With tfprobability
, we can compute uncertainty
estimates for keras
layers. This vignette shows how to do
this for dense layers.
Our example will have two types of uncertainty estimated:
To achieve the former, we have our model learn the spread in the data; to achieve the latter, we use a variational layer that learns a posterior over the weights. Internally, this layer works by minimizing the evidence lower bound (ELBO), thus striving to find an approximative posterior that does two things:
As users, we get to specify the form of the posterior as well as that of the prior. But first let’s generate some data.
library(tensorflow)
# assume it's version 1.14, with eager not yet being the default
tf$compat$v1$enable_v2_behavior()
library(tfprobability)
library(keras)
library(dplyr)
library(tidyr)
library(ggplot2)
# generate the data
x_min <- -40
x_max <- 60
n <- 150
w0 <- 0.125
b0 <- 5
normalize <- function(x) (x - x_min) / (x_max - x_min)
# training data; predictor
x <- x_min + (x_max - x_min) * runif(n) %>% as.matrix()
# training data; target
eps <- rnorm(n) * (3 * (0.25 + (normalize(x)) ^ 2))
y <- (w0 * x * (1 + sin(x)) + b0) + eps
# test data (predictor)
x_test <- seq(x_min, x_max, length.out = n) %>% as.matrix()
Here is a simple, trainable prior (in empirical
Bayesian spirit: A normal distribution where the network may learn
the mean (but not the scale). Alternatively, we could disallow learning
from the data by setting trainable
to
FALSE
.
prior_trainable <-
function(kernel_size,
bias_size = 0,
dtype = NULL) {
n <- kernel_size + bias_size
keras_model_sequential() %>%
layer_variable(n, dtype = dtype, trainable = TRUE) %>%
layer_distribution_lambda(function(t) {
tfd_independent(tfd_normal(loc = t, scale = 1),
reinterpreted_batch_ndims = 1)
})
}
The posterior then is a normal, too:
posterior_mean_field <-
function(kernel_size,
bias_size = 0,
dtype = NULL) {
n <- kernel_size + bias_size
c <- log(expm1(1))
keras_model_sequential(list(
layer_variable(shape = 2 * n, dtype = dtype),
layer_distribution_lambda(
make_distribution_fn = function(t) {
tfd_independent(tfd_normal(
loc = t[1:n],
scale = 1e-5 + tf$nn$softplus(c + t[(n + 1):(2 * n)])
), reinterpreted_batch_ndims = 1)
}
)
))
}
Now for the main model. The variational-dense layer is defined to
have two units, one for the distribution of means and distribution of
scales each. layer_distribution_lambda
then takes their
respective outputs as the mean and scale of the posterior
distribution.
model <- keras_model_sequential() %>%
layer_dense_variational(
units = 2,
make_posterior_fn = posterior_mean_field,
make_prior_fn = prior_trainable,
# scale by the size of the dataset
kl_weight = 1 / n
) %>%
layer_distribution_lambda(function(x)
tfd_normal(loc = x[, 1, drop = FALSE],
scale = 1e-3 + tf$math$softplus(0.01 * x[, 2, drop = FALSE])
)
)
The model is then simply trained to minimize the negative log
likelihood, and fitted like a normal keras
network.
negloglik <- function(y, model) - (model %>% tfd_log_prob(y))
model %>% compile(optimizer = optimizer_adam(0.01), loss = negloglik)
model %>% fit(x, y, epochs = 1000)
Because of the uncertainty in the weights, this model does not predict one line, but an ensemble of lines. Each of these lines has its own opinion about the spread in the data. Here is a way we could display this – each colored line is the mean of a distribution, surrounded by a confidence band indicating +/- two standard deviations.
# each time we ask the model to predict, we get a different line
yhats <- purrr::map(1:100, function(x) model(tf$constant(x_test)))
means <-
purrr::map(yhats, purrr::compose(as.matrix, tfd_mean)) %>% abind::abind()
sds <-
purrr::map(yhats, purrr::compose(as.matrix, tfd_stddev)) %>% abind::abind()
means_gathered <- data.frame(cbind(x_test, means)) %>%
gather(key = run, value = mean_val,-X1)
sds_gathered <- data.frame(cbind(x_test, sds)) %>%
gather(key = run, value = sd_val,-X1)
lines <-
means_gathered %>% inner_join(sds_gathered, by = c("X1", "run"))
mean <- apply(means, 1, mean)
ggplot(data.frame(x = x, y = y, mean = as.numeric(mean)), aes(x, y)) +
geom_point() +
theme(legend.position = "none") +
geom_line(aes(x = x_test, y = mean), color = "violet", size = 1.5) +
geom_line(
data = lines,
aes(x = X1, y = mean_val, color = run),
alpha = 0.6,
size = 0.5
) +
geom_ribbon(
data = lines,
aes(
x = X1,
ymin = mean_val - 2 * sd_val,
ymax = mean_val + 2 * sd_val,
group = run
),
alpha = 0.05,
fill = "grey",
inherit.aes = FALSE
)
Summing up, using layer_dense_variational
we are able to
construct a posterior predictive distribution from an ensemble of
models, where each single model by itself learns the spread in the data.
For some more background narrative on this topic, see Adding
uncertainty estimates to Keras models with
tfprobability.