def calculateProbabilities(markov_table, init_dist):
mc = markov_chain(markov_table, init_dist)
experiments = 500000
steps = 40
visits = 0
for index in range(experiments):
mc.start()
for j in range(steps):
mc.move()
if mc.running_state == 1: visits += 1
probability = visits / experiments
return probability
def mc_est(n: int) -> float:
markov_table = { # Transition Table
0: {
1: .5,
2: .5
}, # from state 0 we move to state 1 with prob 0.5 and to state 2 with 0.5
1: {
0: 1 / 3,
3: 2 / 3
},
2: {
2: 1.
},
3: {
0: .5,
3: .25,
4: .25
},
4: {
4: 1.
}
}
# Initial Distribution
init_dist = {0: 1.} # we start from state 0 with probability 1
mc = markov_chain(markov_table, init_dist)
sample_size = n # Ν
running_total = 0
for i in range(sample_size):
mc.start()
while mc.running_state != 2 and mc.running_state != 4:
mc.move()
running_total += mc.steps # steps it took to be absorbed
mc_estimate = running_total / sample_size
return mc_estimate
States[8]: {States[11]: p, States[12]: q},
States[9]: {States[12]: p, States[16]: q},
States[10]: {States[13]: p, States[14]: q},
States[11]: {States[14]: p, States[15]: q},
States[12]: {States[15]: p, States[16]: q},
States[13]: {States[13]: 1.},
States[14]: {States[13]: p, States[11]: q},
States[15]: {States[11]: p, States[16]: q},
States[16]: {States[16]: 1.}
}
init_probs = {States[0]: 1.0}
# Ok... we are ready know
# Let"s construct a Markov Chain. So let"s call the constructor
mc = markov_chain(markov_table, init_probs)
N = 20000
#print ("N = ",N)
## Experiment parameters
#N = 1000 # number of samples
steps = 0 # the target time
#stepsTrack = []
counter = 0 # to count the number of times the event {X_40 = 1} occurs
## Simulation
for i in range(N):
mc.start() # new experiment
while(mc.running_state not in [States[13],States[16]]):
mc.move()
steps += 1
if mc.running_state == States[13]:
markov_game = {
'0': [('1', p), ('2', 1 - p)],
'1': [('1', p), ('2', 1 - p)],
'2': [('1', p), ('2', 1 - p)]
}
markov_tie = {
'0': [('1', p), ('2', 1 - p)],
'1': [('3', p), ('0', 1 - p)],
'2': [('0', p), ('4', 1 - p)],
'3': [('3', 1)],
'4': [('4', 1)]
}
# Ok... we are ready know
# Let's construct a Markov Chain. So let's call the constructor
# Experiment parameters
N = 100000
counter = 0
for i in xrange(N):
m1 = lib.markov_chain(init_probs, markov_game)
m1.start()
res = game(m1)
if res == 0:
m2 = lib.markov_chain(init_probs, markov_tie)
m2.start()
res = tie(m2)
if res == 1:
counter += 1
# and check if we end up in one of the goal_states
print "So we estimate the Pr to win by", float(counter) / N
('4', 2 / 11.0), ('5', 2 / 11.0), ('6', 2 / 11.0)],
'2': [('1', 2 / 11.0), ('2', 1 / 11.0), ('3', 2 / 11.0),
('4', 2 / 11.0), ('5', 2 / 11.0), ('6', 2 / 11.0)],
'3': [('1', 2 / 11.0), ('2', 2 / 11.0), ('3', 1 / 11.0),
('4', 2 / 11.0), ('5', 2 / 11.0), ('6', 2 / 11.0)],
'4': [('1', 2 / 11.0), ('2', 2 / 11.0), ('3', 2 / 11.0),
('4', 1 / 11.0), ('5', 2 / 11.0), ('6', 2 / 11.0)],
'5': [('1', 2 / 11.0), ('2', 2 / 11.0), ('3', 2 / 11.0),
('4', 2 / 11.0), ('5', 1 / 11.0), ('6', 2 / 11.0)],
'6': [('1', 2 / 11.0), ('2', 2 / 11.0), ('3', 2 / 11.0),
('4', 2 / 11.0), ('5', 2 / 11.0), ('6', 1 / 11.0)],
}
# Ok... we are ready know
# Let's construct a Markov Chain. So let's call the constructor
m = lib.markov_chain(init_probs, markov_table)
# Experiment parameters
N = 100000
steps = int(raw_input())
counter = 0
for i in xrange(N):
