def KMR_Markov_matrix_simultaneous(N, p, epsilon):
"""
Generate the Markov matrix for the KMR model with *simultaneous* move
Parameters
----------
N : int
Number of players
p : float
Level of p-dominance of action 1, i.e.,
the value of p such that action 1 is the BR for (1-q, q) for any q > p,
where q (1-q, resp.) is the prob that the opponent plays action 1 (0, resp.)
epsilon : float
Probability of mutation
Returns
-------
P : numpy.ndarray
Markov matrix for the KMR model with simultaneous move
Notes
-----
For simplicity, the transition probabilities are computed under the assumption
that a player is allowed to be matched to play with himself.
"""
P = np.empty((N+1, N+1), dtype=float)
for n in range(N+1):
P[n, :] = \
(n/N < p) * binom.pmf(range(N+1), N, epsilon/2) + \
(n/N == p) * binom.pmf(range(N+1), N, 1/2) + \
(n/N > p) * binom.pmf(range(N+1), N, 1-epsilon/2)
return P
def nextround(playpos, dround):
thisround = weightmatrix.playpos.filter(round == dround)
nextround = weightmatrix.playpos.filter(round == dround+1)
##Setting picks by round
if dRound % 2 == 0:
m = 8
n = 4
else:
m = 10
n = 5
##Calculating weights based on binomial probability
if dRound % 2 == 0:
pickprob = []
for i in range(m+1):
prob = 0
for l in range(max(0,i-4),min(4,i)+1):
prob = prob + binom.pmf(l,n,thisround)*binom.pmf(i-l,m-n,nextround)
pickprob.append(prob)
else:
pickprob = []
for i in range(m+1):
prob = 0
for l in range(max(0,i-5),min(5,i)+1):
prob = prob + binom.pmf(l,n,thisround)*binom.pmf(i-l,m-n,nextround)
pickprob.append(prob)
##Setting probabilities to be cumulative, i.e. now it will describe the probability that
##that player will still be available by next round
for i in range(1,len(pickprob)):
pickprob[i] = pickprob[i-1] + pickprob[i]
return pickprob
def plotMLE():
"""
Plots a binomial distribution, and another with parameters estimated from samples
from the first distribution.
"""
# Number of samples
NumSamples = 10
# Number of independent experiments
N = 10
# Probability of success
mu = 0.2
# Plot the true binomial distribution
x = xrange(N + 1)
plt.plot(binom.pmf(x, N, mu), label="True Distribution")
# Estimate the success probability from samples
muML = sum(np.random.binomial(N, mu, NumSamples)) * 1.0 / N / NumSamples
# Plot the maximum likelihood distribution
plt.plot(binom.pmf(x, N, muML), label="MLE")
plt.legend()
plt.show()
def KMR_2x2_P_simultaneous(N, p, epsilon):
P = np.empty((N+1, N+1), dtype=float)
for x in range(N+1):
P[x,:] = /
(x/N < p) * binom.pmf(range(N+1), N, epsilon/2) + /
(x/N == p) * binom.pmf(range(N+1), N, 1/2) + /
(x/N > p) * binom.pmf(range(N+1), N, 1-epsilon/2) +/
def set_move(self):
self.X = np.zeros((self.n+1,self.n+1))
if self.mode == 'sequential': # 逐次改訂
expay0 = np.empty(2)
expay1 = np.empty(2)
for k in range(1,self.n):
#直前まで0だった人が選ばれた時の行動選択による期待利得
expay0[0] = self.one_pay[0][0]*(self.n-k-1)/(self.n-1)+self.one_pay[0][1]*k/(self.n-1)
expay0[1] = self.one_pay[1][0]*(self.n-k-1)/(self.n-1)+self.one_pay[1][1]*k/(self.n-1)
#直前まで1だった人が選ばれた時の行動選択による期待利得
expay1[0] = self.one_pay[0][0]*(self.n-k)/(self.n-1)+self.one_pay[0][1]*(k-1)/(self.n-1)
