In boettiger-lab/pomdpplus: Planning and Learning in Uncertain Systems

library("sarsop")
library("pomdpplus")
library("ggplot2")
library("tidyr")
library("dplyr")
knitr::opts_chunk$set(cache = TRUE)

Problem definition

## Not run, all parameters read from log instead

states <- 0:21
actions <- states
obs <- states
sigma_g <- 0.1
sigma_m <- sigma_g
reward_fn <- function(x,h) pmin(x,h)
discount <- 0.95

K1 <- function(x, h, r = 1, K = 10){
  s <- pmax(x - h, 0)
  s * exp(r * (1 - s / K) )
}
K2 <- function(x, h, r = 1, K = 15){
  s <- pmax(x - h, 0)
  s * exp(r * (1 - s / K) )
}
models <- lapply(list(K1, K2), function(f) fisheries_matrices(states, actions, obs, reward_fn, f, sigma_g, sigma_m))

## Potentially intensive step
alphas <- sarsop_plus(models, discount, precision = 1)

Alternately, we could import parameters from log

log_dir <- "https://raw.githubusercontent.com/cboettig/pomdp-solutions-library/master/library"
meta <- sarsop::meta_from_log(data.frame(model ="ricker", r = 1, K = c(10, 15)), log_dir)[1:2,]
meta
setup <- meta[1,]
states <- 0:(setup$n_states - 1)
actions <- states
obs <- states
sigma_g <- setup$sigma_g
sigma_m <- setup$sigma_m
reward_fn <- function(x,h) pmin(x,h)
discount <- setup$discount 
models <- models_from_log(meta, reward_fn)
alphas <- alphas_from_log(meta, log_dir)

Planning solution

We can compare results for a different priors over states. For simplicity of interpretation, we assume the model is known to be model 2 (model prior (0,1)) As expected, the policy is much more conservative when prior belief is lower!

low <-  compute_plus_policy(alphas, models, c(0, 1), models[[2]]$observation[,4,1])
ave <-  compute_plus_policy(alphas, models, c(0, 1), models[[2]]$observation[,10,1])
unif <- compute_plus_policy(alphas, models, c(0, 1))
high <- compute_plus_policy(alphas, models, c(0, 1), models[[2]]$observation[,15,1])
df <- dplyr::bind_rows(low, ave, unif, high, .id = "prior")


ggplot(df, aes(states[state], states[state] - actions[policy], col = prior, pch = prior)) + 
  geom_point(alpha = 0.5, size = 3) + 
  geom_line()

Use alphas to compute policy given model priors, for comparison:

compare_policies <- function(alphas, models){
  low <-  compute_plus_policy(alphas, models, c(1, 0))
  unif <- compute_plus_policy(alphas, models, c(1/2, 1/2))
  high <- compute_plus_policy(alphas, models, c(0, 1))
  dplyr::bind_rows(low, unif, high, .id = "prior")
}

df <- compare_policies(alphas, models)
ggplot(df, aes(states[state], states[state] - actions[policy], col = prior, pch = prior)) + 
  geom_point(alpha = 0.5, size = 3) + 
  geom_line()

set.seed(123)
out <- sim_plus(models = models, discount = discount,
                x0 = 20, a0 = 1, Tmax = 100, 
                true_model = models[[2]], 
                alphas = alphas)


out$df %>% 
  dplyr::select(-value) %>% 
  tidyr::gather(variable, stock, -time) %>% 
  ggplot(aes(time, stock, color = variable)) + geom_line()  + geom_point()

Evolution of the belief state:

Tmax <-length(out$state_posterior[,1])
out$state_posterior %>% data.frame(time = 1:Tmax) %>% 
  tidyr::gather(state, probability, -time, factor_key =TRUE) %>% 
  dplyr::mutate(state = as.numeric(state)) %>% 
  ggplot(aes(state, probability, group = time, alpha = time)) + geom_line()

Final 20 belief states continue to move around:

Tmax <-length(out$state_posterior[,1])
out$state_posterior %>% data.frame(time = 1:Tmax) %>% 
  tidyr::gather(state, probability, -time, factor_key =TRUE) %>% 
  dplyr::mutate(state = as.numeric(state)) %>% 
  dplyr::filter(time > 80) %>% 
  ggplot(aes(state, probability, group = time, alpha = time)) + geom_line()

Model posterior converges more quickly to the true model (examining first 15 probabilities already shows model 2 probability nearly 1)

out$model_posterior %>% data.frame(time = 1:Tmax) %>% 
  tidyr::gather(model, probability, -time, factor_key =TRUE) %>% 
  dplyr::filter(time < 50) %>% 
  ggplot(aes(model, probability, group = time, alpha = time)) + geom_point()