Hierarchical models of any complexity may be specified using
tfd_joint_distribution_sequential(). As hinted at by that
function’s name, it builds a representation of a joint distribution
where every component may optionally depend on components declared
before it.
The model is then fitted to data using some form of Monte Carlo algorithm – Hamiltonian Monte Carlo (HMC), in most cases. Supplementing Monte Carlo methods is an implementation of Variational Inference (VI), but we don’t cover VI in this document.
We illustrate the process by example, using the reedfrogs
dataset from Richard McElreath’s rethinking package. Each
row in the dataset describes one tadpole tank, with its initial count of
inhabitants (density) and number of survivors
(surv).
# assume it's version 1.14, with eager not yet being the default
library(tensorflow)
tf$enable_v2_behavior()
library(tfprobability)
library(rethinking)
library(zeallot)
library(purrr)
data("reedfrogs")
d <- reedfrogs
str(d)'data.frame': 48 obs. of 5 variables:
$ density : int 10 10 10 10 10 10 10 10 10 10 ...
$ pred : Factor w/ 2 levels "no","pred": 1 1 1 1 1 1 1 1 2 2 ...
$ size : Factor w/ 2 levels "big","small": 1 1 1 1 2 2 2 2 1 1 ...
$ surv : int 9 10 7 10 9 9 10 9 4 9 ...
$ propsurv: num 0.9 1 0.7 1 0.9 0.9 1 0.9 0.4 0.9 ...We port to tfprobability the partially-pooled model
presented in McElreath’s book. With partial pooling, each tank gets its
own probability of survival.
In the model specification, we list the global priors first; then comes the intermediate layer yielding the per-tank priors; finally we have the likelihood which in this case is a binomial:
n_tadpole_tanks <- nrow(d)
n_surviving <- d$surv
n_start <- d$density
model <- tfd_joint_distribution_sequential(
list(
# a_bar, the prior for the mean of the normal distribution of per-tank logits
tfd_normal(loc = 0, scale = 1.5),
# sigma, the prior for the variance of the normal distribution of per-tank logits
tfd_exponential(rate = 1),
# normal distribution of per-tank logits
# parameters sigma and a_bar refer to the outputs of the above two distributions
function(sigma, a_bar)
tfd_sample_distribution(
tfd_normal(loc = a_bar, scale = sigma),
sample_shape = list(n_tadpole_tanks)
),
# binomial distribution of survival counts
# parameter l refers to the output of the normal distribution immediately above
function(l)
tfd_independent(
tfd_binomial(total_count = n_start, logits = l),
reinterpreted_batch_ndims = 1
)
)
)Our model technically being a distribution, we can verify it conforms to our expectations by sampling from it:
s <- model %>% tfd_sample(2)
s[[1]]
tf.Tensor([2.1276963 0.26374984], shape=(2,), dtype=float32)
[[2]]
tf.Tensor([1.0527238 2.0026767], shape=(2,), dtype=float32)
[[3]]
tf.Tensor(
[[ 5.3084397e-01 4.1868687e-03 6.5364146e-01 2.2994227e+00
...
2.0958326e+00 8.9087760e-01 1.6273866e+00 2.7854009e+00]
[-5.5288523e-01 1.0414324e+00 -1.3420627e-01 2.5128570e+00
...
-6.6325682e-01 3.0505228e+00 8.1649482e-01 1.0340663e+00]], shape=(2, 48), dtype=float32)
[[4]]
tf.Tensor(
[[ 7. 6. 7. 10. 10. 8. 10. 9. 7. 10. 9. 10. 10. 7. 9. 10. 22. 25.
17. 22. 17. 19. 21. 22. 19. 19. 19. 25. 23. 25. 23. 15. 32. 33. 32. 34.
35. 34. 28. 33. 33. 32. 26. 31. 33. 30. 31. 33.]
[ 2. 8. 4. 10. 6. 1. 8. 3. 7. 9. 1. 0. 5. 10. 4. 5. 2. 21.
1. 14. 4. 14. 9. 6. 12. 0. 20. 19. 1. 15. 15. 7. 30. 7. 12. 4.
23. 3. 16. 34. 35. 5. 14. 10. 20. 32. 19. 24.]], shape=(2, 48), dtype=float32)Another useful correctness check is that it yields a scalar log likelihood:
model %>% tfd_log_prob(s)tf.Tensor([-149.4476 -193.44107], shape=(2,), dtype=float32)`
Besides the model, we need to specify the loss, which here is just the joint log likelihood of the parameters and the target variable:
logprob <- function(a, s, l)
model %>% tfd_log_prob(list(a, s, l, n_surviving))Now we can set up HMC sampling, making use of
mcmc_simple_step_size_adaptation for dynamic step size
evolution based on a desired acceptance probability.
# number of steps after burnin
n_steps <- 500
# number of chains
n_chain <- 4
# number of burnin steps
n_burnin <- 500
hmc <- mcmc_hamiltonian_monte_carlo(
target_log_prob_fn = logprob,
num_leapfrog_steps = 3,
# one step size for each parameter
step_size = list(0.1, 0.1, 0.1),
) %>%
mcmc_simple_step_size_adaptation(target_accept_prob = 0.8,
num_adaptation_steps = n_burnin)The actual sampling should run on the TensorFlow graph for
performance. So if we’re executing in eager mode, we wrap the call in
tf_function:
# initial values to start the sampler
c(initial_a, initial_s, initial_logits, .) %<-% (model %>% tfd_sample(n_chain))
# optionally retrieve metadata such as acceptance ratio and step size
trace_fn <- function(state, pkr) {
list(pkr$inner_results$is_accepted,
pkr$inner_results$accepted_results$step_size)
}
run_mcmc <- function(kernel) {
kernel %>% mcmc_sample_chain(
num_results = n_steps,
num_burnin_steps = n_burnin,
current_state = list(initial_a, tf$ones_like(initial_s), initial_logits),
trace_fn = trace_fn
)
}
run_mcmc <- tf_function(run_mcmc)
res <- run_mcmc(hmc)Now res$all_states contains the samples from the four
chains, while res$trace has the diagnostic output.
mcmc_trace <- res$all_statesIn our example, we have three levels of learned parameters (the two “hyperpriors” and the per-tank prior), so the samples come as a list of three. For each distribution, the first dimension reflects the number of samples per chain, the second, the number of chains and the third, the number of parameters in the chain.
map(mcmc_trace, ~ compose(dim, as.array)(.x))[[1]]
[1] 500 4
[[2]]
[1] 500 4
[[3]]
[1] 500 4 48We can obtain the rhat value, as well as the effective
sample size, using mcmc_potential_scale_reduction and
mcmc_effective_sample_size, respectively:
mcmc_potential_scale_reduction(mcmc_trace)
mcmc_effective_sample_size(mcmc_trace)These again are returned as lists of three.
Rounding up on diagnostic output, we may inspect individual
acceptance in res$trace[[1]] and step sizes in
res$trace[[2]].
For ways to plot the samples and create summary output, as well as some background narrative, see Tadpoles on TensorFlow: Hierarchical partial pooling with tfprobability and its follow-up, Hierarchical partial pooling, continued: Varying slopes models with TensorFlow Probability on the TensorFlow for R blog.