expay1[1] = self.one_pay[1][0]*(self.n-k)/(self.n-1)+self.one_pay[1][1]*(k-1)/(self.n-1)
if expay1[0] > expay1[1]:
self.X[k][k-1]=(k/self.n)*(1-self.epsi*0.5) #k人からk-1人になる確率
self.X[k][k]=(k/self.n)*self.epsi*0.5
elif expay1[0]==expay1[1]:
self.X[k][k-1]=(k/self.n)*0.5
self.X[k][k]=(k/self.n)*0.5
else:
self.X[k][k-1]= (k/self.n)*self.epsi*0.5
self.X[k][k] = (k/self.n)*(1-self.epsi*0.5)
if expay0[1]>expay0[0]:
self.X[k][k+1]=((self.n-k)/self.n)*(1-self.epsi*0.5) #k人からk+1人になる確率
self.X[k][k] += ((self.n-k)/self.n)*self.epsi*0.5 #X[k][k]は上でも定めているので上書きでなく加えている
elif expay0[1]==expay0[0]:
self.X[k][k+1] = ((self.n-k)/self.n)*0.5
self.X[k][k] += ((self.n-k)/self.n)*0.5
else:
self.X[k][k+1] = ((self.n-k)/self.n)*self.epsi*0.5
self.X[k][k] += ((self.n-k)/self.n)*(1-self.epsi*0.5)
self.X[0][0] = 1-self.epsi*0.5
self.X[0][1] = self.epsi*0.5
self.X[self.n][self.n-1] = self.epsi*0.5
self.X[self.n][self.n] = 1-self.epsi*0.5
elif self.mode == 'simultaneous': # 同時改訂
list=[]
for i in range(self.n+1):
list.append(i)
expay = np.empty(2)
for k in range(0,self.n+1):
#各人の行動選択による期待利得
expay[0] = self.one_pay[0][0]*(self.n-k)/(self.n)+self.one_pay[0][1]*k/(self.n)
expay[1] = self.one_pay[1][0]*(self.n-k)/(self.n)+self.one_pay[1][1]*k/(self.n)
if expay[0] > expay[1]:
self.X[k] = binom.pmf(list,self.n,self.epsi)
elif expay[0] == expay[1]:
self.X[k] = binom.pmf(list,self.n,0.5)
else:
self.X[k] = binom.pmf(list,self.n,1-self.epsi)
else:
print 'The mode '+ self.mode+'is Unknown. Check your input.'
def compute_posteriors(c, n, p_H1=0.5, iterations=1000, verbose=True):
p_H2 = 1.0 - p_H1
p_data_given_H1 = binomial_distribution.pmf(c, n, 0.5)
if verbose: print "p(data|H_1) =", p_data_given_H1
# Montecarlo:
low = 0.5
high = 1.0
pi = np.random.uniform(low=0.5, high=1.0, size=iterations)
p_data_given_H2 = (binomial_distribution.pmf(c, n, pi) * 1.0 / (high-low)).mean()
if verbose: print "p(data|H_2) =", p_data_given_H2
p_H1_given_data = (p_data_given_H1 * p_H1) / (p_data_given_H1 * p_H1 + p_data_given_H2 * p_H2)
p_H2_given_data = (p_data_given_H2 * p_H2) / (p_data_given_H1 * p_H1 + p_data_given_H2 * p_H2)
return p_H1_given_data, p_H2_given_data
def calculate(self, option):
if(option.payoffType is PayoffType.call):
for i in range(node_number):
spot_T = spot * (u ** (nreps - i)) * (d ** (i))
call_T += option.payoff(spot_T, strike) * binom.pmf(nreps - i, nreps, pu)
call_price = call_T * np.exp(-rate * expiry)
return call_price
elif(option.payoffType is PayoffType.put):
for i in range(node_number):
spot_T = spot * (u ** (nreps - i)) * (d ** (i))
put_T += option.payoff(spot_T, strike) * binom.pmf(nreps - i, nreps, pd)
put_price = put_T * np.exp(-rate * expiry)
return put_price
else:
raise ValueError("Wrong Payoff Type")
def test_remaining_gain_sample_gtps():
n = len(ground_truth_pairs)
gtps = sorted(gtp_scores.remaining_gain_sample_gtps(max_n=n))
assert len(gtps) == n
# if we draw everything the results should always be everything
assert gtps == sorted(gtp_scores.remaining_gain_sample_gtps(max_n=n))
# if we don't draw everything it's quite unlikely we get the same result
gtps = gtp_scores.remaining_gain_sample_gtps(max_n=5)
