv` layer is used in the propagation module; otherwise, a regular `GraphConv` layer is used.
`probs`	(applicable only if `estimator_type = "interval"`) probability levels defining the lower and upper endpoints of the posterior credible interval.

Value

the initialised neural estimator, a JuliaProxy object

Examples

## Not run: 
library("NeuralEstimators")
p = 2
initialise_estimator(p, architecture = "MLP")
initialise_estimator(p, architecture = "GNN")

## 1D convolution              
initialise_estimator(p, architecture = "CNN", kernel_size = list(10, 5, 3))

## 2D convolution
initialise_estimator(p, architecture = "CNN", 
                     kernel_size = list(c(10, 10), c(5, 5), c(3, 3)))
## End(Not run)
## Not run: 
library("NeuralEstimators")
p = 2
initialise_estimator(p, architecture = "MLP")
initialise_estimator(p, architecture = "GNN")

## 1D convolution              
initialise_estimator(p, architecture = "CNN", kernel_size = list(10, 5, 3))

## 2D convolution
initialise_estimator(p, architecture = "CNN", 
                     kernel_size = list(c(10, 10), c(5, 5), c(3, 3)))
## End(Not run)

load a saved state of a neural estimator

Description

load a saved state of a neural estimator

Usage

loadstate(estimator, filename)
loadstate(estimator, filename)

Arguments

`estimator`	the neural estimator that we wish to load the state into
`filename`	file (including absolute path) of the neural-network state in a `bson` file

Value

estimator updated with the saved state

load a collection of saved weights of a neural estimator

Description

load a collection of saved weights of a neural estimator

Usage

loadweights(estimator, filename)
loadweights(estimator, filename)

Arguments

`estimator`	the neural estimator that we wish to load weights into
`filename`	file (including absolute path) of the neural-network weights saved as a `bson` file

Value

estimator updated with the saved weights

Maximum a posteriori estimation

Description

Given data Z, a neural likelihood-to-evidence-ratio estimator, and a prior, computes the implied approximate maximum a posteriori (MAP) estimate

If a vector theta0 of initial parameter estimates is given, the approximate posterior density is maximised by gradient descent. Otherwise, if a matrix of parameters theta_grid is given, the approximate posterior density is maximised by grid search.

Usage

mapestimate(
  estimator,
  Z,
  prior = NULL,
  theta_grid = NULL,
  theta0 = NULL,
  use_gpu = TRUE
)
mapestimate(
  estimator,
  Z,
  prior = NULL,
  theta_grid = NULL,
  theta0 = NULL,
  use_gpu = TRUE
)

Arguments

`estimator`	a neural likelihood-to-evidence-ratio estimator
`Z`	data; it's format should be amenable to the architecture of `estimator`
`prior`	the prior (default uniform), specified as a Julia or R function
`theta_grid`	a (fine) gridding of the parameter space, given as a matrix with p rows, where p is the number of parameters in the model
`theta0`	a vector of initial parameter estimates
`use_gpu`	a boolean indicating whether to use the GPU if it is available (default true)

Value

a p × K matrix of MAP estimates, where p is the number of parameters in the statistical model and K is the number of data sets provided in Z

Maximum likelihood estimation

Description

Given data Z and a neural likelihood-to-evidence-ratio estimator, computes the implied approximate maximum-likelihood estimate

If a vector theta0 of initial parameter estimates is given, the approximate likelihood is maximised by gradient descent. Otherwise, if a matrix of parameters theta_grid is given, the approximate likelihood is maximised by grid search.

Usage

mlestimate(estimator, Z, theta_grid = NULL, theta0 = NULL, use_gpu = TRUE)
mlestimate(estimator, Z, theta_grid = NULL, theta0 = NULL, use_gpu = TRUE)

Arguments

`estimator`	a neural likelihood-to-evidence-ratio estimator
`Z`	data; it's format should be amenable to the architecture of `estimator`
`theta_grid`	a (fine) gridding of the parameter space, given as a matrix with p rows, where p is the number of parameters in the model
`theta0`	a vector of initial parameter estimates
`use_gpu`	a boolean indicating whether to use the GPU if it is available (default true)

Value

a p × K matrix of maximum-likelihood estimates, where p is the number of parameters in the statistical model and K is the number of data sets provided in Z

Plot the empirical sampling distribution of an estimator.

