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demo-pomdp.py
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demo-pomdp.py
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from memo import memo
import jax
import jax.numpy as np
from icecream import ic
from functools import cache
''' Baby POMDP '''
# https://algorithmsbook.com/files/appendix-f.pdf
S = [0, 1] # hungry, sated
A = [0, 1, 2] # feed, sing, ignore
O = [0, 1] # crying, quiet
B = np.linspace(0, 1, 50) # P(hungry)
@jax.jit
def get_belief(b, s):
return np.array([b, 1 - b])[s]
@jax.jit
def Tr(s, a, s_):
z = np.array([ # P(hungry | s, a)
[0.0, 1.0, 1.0], # if hungry
[0.0, 0.1, 0.1] # if sated
])[s, a]
return np.array([z, 1 - z])[s_]
@jax.jit
def Obs(o, s, a):
z = np.array([ # P(cry | s, a)
[0.8, 0.9, 0.8], # if hungry
[0.1, 0.0, 0.1] # if sated
])[s, a]
return np.array([z, 1 - z])[o]
@jax.jit
def R(s, a):
return (
np.array([-10, 0])[s] +
np.array([-5, -0.5, 0])[a]
)
@cache
@memo
def V[b: B](t):
cast: [alice, env, future_alice]
alice: knows(b)
alice: thinks[
env: knows(b),
env: chooses(s in S, wpp=get_belief(b, s))
]
alice: chooses(a in A, wpp=π[b, a](t))
alice: thinks[
env: knows(a),
env: chooses(s_ in S, wpp=Tr(s, a, s_)),
env: chooses(o in O, wpp=Obs(o, s_, a))
]
return E[ alice[
E[ R(env.s, a) ] + (0.0 if t <= 0 else 0.9 * imagine[
future_alice: observes [env.o] is env.o,
future_alice: chooses(b_ in B, wpp=exp(-100.0 * abs(E[env.s_ == 0] - b_))),
E[ future_alice[ V[b_](t - 1) ] ]
])
] ]
@cache
@memo
def π[b: B, a: A](t):
cast: [alice, env, future_alice]
alice: knows(b)
alice: thinks[
env: knows(b),
env: chooses(s in S, wpp=get_belief(b, s))
]
alice: chooses(
a in A,
to_maximize=(
(E[ R(env.s, a) ] + (0.0 if t <= 0 else 0.9 * imagine[
env: knows(a),
env: chooses(s_ in S, wpp=Tr(s, a, s_)),
env: chooses(o in O, wpp=Obs(o, s_, a)),
future_alice: thinks[
env: knows(a),
env: chooses(s_ in S, wpp=Tr(s, a, s_)),
env: chooses(o in O, wpp=Obs(o, s_, a))
],
future_alice: observes [env.o] is env.o,
future_alice: chooses(b_ in B, wpp=exp(-100.0 * abs(E[env.s_ == 0] - b_))),
E[V[future_alice.b_](t - 1)],
]
)
)
),
)
return E[ alice.a == a ]
@memo
def belief_update[b: B, b_: B, a: A, o: O]():
cast: [alice, env]
alice: knows(b)
alice: knows(a)
alice: thinks[
env: knows(b),
env: chooses(s in S, wpp=get_belief(b, s)),
env: knows(a),
env: chooses(o in O, wpp=Obs(o, s, a))
]
alice: observes [env.o] is o
alice: chooses(b_ in B, wpp=exp(-100.0 * abs(E[env.s == 0] - b_)))
return E[alice.b_ == b_]
from matplotlib import pyplot as plt
plt.figure(figsize=(3, 2))
v = V(10)
p = π(10)
plt.plot(B[p[:, 0] == 1], v[p[:, 0] == 1], label='feed')
plt.plot(B[p[:, 1] == 1], v[p[:, 1] == 1], ':', label='sing')
plt.plot(B[p[:, 2] == 1], v[p[:, 2] == 1], '--', label='ignore')
plt.legend()
plt.xlabel('Belief state, P(hungry)')
plt.ylabel('Long-term reward')
plt.title('Crying baby POMDP solution')
plt.tight_layout()
plt.savefig('../paper/fig/pomdp.pdf')