assert len(gtps) == 5
assert gtps != gtp_scores.remaining_gain_sample_gtps(max_n=5)
# make sure we never get items that are fully covered already
gtp_scores.gtp_max_precisions[ground_truth_pairs[0]] = 1
c = Counter()
k = 100
n = 128
for i in range(k):
c.update(gtp_scores.remaining_gain_sample_gtps(max_n=n))
assert ground_truth_pairs[0] not in c
assert sum(c.values()) == k * n
# count how many aren't in gtps
c_not_in = 0
for gtp in ground_truth_pairs[1:]:
if gtp not in c:
c_not_in += 1
assert c_not_in < 2, \
"it's very unlikely that 2 gtps weren't in our %d samples, " \
"but %d are not" % (k, c_not_in)
# near end simulation
gtpe_scores = gtp_scores.copy_reset()
# set all scores to 1 --> remaining gains to 0
gtpe_scores.gtp_max_precisions = gtpe_scores.get_remaining_gains()
high_prob, low_prob = random.sample(gtpe_scores.ground_truth_pairs, 2)
# high and low prob refer to the remaining gains and the expected probs to
# be selected by remaining gain samples...
gtpe_scores.gtp_max_precisions[high_prob] = 0.1
gtpe_scores.gtp_max_precisions[low_prob] = 0.9
assert gtpe_scores.remaining_gain == 1
c = Counter()
for i in range(100):
c.update(gtpe_scores.remaining_gain_sample_gtps(max_n=1))
assert len(c) == 2
assert sum(c.values()) == 100
assert (binom.pmf(c[high_prob], 100, .9) > 0.001 and
binom.pmf(c[low_prob], 100, .1) > 0.001), \
'expected that high_prob item is drawn with a 9:1 chance, but got:\n' \
'high: %d, low: %d' % (c[high_prob], c[low_prob])
def triplet_prob(Nw,t,ptriplet,psingle):
weights = np.zeros(Nw+1)
# number of non-singles must be greater than or equal to t
for ns in xrange(t,Nw+1):
weights[ns] = 1.0 - binom.cdf(t-1,ns,ptriplet)
return sum(weights * binom.pmf(np.linspace(0,Nw,Nw+1),Nw,(1.0-psingle)))
def smooth(arr, n):
avgs = empty(len(arr))
pm = binom.pmf(xrange(n * 2), n * 2, 0.5)
for i in xrange(len(avgs)):
subarr = zero_padded(arr, i - n, i + n)
avgs[i] = sum(map(operator.mul, pm, subarr))
return avgs
def E_V(x, u, t, V): wl = list(range(0, n+1)) wpl = np.array(binom.pmf(wl, n, p)) xl = [f(x, u, w) for w in wl] vl = np.array([V[t + 1, xn] for xn in xl]) ev = sum(wpl * vl) return ev
def update_figure(sample_data):
# first sample from current posterior beta distribution
sample_data = json.loads(sample_data)
p = sample_data['sample']
x_binom_pdf = np.arange(0,
10)
y_binom_pdf = binom.pmf(x_binom_pdf, n, p)
trace = go.Scatter(
x=x_binom_pdf,
y=y_binom_pdf
)
layout_posterior = go.Layout(
xaxis={'title': 'X'},
yaxis={'title': 'Prob X', 'range': [0, 1]},
margin={'l': 40, 'b': 40, 't': 10, 'r': 10},
legend={'x': 0, 'y': 1}
)
fig2 = dict(data=[trace], layout=layout_posterior)
return fig2
def get_risk1(adata, pp1, pp2, i, li, turk_uncer, r, crowd_type = 'single', pp3 = None):
"""
Risk after doing action 1 (query the crowd)
with item i and get label li
"""
unlab_risk = get_unl_risk(adata, pp1, [i], li, r = r, pp3 = pp3)
#for item i
if crowd_type == 'all5':
# generate 5 crowd labels from the crowd accuracy model:
# assume li is the true label
# eval the prob that 0,1,2,..5 crowd mems say li
pr = [[0,0],[0,0]]