Description

Plot the empirical sampling distribution of an estimator.

Usage

plotdistribution(
  df,
  type = c("box", "density", "scatter"),
  parameter_labels = NULL,
  estimator_labels = ggplot2::waiver(),
  truth_colour = "black",
  truth_size = 8,
  truth_line_size = NULL,
  pairs = FALSE,
  upper_triangle_plots = NULL,
  legend = TRUE,
  return_list = FALSE,
  flip = FALSE
)
plotdistribution(
  df,
  type = c("box", "density", "scatter"),
  parameter_labels = NULL,
  estimator_labels = ggplot2::waiver(),
  truth_colour = "black",
  truth_size = 8,
  truth_line_size = NULL,
  pairs = FALSE,
  upper_triangle_plots = NULL,
  legend = TRUE,
  return_list = FALSE,
  flip = FALSE
)

Arguments

`df`	a long form data frame containing fields `estimator`, `parameter`, `estimate`, `truth`, and a column (e.g., `replicate`) to uniquely identify each observation.
`type`	string indicating whether to plot kernel density estimates for each individual parameter (`type = "density"`) or scatter plots for all parameter pairs (`type = "scatter"`).
`parameter_labels`	a named vector containing parameter labels.
`estimator_labels`	a named vector containing estimator labels.
`truth_colour`	the colour used to denote the true parameter value.
`truth_size`	the size of the point used to denote the true parameter value (applicable only for `type = "scatter"`).
`truth_line_size`	the size of the cross-hairs used to denote the true parameter value. If `NULL` (default), the cross-hairs are not plotted. (applicable only for `type = "scatter"`).
`pairs`	logical; should we combine the scatter plots into a single pairs plot (applicable only for `type = "scatter"`)?
`upper_triangle_plots`	an optional list of plots to include in the uppertriangle of the pairs plot.
`legend`	Flag; should we include the legend (only applies when constructing a pairs plot)
`return_list`	Flag; should the parameters be split into a list?
`flip`	Flag; should the boxplots be "flipped" using `coord_flip()` (default `FALSE`)?

Value

a list of 'ggplot' objects or, if pairs = TRUE, a single 'ggplot'.

Examples

## Not run: 
# In the following, we have two estimators and, for each parameter, 50 estimates
# from each estimator.

estimators <- c("Estimator 1", "Estimator 2")
estimator_labels <- c("Estimator 1" = expression(hat(theta)[1]("·")),
                      "Estimator 2" = expression(hat(theta)[2]("·")))

# Single parameter:
df <- data.frame(
  estimator = estimators, truth = 0, parameter = "mu",
  estimate  = rnorm(2*50),
  replicate = rep(1:50, each = 2)
)
parameter_labels <- c("mu" = expression(mu))
plotdistribution(df)
plotdistribution(df, type = "density")
plotdistribution(df, parameter_labels = parameter_labels, estimator_labels = estimator_labels)

# Two parameters:
df <- rbind(df, data.frame(
  estimator = estimators, truth = 1, parameter = "sigma",
  estimate  = rgamma(2*50, shape = 1, rate = 1),
  replicate = rep(1:50, each = 2)
))
parameter_labels <- c(parameter_labels, "sigma" = expression(sigma))
plotdistribution(df, parameter_labels = parameter_labels)
plotdistribution(df, parameter_labels = parameter_labels, type = "density")
plotdistribution(df, parameter_labels = parameter_labels, type = "scatter")

