There are a few accuracy measures available in fabletools which allow you to evaluate the accuracy of intervals and distributions.
For intervals, the winkler() score is available.
For distributions, percentile_score() and CRPS() are available.
Explanations of how winkler() and percentile_score() are computed is available here: https://robjhyndman.com/papers/forecasting_state_of_the_art.pdf
There should be plenty of resources online to learn about continuous ranked probability scores (CRPS()).
Commonly used (and implemented) accuracy measures are organised into lists named interval_accuracy_measures and distribution_accuracy_measures, and I have used these below. However it is also possible to create your own list of accuracy measures to use.
library(tsibble)
library(fable)
library(dplyr)
us_deaths <- as_tsibble(USAccDeaths)
us_deaths %>%
# Withold a test set of one year
filter(index < yearmonth("1978 Jan")) %>%
# Model the training data
model(ETS(value)) %>%
# Forecast the test set
forecast(h = "1 year") %>%
# Compute interval/distribution accuracy
accuracy(us_deaths, measures = c(interval_accuracy_measures, distribution_accuracy_measures))
#> # A tibble: 1 x 5
#> .model .type winkler percentile CRPS
#> <chr> <chr> <dbl> <dbl> <dbl>
#> 1 ETS(value) Test 2036. 91.6 181.
Created on 2020-01-28 by the reprex package (v0.3.0)