for i in range(2):
for j in range(2):
pr[i][j] = adata.count[i][j] * 1.0 / (adata.count[i][j] + adata.count[1-i][j])
i_risk = 0
for x in range(6):
p = binom.pmf(x, 5, pr[li][li])
new_labels = [li]*x
p_true = prob_from_crowd_label2(i, adata, pp = None, labels = new_labels)
this_i_risk = p_true[1-li] * r[1-li][li]
i_risk += p * this_i_risk
else:
new_labels = [li]
p_true = prob_from_crowd_label2(i, adata, pp = None, labels = new_labels)
i_risk = p_true[1-li] * r[1-li][li]
#res += r[0][1] * pp[i][1] * (li == 0) \
# + r[1][0] * pp[i][0] * (li == 1)
# for labelled items
#pp = pp2
crowd_risk = get_crowd_risk(adata, pp2, turk_uncer, r = r)
return unlab_risk + i_risk + crowd_risk
def posteriorHiddenMemberProbs(data, params_old):
"""
:type data: Data
:param data:
:type params_old: Paramters
:param params_old:
:return:
"""
n = data.n
m = data.m
memberProbs = softmaxForData(data.Xm, params_old.Alpha, params_old.a, True)
choosingProbs = sigmoidForData(data.Xr, params_old.Beta, params_old.b)
if isinstance(m, int):
if m == 1: # binary response
observedLikelihoods = np.abs((1. - data.y).reshape(n, 1) - choosingProbs)
else:
observedLikelihoods = binom.pmf(data.y.reshape(n, 1),
np.repeat(m, n).reshape(n, 1),
choosingProbs)
else:
raise("Function is not yet implemented for non-integer 'm'.")
jointProbs = np.multiply(memberProbs, observedLikelihoods)
scales = jointProbs.max(axis=1).reshape(n, 1)
scaledJointProbs = jointProbs / scales
return scaledJointProbs / scaledJointProbs.sum(axis=1).reshape(n, 1)
def dbinom(x, size=1, prob=0.5, log=False):
"""
============================================================================
dbinom()
============================================================================
Density Function for the binomial distribution.
Returns the probability of getting "x" successes out of "size" number of
trials, given a probability of "prob" for each success.
USAGE:
dbinom(x, size, prob=0.5, log=False)
pbinom(q, size, prob=0.5, lowertail=True, log=False)
qbinom(p, size, prob=0.5, lowertail=True, log=False)
rbinom(n=1, size=1, prob=0.5)
:param x: int. or array of ints. The number of successes
:param size: int. Number of trials
:param prob: float. Probability of a success
:param log: bool. take the log?
:return:
============================================================================
"""
if log:
# note, scipy flips meaning of n and size
return binom.logpmf(x, n=size, p=prob, loc=0)
else:
# note, scipy flips meaning of n and size
return binom.pmf(x, n=size, p=prob)
def observedLogLikelihood(data, params):
"""
:type data: Data
:param data:
:type params: Paramters
:param params:
:return:
"""
n = data.n
m = data.m
memberProbs = softmaxForData(data.Xm, params.Alpha, params.a, True)
choosingProbs = sigmoidForData(data.Xr, params.Beta, params.b)
if isinstance(m, int):
if m == 1:
observedLikelihoods = np.abs((1. - data.y).reshape(n, 1) - choosingProbs)
else:
observedLikelihoods = binom.pmf(data.y.reshape(n, 1),
np.repeat(m, n).reshape(n, 1),
choosingProbs)
else:
raise("Function is not yet implemented for non-integer 'm'.")