# Three parameters:
df <- rbind(df, data.frame(
  estimator = estimators, truth = 0.25, parameter = "alpha",
  estimate  = 0.5 * runif(2*50),
  replicate = rep(1:50, each = 2)
))
parameter_labels <- c(parameter_labels, "alpha" = expression(alpha))
plotdistribution(df, parameter_labels = parameter_labels)
plotdistribution(df, parameter_labels = parameter_labels, type = "density")
plotdistribution(df, parameter_labels = parameter_labels, type = "scatter")
plotdistribution(df, parameter_labels = parameter_labels, type = "scatter", pairs = TRUE)

# Pairs plot with user-specified plots in the upper triangle:
upper_triangle_plots <- lapply(1:3, function(i) {
  x = rnorm(10)
  y = rnorm(10)
  shape = sample(c("Class 1", "Class 2"), 10, replace = TRUE)
  ggplot() +
    geom_point(aes(x = x, y = y, shape = shape)) + 
    labs(shape = "") +
    theme_bw()
})
plotdistribution(
    df, 
    parameter_labels = parameter_labels, estimator_labels = estimator_labels,
    type = "scatter", pairs = TRUE, upper_triangle_plots = upper_triangle_plots
    )
## End(Not run)
## Not run: 
# In the following, we have two estimators and, for each parameter, 50 estimates
# from each estimator.

estimators <- c("Estimator 1", "Estimator 2")
estimator_labels <- c("Estimator 1" = expression(hat(theta)[1]("·")),
                      "Estimator 2" = expression(hat(theta)[2]("·")))

# Single parameter:
df <- data.frame(
  estimator = estimators, truth = 0, parameter = "mu",
  estimate  = rnorm(2*50),
  replicate = rep(1:50, each = 2)
)
parameter_labels <- c("mu" = expression(mu))
plotdistribution(df)
plotdistribution(df, type = "density")
plotdistribution(df, parameter_labels = parameter_labels, estimator_labels = estimator_labels)

# Two parameters:
df <- rbind(df, data.frame(
  estimator = estimators, truth = 1, parameter = "sigma",
  estimate  = rgamma(2*50, shape = 1, rate = 1),
  replicate = rep(1:50, each = 2)
))
parameter_labels <- c(parameter_labels, "sigma" = expression(sigma))
plotdistribution(df, parameter_labels = parameter_labels)
plotdistribution(df, parameter_labels = parameter_labels, type = "density")
plotdistribution(df, parameter_labels = parameter_labels, type = "scatter")

# Three parameters:
df <- rbind(df, data.frame(
  estimator = estimators, truth = 0.25, parameter = "alpha",
  estimate  = 0.5 * runif(2*50),
  replicate = rep(1:50, each = 2)
))
parameter_labels <- c(parameter_labels, "alpha" = expression(alpha))
plotdistribution(df, parameter_labels = parameter_labels)
plotdistribution(df, parameter_labels = parameter_labels, type = "density")
plotdistribution(df, parameter_labels = parameter_labels, type = "scatter")
plotdistribution(df, parameter_labels = parameter_labels, type = "scatter", pairs = TRUE)

# Pairs plot with user-specified plots in the upper triangle:
upper_triangle_plots <- lapply(1:3, function(i) {
  x = rnorm(10)
  y = rnorm(10)
  shape = sample(c("Class 1", "Class 2"), 10, replace = TRUE)
  ggplot() +
    geom_point(aes(x = x, y = y, shape = shape)) + 
    labs(shape = "") +
    theme_bw()
})
plotdistribution(
    df, 
    parameter_labels = parameter_labels, estimator_labels = estimator_labels,
    type = "scatter", pairs = TRUE, upper_triangle_plots = upper_triangle_plots
    )
## End(Not run)

Plot estimates vs. true values.

Description

Plot estimates vs. true values.

Usage

plotestimates(
  df,
  estimator_labels = ggplot2::waiver(),
  parameter_labels = NULL
)
plotestimates(
  df,
  estimator_labels = ggplot2::waiver(),
  parameter_labels = NULL
)

Arguments

`df`	a long form data frame containing fields `estimator`, `parameter`, `estimate`, and `truth`.
`estimator_labels`	a named vector containing estimator labels.
`parameter_labels`	a named vector containing parameter labels.