# Let's initiate the running state for t=0.
# We must do this every time we want to restart the Chain
m.start()
# Let the markov chain move by 20 steps
for j in xrange(steps):
m.move()
# and check if we end up in one of the goal_states
if m.running_state == '1':
counter += 1
'0-30': {'15-30': p, '0-40': 1 - p},
'40-0': {'GameA': p, '40-15': 1 - p}, '30-15': {'40-15': p, 'Deuce': 1 - p}, # 4th row
'15-30': {'Deuce': p, '15-40': 1 - p}, '0-40': {'15-40': p, 'GameB': 1 - p},
'40-15': {'GameA': p, 'AdvA': 1 - p}, 'Deuce': {'AdvA': p, 'AdvB': 1 - p}, '15-40': {'AdvB': p, 'GameB': 1 - p},
# 5th row
'GameA': {'GameA': 1.}, 'AdvA': {'GameA': p, 'Deuce': 1 - p}, # 6th row
'AdvB': {'Deuce': p, 'GameB': 1 - p}, 'GameB': {'GameB': 1.}
}
initial_dist = {'0-0': 1.} # Every game starts from 0-0
# Markov Chain construction
mc = markov_chain(markov_table, initial_dist)
# Experiment parameters
N: int = 1000 # number of samples
M: int = 500 # Sample size
p_list = [] # List of the phat estimates
# Simulation
for j in range(M):
counter = 0
for i in range(N):
mc.start() # new experiment
while True:
mc.move()
if mc.running_state == 'GameA':
def TennisWinProb(
p: float, n: int
) -> float: # Function to compute probability of PlayerA winning, given
# probability of point p and number of samples n.
markov_table = {
'0-0': {
'15-0': p,
'0-15': 1 - p
}, # 1st row of the pic
'15-0': {
'30-0': p,
'15-15': 1 - p
},
'0-15': {
'15-15': p,
'0-30': 1 - p
}, # 2nd row
'30-0': {
'40-0': p,
'30-15': 1 - p
},
'15-15': {
'30-15': p,
'15-30': 1 - p
}, # 3rd row
'0-30': {
'15-30': p,
'0-40': 1 - p
},
'40-0': {
'GameA': p,
'40-15': 1 - p
},
'30-15': {
'40-15': p,
'Deuce': 1 - p
}, # 4th row
'15-30': {
'Deuce': p,
'15-40': 1 - p
},
'0-40': {
'15-40': p,
'GameB': 1 - p
},
'40-15': {
'GameA': p,
'AdvA': 1 - p
},
'Deuce': {
'AdvA': p,
'AdvB': 1 - p
},
'15-40': {
'AdvB': p,
'GameB': 1 - p
},
# 5th row
'GameA': {
'GameA': 1.
},
'AdvA': {
'GameA': p,
'Deuce': 1 - p
}, # 6th row
'AdvB': {
'Deuce': p,
'GameB': 1 - p
},
'GameB': {
'GameB': 1.
}
}
initial_dist = {'0-0': 1.} # Every game starts from 0-0
# Markov Chain construction
mc = markov_chain(markov_table, initial_dist)
counter = 0
# Simulation
for i in range(n):
mc.start() # new experiment
while True:
mc.move()
if mc.running_state == 'GameA':
counter += 1
break
elif mc.running_state == 'GameB':
break
phat: float = counter / n
return phat