jointProbs = np.multiply(memberProbs, observedLikelihoods)
return np.log(jointProbs.sum(axis=1)).sum()
def binomial_distribution_vectors(n, p, v_size=None):
pmf = np.array([binom.pmf(i, n, p) for i in range(n+1)])
cdf = np.array([binom.cdf(i, n, p) for i in range(n+1)])
if (v_size != None) and (v_size > n+1):
# padding to the right to reach v_size length
pmf = np.append(pmf, np.zeros(v_size-n))
cdf = np.append(cdf, np.ones(v_size-n))
return pmf,cdf
def get_value(x, distr):
if distr['distr_name'] == 'binomial':
if lookup_pmf.has_key(x):
return lookup_pmf[x]
else:
v = binom.pmf(x, distr['n'], distr['p'])
lookup_pmf[x] = v
return v
def likelihood(self,hypo,data):
idx,k = data
N,f = hypo
N = N * ((1-f) ** (idx-1)) #1-based
if N < k:
return 0
val = binom.pmf(k,N,f)
return val
def pdf_integral(p1, data):
# calculate pdens for range of p1
xj, nj, c, p2, var = data
dens = pdf(p1, data=data[2:])
return dens * binom.pmf(xj, nj, p=p1)
def json_prob_anc(self):
"""return the probability this individual is an X ancestor"""
breakpoints = np.arange(1e3) # how many breakpoints there are
prob = 1 - np.sum(poisson.pmf(breakpoints, (self.xnrec-1)*XGENLEN) *
binom.pmf(0, n=breakpoints+1, p=1/2.0**(self.xnrec-1)))
if np.isnan(prob):
prob = 1
all_slots = dict([(s, getattr(self, s)) for s in self.__slots__])
return dict(prob_anc=prob, **all_slots)
def __init__ (self, dice_type, num_dice, use_focus=False, use_target_lock=False):
self.num_dice = num_dice
self.results = [ x for x in range(num_dice+1)]
self.probability_of_success = 0
if dice_type == DiceType.RED:
self.probability_of_success = get_hit_prob(use_focus, use_target_lock)
else:
self.probability_of_success = get_evade_prob(use_focus)
self.pmf = binom.pmf( self.results, num_dice, self.probability_of_success)
def compare_one(col, cons_aa, aln_size, weights, aa_freqs, pseudo_size):
"Column probability using the ball-in-urn model."
col_counts = count_col(col, weights, aa_freqs, pseudo_size)
col_tot = col_counts['S'] + col_counts['T'] + col_counts['Y']
p_j = (aa_freqs['S'] + aa_freqs['T'] + aa_freqs['Y'])
# pvalue = binom.pmf(range(cons_count, aln_size + 1),
# aln_size, p_j).sum()
pvalue = binom.pmf(range(int(math.ceil(col_tot)), aln_size + 2),
aln_size + 1, p_j).sum()
return pvalue
def binomialDistanceProb(delta, alpha, beta, prob):
if alpha - beta == 0:
if delta == 0:
return 1
else:
# This causes a gradient of 0 among "impossible" distance
# values - making gradient ascent impossible. For cases where
# gradient ascent is needed, use gradientBinomialDistanceProb
return 0
return binom.pmf(delta, alpha - beta, prob)
def gradientBinomialDistanceProb(self, delta, alpha, beta, prob):
if alpha - beta == 0:
if delta == 0:
return 1
else:
return self.minimumDistanceProb**(1+delta)
return max(min(
binom.pmf(delta, alpha - beta, prob),
self.maximumDistanceProb),
self.minimumDistanceProb)
def compare_one(col, cons_aa, aln_size, weights, aa_freqs, pseudo_size):
"Column probability using the ball-in-urn model."
# cons_count = col.count(cons_aa)
cons_count = count_col(col, weights)[cons_aa]
p_j = aa_freqs[cons_aa]
cons_count_i = int(ceil(cons_count))
size_i = int(ceil(aln_size))
# pvalue = float(cons_count_i)/len(col)
pvalue = binom.pmf(range(cons_count_i, size_i + 1), size_i, p_j).sum()
return pvalue
def computeLPPD_components(gam,pnull,y,N,x):
'''Returns log-posterior predictive density (LPPD) for extracted gam,pnull from binomial model'''
from scipy.stats import binom
ll = 0
lppd = []
for i in xrange(5):
ll = binom.pmf(y[i], N[i],
logist(x[i]*gam + (1-gam)*log_odds(pnull)))
lppd.append(np.log(np.mean(ll)))
return np.array(lppd)
def mk_matrix1(N, ep, p): #同時改訂
P = np.empty((N+1, N+1))
for i in range(N):
if i/N < p:
pro = ep/2
elif i/N == p:
pro = 1/2
else:
pro = 1-ep/2
P[i] = binom.pmf(range(N+1), N, pro)
return P
def likelihood(p_in):
"""
A function to return the likelihood for a given value of p.