Value

a 'ggplot' of the estimates for each parameter against the true value.

Examples

## Not run: 
K <- 50
df <- data.frame(
  estimator = c("Estimator 1", "Estimator 2"), 
  parameter = rep(c("mu", "sigma"), each = K),
  truth = 1:(2*K), 
  estimate = 1:(2*K) + rnorm(4*K)
)
estimator_labels <- c("Estimator 1" = expression(hat(theta)[1]("·")),
                      "Estimator 2" = expression(hat(theta)[2]("·")))
parameter_labels <- c("mu" = expression(mu), "sigma" = expression(sigma))

plotestimates(df,  parameter_labels = parameter_labels, estimator_labels)
## End(Not run)
## Not run: 
K <- 50
df <- data.frame(
  estimator = c("Estimator 1", "Estimator 2"), 
  parameter = rep(c("mu", "sigma"), each = K),
  truth = 1:(2*K), 
  estimate = 1:(2*K) + rnorm(4*K)
)
estimator_labels <- c("Estimator 1" = expression(hat(theta)[1]("·")),
                      "Estimator 2" = expression(hat(theta)[2]("·")))
parameter_labels <- c("mu" = expression(mu), "sigma" = expression(sigma))

plotestimates(df,  parameter_labels = parameter_labels, estimator_labels)
## End(Not run)

computes a Monte Carlo approximation of an estimator's Bayes risk

Description

computes a Monte Carlo approximation of an estimator's Bayes risk

Usage

risk(
  assessment,
  loss = function(x, y) abs(x - y),
  average_over_parameters = FALSE,
  average_over_sample_sizes = TRUE
)
risk(
  assessment,
  loss = function(x, y) abs(x - y),
  average_over_parameters = FALSE,
  average_over_sample_sizes = TRUE
)

Arguments

`assessment`	an object returned by `assess()` (or the `estimates` data frame of this object)
`loss`	a binary operator defining the loss function (default absolute-error loss)
`average_over_parameters`	if `TRUE`, the loss is averaged over all parameters; otherwise (default), the loss is averaged over each parameter separately
`average_over_sample_sizes`	if `TRUE` (default), the loss is averaged over all sample sizes (the column `m` in `df`); otherwise, the loss is averaged over each sample size separately

Value

a dataframe giving an estimate of the Bayes risk

computes a Monte Carlo approximation of an estimator's root-mean-square error (RMSE)

Description

computes a Monte Carlo approximation of an estimator's root-mean-square error (RMSE)

Usage

rmse(assessment, ...)
rmse(assessment, ...)

Arguments

`assessment`	an object returned by `assess()` (or the `estimates` data frame of this object)
`...`	optional arguments inherited from `risk` (excluding the argument `loss`)

Value

a dataframe giving the estimated RMSE

sampleposterior

Description

Given data Z, a neural likelihood-to-evidence-ratio estimator, and a prior, draws samples from the implied approximate posterior distribution

Currently, the sampling algorithm is based on a fine-gridding theta_grid of the parameter space. The approximate posterior density is evaluated over this grid, which is then used to draw samples. This is very effective when making inference with a small number of parameters. For models with a large number of parameters, other sampling algorithms may be needed (please feel free to contact the package maintainer for discussion).

Usage

sampleposterior(
  estimator,
  Z,
  theta_grid,
  N = 1000,
  prior = NULL,
  use_gpu = TRUE
)
sampleposterior(
  estimator,
  Z,
  theta_grid,
  N = 1000,
  prior = NULL,
  use_gpu = TRUE
)

Arguments

`estimator`	a neural likelihood-to-evidence-ratio estimator
`Z`	data; it's format should be amenable to the architecture of `estimator`
`theta_grid`	a (fine) gridding of the parameter space, given as a matrix with p rows, where p is the number of parameters in the model
`N`	number of samples to draw (default 1000)
`prior`	the prior (default uniform), specified as a Julia or R function
`use_gpu`	a boolean indicating whether to use the GPU if it is available (default true)

Value

a p × N matrix of posterior samples, where p is the number of parameters in the model. If multiple data sets are given in Z, a list of posterior samples will be returned

save the state of a neural estimator

Description

save the state of a neural estimator

Usage

savestate(estimator, filename)
savestate(estimator, filename)

Arguments

`estimator`	the neural estimator that we wish to save
`filename`	file in which to save the neural-network state as a `bson` file

Value

No return value, called for side effects

tanhloss

Description

For k > 0, defines Julia code that defines the loss function,

$L(\hat{\theta}, \theta) = \tanh\left(\frac{|\hat{\theta} - \theta|}{k}\right),$

which approximates the 0-1 loss as k tends to zero.

The resulting string is intended to be used in the function train, but can also be converted to a callable function using juliaEval.

Usage

tanhloss(k)
tanhloss(k)

Arguments

`k`	Positive numeric value that controls the smoothness of the approximation.

Value

String defining the tanh loss function in Julia code.

Train a neural estimator

Description

The function caters for different variants of "on-the-fly" simulation. Specifically, a sampler can be provided to continuously sample new parameter vectors from the prior, and a simulator can be provided to continuously simulate new data conditional on the parameters. If provided with specific sets of parameters (theta_train and theta_val) and/or data (Z_train and Z_val), they will be held fixed during training.

Note that using R functions to perform "on-the-fly" simulation requires the user to have installed the Julia package RCall.

Usage

train(
  estimator,
  sampler = NULL,
  simulator = NULL,
  theta_train = NULL,
  theta_val = NULL,
  Z_train = NULL,
  Z_val = NULL,
  m = NULL,
  M = NULL,
  K = 10000,
  xi = NULL,
  loss = "absolute-error",
  learning_rate = 1e-04,
  epochs = 100,
  batchsize = 32,
  savepath = "",
  stopping_epochs = 5,
  epochs_per_Z_refresh = 1,
  epochs_per_theta_refresh = 1,
  simulate_just_in_time = FALSE,
  use_gpu = TRUE,
  verbose = TRUE
)
train(
  estimator,
  sampler = NULL,
  simulator = NULL,
  theta_train = NULL,
  theta_val = NULL,
  Z_train = NULL,
  Z_val = NULL,
  m = NULL,
  M = NULL,
  K = 10000,
  xi = NULL,
  loss = "absolute-error",
  learning_rate = 1e-04,
  epochs = 100,
  batchsize = 32,
  savepath = "",
  stopping_epochs = 5,
  epochs_per_Z_refresh = 1,
  epochs_per_theta_refresh = 1,
  simulate_just_in_time = FALSE,
  use_gpu = TRUE,
  verbose = TRUE
)