"""
if (testPrior): # Allows us to test the prior, by ignoring the data
return 1
elif (not testPrior): # Returns a likelihood score for actual data
if (p_in >= 0 and p_in <= 1):
return binom.pmf(data,n,p_in)
else:
return 0 # Returns value of 0 (rather than non) if p outside bounds.
def binomial_variance(n, p):
#TODO: should not be here
global alt_var_cnt
n1 = np.floor(n*p)
diff = binom.pmf(np.arange(0, n1), n, p)
s = np.sum(diff * np.power(np.arange(0, n1) - n*p, 2))
old_var = n*p*(1-p)
ret = (old_var - s) / (1- np.sum(diff))
alt_var_cnt += 1
return ret
def plot_expected_prices(
und_vals: List[List[float]],
p: float,
ticker: str,
expiration: str,
external_axes: Optional[List[plt.Axes]] = None,
) -> None:
"""Plot expected prices of the underlying asset at expiration
Parameters
----------
und_vals : List[List[float]]
The expected underlying values at the expiration date
p : float
The probability of the stock price moving upward each round
ticker : str
The ticker of the option's underlying asset
expiration : str
The expiration for the option
external_axes : Optional[List[plt.Axes]], optional
External axes (1 axis is expected in the list), by default None
"""
up_moves = list(range(len(und_vals[-1])))
up_moves.reverse()
probs = [100 * binom.pmf(r, len(up_moves), p) for r in up_moves]
if external_axes is None:
_, ax = plt.subplots(figsize=plot_autoscale(), dpi=cfp.PLOT_DPI)
else:
if len(external_axes) != 1:
logger.error("Expected list of one axis item.")
console.print("[red]Expected list of one axis item./n[/red]")
return
(ax,) = external_axes
ax.set_title(f"Probabilities for ending prices of {ticker} on {expiration}")
ax.xaxis.set_major_formatter("${x:1.2f}")
ax.yaxis.set_major_formatter(mtick.PercentFormatter())
ax.plot(und_vals[-1], probs)
theme.style_primary_axis(ax)
if external_axes is None:
theme.visualize_output()
def euro_binomial_pricer(S, K, r, v, q, T, n, payoff, verbose=True):
nodes = n + 1
h = T / n
u = np.exp((r - q) * h + v * np.sqrt(h))
d = np.exp((r - q) * h - v * np.sqrt(h))
pstar = (np.exp((r - q) * h) - d) / (u - d)
price = 0.0
for i in range(nodes):
prob = binom.pmf(i, n, pstar)
spotT = S * (u**i) * (d**(n - i))
po = payoff(spotT, K)
price += po * prob
if verbose:
print(f"({spotT:0.4f}, {po:0.4f}, {prob:0.4f})")
price *= np.exp(-r * T)
return price
def binomialPricer(option, S, r, v, q, n):
nodes = n + 1
T = option.expiry
K = option.strike
h = T / n
u = np.exp((r - q) * h + v * np.sqrt(h))
d = np.exp((r - q) * h - v * np.sqrt(h))
pstar = (np.exp((r - q) * h) - d) / (u - d)
price = 0.0
for i in range(nodes):
prob = binom.pmf(i, n, pstar)
spotT = S * (u**i) * (d**(n - i))
po = option.payoff(spotT)
price += po * prob
price *= np.exp(-r * T)
return price
def binom_cal():
n = int(e.get())
p = float(e1.get())
x1 = int(e2.get())
x2 = int(e3.get())
global x
x = np.arange(x1, x2)
global r
r = binom.pmf(x, n, p)
print(r)
total = 0
for add in r:
#total = 0
total += add
print(total)
global e4
e4 = Entry(root)
e4.grid(row=5, column=1)
e4.insert(10, str(total))
def Binomial():
binomValues = list(map(float, txt.get().split(',')))
n = np.float_(binomValues[0])
p = np.float_(binomValues[1])
x = np.float_(binomValues[2])
x = np.arange(x) #it creates list [0,1,2,3]
print(x)
result = binom.pmf(x, n, p) #calculate binomial function
print(
result
) #printing list of result at each value of x ## result at y axis ##probability always betwenn 0 to 1 so on y AXIS value tkes between 0 and 1
plt.plot(
x, result, 'o-'
) # at index zero of result of x=0 pmf calculated pmf= (n!/(x!*(n-x)!)) p^x*(1-p)^(1-p)
plt.title("Binomial graph graph")
plt.xlabel("Number of success") # x-axis lable
plt.ylabel("probability of sucess") # y axis lable
plt.show() #graph show
def next_round_prob_bravo(self, margin, round_size_prev, round_size, kmin_first, kmin, prob_table_prev):
"""
Parameters
----------
:param margin: margin of a given race (float in [0, 1])
:param round_size_prev: the size of the previous round
:param round_size: the size of the current round
:param kmin_first: the kmin for the first round
:param kmin: the value of previous kmin
:param prob_table_prev: the probability distribution at the begining of the current round
:return: prob_table: the probability distribution at the end of the current round is returned
"""
prob_table = [0] * (round_size + 1)
for i in range(kmin + 1):
for j in range(min(round_size + 1, round_size - round_size_prev + kmin + 1)):
prob_table[j] = prob_table[j] + binom.pmf(j-i, round_size - round_size_prev, (1+margin)/2) * prob_table_prev[i]
return prob_table
def _jc_posterior_ng(COUNT_MATRIX, DIST_SAMPLES):
MATRIX_SUM = COUNT_MATRIX.sum()
MATRIX_DIAGONAL_SUM = COUNT_MATRIX.diagonal().sum()
P = np.exp(-DIST_SAMPLES / 100)
likelihood = binom.pmf(MATRIX_DIAGONAL_SUM, MATRIX_SUM, P)
# This code is not commented in Octave
# if (any(isnan(likelihood)))
# # In case the binomial is tricky to compute, approx with normal distr!
# likelihood = normpdf(k, tot .* p, tot .* p .* (1 .- p));
# endif
likelihood[0] /= 2
likelihood[-1] /= 2
POSTERIOR_VEC = likelihood / (likelihood.sum() *
(DIST_SAMPLES[1] - DIST_SAMPLES[0]))
return POSTERIOR_VEC
def binomfortour(k=11):
toursize_max = 60
toursize = np.arange(1, toursize_max,1)
#print(toursize)
popsize = 800
probabilities = toursize/popsize
#print(probabilities)
test = binom.pmf(k=k, n=popsize, p=probabilities) #P(X = k)
test1 = binom.cdf(k=k, n=popsize, p=probabilities) #P(X <= k)
test2 = 1 - binom.cdf(k=k-1, n=popsize, p=probabilities) #P(X >= k)
#print(np.round(test, decimals=2))
#print(np.round(test1, decimals=2))
#print(np.round(test2, decimals=2))
plt.bar(x=toursize, height=test1, width=0.6)
xticks = np.arange(5, toursize_max, 5)
plt.xticks(xticks, xticks)
plt.title("P(X >= "+str(k)+")")
plt.show()
def binomial_pmf_theoretical(params):
""" Computes the PMF for a binomial distribution using the explicit formula
Args:
params: An object containing two attributes: n and p. These are the distribution parameters
Returns:
A pandas data frame with 2 columns called 'k' and 'prob_thr'. The first column contains integers
from 0 to n representing the number of successful trials. The second column contains the
probabilities Pr[X=k].