Arguments

`estimator`	a neural estimator
`sampler`	a function that takes an integer `K`, samples `K` parameter vectors from the prior, and returns them as a px`K` matrix
`simulator`	a function that takes a px`K` matrix of parameters and an integer `m`, and returns `K` simulated data sets each containing `m` independent replicates
`theta_train`	a set of parameters used for updating the estimator using stochastic gradient descent
`theta_val`	a set of parameters used for monitoring the performance of the estimator during training
`Z_train`	a simulated data set used for updating the estimator using stochastic gradient descent
`Z_val`	a simulated data set used for monitoring the performance of the estimator during training
`m`	vector of sample sizes. If `NULL` (default), a single neural estimator is trained, with the sample size inferred from `Z_val`. If `m` is a vector of integers, a sequence of neural estimators is constructed for each sample size; see the Julia documentation for `trainx()` for further details
`M`	deprecated; use `m`
`K`	the number of parameter vectors sampled in the training set at each epoch; the size of the validation set is set to `K`/5.
`xi`	a list of objects used for data simulation (e.g., distance matrices); if it is provided, the parameter sampler is called as `sampler(K, xi)`.
`loss`	the loss function: a string ('absolute-error' for mean-absolute-error loss or 'squared-error' for mean-squared-error loss), or a string of Julia code defining the loss function. For some classes of estimators (e.g., `QuantileEstimator` and `RatioEstimator`), the loss function does not need to be specified.
`learning_rate`	the learning rate for the optimiser ADAM (default 1e-3)
`epochs`	the number of epochs to train the neural network. An epoch is one complete pass through the entire training data set when doing stochastic gradient descent.
`batchsize`	the batchsize to use when performing stochastic gradient descent, that is, the number of training samples processed between each update of the neural-network parameters.
`savepath`	path to save the trained estimator and other information; if null (default), nothing is saved. Otherwise, the neural-network parameters (i.e., the weights and biases) will be saved during training as `bson` files; the risk function evaluated over the training and validation sets will also be saved, in the first and second columns of `loss_per_epoch.csv`, respectively; the best parameters (as measured by validation risk) will be saved as `best_network.bson`.
`stopping_epochs`	cease training if the risk doesn't improve in this number of epochs (default 5).
`epochs_per_Z_refresh`	integer indicating how often to refresh the training data
`epochs_per_theta_refresh`	integer indicating how often to refresh the training parameters; must be a multiple of `epochs_per_Z_refresh`
`simulate_just_in_time`	flag indicating whether we should simulate "just-in-time", in the sense that only a `batchsize` number of parameter vectors and corresponding data are in memory at a given time
`use_gpu`	a boolean indicating whether to use the GPU if one is available
`verbose`	a boolean indicating whether information, including empirical risk values and timings, should be printed to the console during training.

Value

a trained neural estimator or, if m is a vector, a list of trained neural estimators

Examples

## Not run: 
# Construct a neural Bayes estimator for replicated univariate Gaussian 
# data with unknown mean and standard deviation. 

# Load R and Julia packages
library("NeuralEstimators")
library("JuliaConnectoR")
juliaEval("using NeuralEstimators, Flux, Distributions")

# Define the neural-network architecture
estimator <- juliaEval('
 d = 1    # dimension of each replicate
 p = 2    # number of parameters in the model
 w = 32   # width of each layer
 psi = Chain(Dense(d, w, relu), Dense(w, w, relu))
 phi = Chain(Dense(w, w, relu), Dense(w, p))
 deepset = DeepSet(psi, phi)
 estimator = PointEstimator(deepset)
')

# Sampler from the prior
sampler <- function(K) {
  mu    <- rnorm(K)      # Gaussian prior for the mean
  sigma <- rgamma(K, 1)  # Gamma prior for the standard deviation
  theta <- matrix(c(mu, sigma), byrow = TRUE, ncol = K)
  return(theta)
}

# Data simulator
simulator <- function(theta_set, m) {
  apply(theta_set, 2, function(theta) {
    t(rnorm(m, theta[1], theta[2]))
  }, simplify = FALSE)
}

# Train using fixed parameter and data sets 
theta_train <- sampler(10000)
theta_val   <- sampler(2000)
m <- 30 # number of iid replicates
Z_train <- simulator(theta_train, m)
Z_val   <- simulator(theta_val, m)
estimator <- train(estimator, 
                   theta_train = theta_train, 
                   theta_val = theta_val, 
                   Z_train = Z_train, 
                   Z_val = Z_val)
                   
# Train using simulation on-the-fly (requires Julia package RCall)
estimator <- train(estimator, sampler = sampler, simulator = simulator, m = m)

##### Simulation on-the-fly using Julia functions ####

# Defining the sampler and simulator in Julia can improve computational 
# efficiency by avoiding the overhead of communicating between R and Julia. 
# Julia is also fast (comparable to C) and so it can be useful to define 
# these functions in Julia when they involve for-loops. 

# Parameter sampler
sampler <- juliaEval("
      function sampler(K)
      	mu = rand(Normal(0, 1), K)
      	sigma = rand(Gamma(1), K)
      	theta = hcat(mu, sigma)'
      	return theta
      end")

# Data simulator
simulator <- juliaEval("
      function simulator(theta_matrix, m)
      	Z = [rand(Normal(theta[1], theta[2]), 1, m) for theta in eachcol(theta_matrix)]
      	return Z
      end")

# Train the estimator
estimator <- train(estimator, sampler = sampler, simulator = simulator, m = m)
## End(Not run)
## Not run: 
# Construct a neural Bayes estimator for replicated univariate Gaussian 
# data with unknown mean and standard deviation. 

# Load R and Julia packages
library("NeuralEstimators")
library("JuliaConnectoR")
juliaEval("using NeuralEstimators, Flux, Distributions")

# Define the neural-network architecture
estimator <- juliaEval('
 d = 1    # dimension of each replicate
 p = 2    # number of parameters in the model
 w = 32   # width of each layer
 psi = Chain(Dense(d, w, relu), Dense(w, w, relu))
 phi = Chain(Dense(w, w, relu), Dense(w, p))
 deepset = DeepSet(psi, phi)
 estimator = PointEstimator(deepset)
')

# Sampler from the prior
sampler <- function(K) {
  mu    <- rnorm(K)      # Gaussian prior for the mean
  sigma <- rgamma(K, 1)  # Gamma prior for the standard deviation
  theta <- matrix(c(mu, sigma), byrow = TRUE, ncol = K)
  return(theta)
}

# Data simulator
simulator <- function(theta_set, m) {
  apply(theta_set, 2, function(theta) {
    t(rnorm(m, theta[1], theta[2]))
  }, simplify = FALSE)
}

# Train using fixed parameter and data sets 
theta_train <- sampler(10000)
theta_val   <- sampler(2000)
m <- 30 # number of iid replicates
Z_train <- simulator(theta_train, m)
Z_val   <- simulator(theta_val, m)
estimator <- train(estimator, 
                   theta_train = theta_train, 
                   theta_val = theta_val, 
                   Z_train = Z_train, 
                   Z_val = Z_val)
                   
# Train using simulation on-the-fly (requires Julia package RCall)
estimator <- train(estimator, sampler = sampler, simulator = simulator, m = m)

##### Simulation on-the-fly using Julia functions ####

# Defining the sampler and simulator in Julia can improve computational 
# efficiency by avoiding the overhead of communicating between R and Julia. 
# Julia is also fast (comparable to C) and so it can be useful to define 
# these functions in Julia when they involve for-loops. 

# Parameter sampler
sampler <- juliaEval("
      function sampler(K)
      	mu = rand(Normal(0, 1), K)
      	sigma = rand(Gamma(1), K)
      	theta = hcat(mu, sigma)'
      	return theta
      end")

# Data simulator
simulator <- juliaEval("
      function simulator(theta_matrix, m)
      	Z = [rand(Normal(theta[1], theta[2]), 1, m) for theta in eachcol(theta_matrix)]
      	return Z
      end")

# Train the estimator
estimator <- train(estimator, sampler = sampler, simulator = simulator, m = m)
## End(Not run)

Package 'NeuralEstimators'

Help Index

NeuralEstimators: Likelihood-Free Parameter Estimation using Neural Networks

Description

Author(s)

assess a neural estimator

Description

Usage

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computes a Monte Carlo approximation of an estimator's bias

Description

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bootstrap

Description

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Arguments

Value

encodedata

Description

Usage

Arguments

Value

Examples

estimate

Description

Usage

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Value

Examples

Initialise a neural estimator

Description

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Value

Examples

load a saved state of a neural estimator

Description

Usage

Arguments

Value

load a collection of saved weights of a neural estimator

Description

Usage

Arguments

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See Also

Maximum a posteriori estimation

Description

Usage

Arguments

Value

See Also

Maximum likelihood estimation

Description

Usage

Arguments

Value

See Also

Plot the empirical sampling distribution of an estimator.

Description

Usage

Arguments

Value

Examples

Plot estimates vs. true values.

Description

Usage

Arguments

Value

Examples

computes a Monte Carlo approximation of an estimator's Bayes risk

Description

Usage

Arguments

Value