"""
xvals = range(
0, params.n + 1
) # X is the random variable in Pr[X=k]. It represents number of successes
probs = [binom.pmf(k=v, n=params.n, p=params.p)
for v in xvals] # Generate p(k) for each possible value of k
# Create a pandas dataframe with the results and return
res_df = pd.DataFrame(data=list(zip(xvals, probs)),
columns=["k", "prob_thr"])
return res_df
def calculate_cVaR(percentile, n, q):
"""
Calculate the conditional value at risk(cVaR) given confidence level and discrete distribution
@percentile::float: confidence level, 0 <= percentile <= 1
@n::int; maximum number of trials
@q::int; the probability of getting heads
Return::int: the cVaR
"""
cVaR = 0
mul = -1 / (1 - percentile) # the multiplier in the front
mu = n * q
currVaR = 0 - n * q # right shift to standard binomial distribution
while currVaR <= -calculate_VaR(percentile, n, q):
x = currVaR + mu # shift back
cVaR += mul * currVaR * binom.pmf(x, n, q)
currVaR += 1
return cVaR
def european_binomial(option, rate, vol, div, steps):
strike = option.strike
expiry = option.expiry
spot = option.spot
call_t = 0.0
spot_t = 0.0
option_t = 0.0
h = expiry / steps
u = np.exp((rate - div) * h + vol * np.sqrt(h))
d = np.exp((rate - div) * h - vol * np.sqrt(h))
num_nodes = steps + 1
pstar = (np.exp(rate * h) - d) / (u - d)
for i in range(num_nodes):
spot_t = spot * (u**(steps - i)) * (d**(i))
option_t += option.payoff(spot_t) * binom.pmf(steps - i, steps, pstar)
option_t *= np.exp(-rate * option.expiry)
return option_t
def _create_mdp(self, path, prob, size):
def _fill_reward(x):
if x == 0:
return 0.1
elif x == size - 1:
return 1
else:
return 0
state_generator = StateGenerator('name', 'reward')
action_generator = ActionGenerator('name')
states = [
state_generator.generate_state(name="s" + str(x),
reward=_fill_reward(x))
for x in range(size)
]
actions = [
action_generator.generate_action(name=name)
for name in ['LEFT', 'RIGHT']
]
# Initializing the states of the mdp with binomial distribution
init_states = dict(
zip(states,
[0] + [binom.pmf(i, size - 3, prob)
for i in range(size - 2)] + [0]))
mdp = MDPModel('MdpEnvLinVariable') \
.add_states(states) \
.add_actions(actions) \
.add_init_states(init_states) \
.add_final_states([states[x] for x in [0, size - 1]], 100)
for i in range(size - 2):
mdp \
.add_transition(states[i + 1], actions[0], {states[i]: 1}) \
.add_transition(states[i + 1], actions[1], {states[i + 2]: 1})
# Visualize the MDP
mdp.finalize()
# mdp.visualize(file="{0}/{1}".format(path, self.__class__.__name__))
return mdp
def _classify_position(self, tx_indels: int, mock_indels: int,
edit_prior: Pr, binom_p: Pr) -> bool:
"""
Classify for each position if it Edit or not.
:param tx_indels:
:param mock_indels:
:param edit_prior:
:param binom_p:
:return: Edit or not bool
"""
total_reads = self._n_reads_tx + self._n_reads_mock
no_edit_prior = 1 - edit_prior
# if no indels in treatment, return True (same results as if run the classifier)
if tx_indels == 0:
return False
# In extreme cases both posteriors are smaller than python minimum number (1e-325).
# The heuristic to solve this rare event, is to divide by 10 both the treatment and the mock indels.
for idx in range(5):
total_indels = tx_indels + mock_indels
# Likelihood function computation
no_edit_likelihood = hypergeom.pmf(tx_indels, total_reads,
total_indels, self._n_reads_tx)
edit_likelihood = binom.pmf(k=tx_indels, n=total_indels, p=binom_p)
# Posterior probability computation
no_edit_post = no_edit_prior * no_edit_likelihood
edit_post = edit_prior * edit_likelihood
# Regular case - Both different from zero
if (edit_post != 0) or (no_edit_post != 0):
edit = edit_post > no_edit_post
return edit
else:
tx_indels = tx_indels // 10
mock_indels = mock_indels // 10
raise ClassificationFailed()
def make_3D(t, N, F, Tf, Ts, R, a, eta, Q, T, St):
dt = t[1] - t[0]
r = np.arange(t[0], t[-1] + dt, dt / 100)
dr = r[1] - r[0]
n = np.arange(np.floor(N - 4 * np.sqrt(N)), np.ceil(N + 4 * np.sqrt(N)))
pois = poisson.pmf(n, N)
nu = np.arange(30)
model = np.sum(((1 - R) * Promt(r, F, Tf, Ts, T, St) + R *
(1 - eta) * Recomb(r, F, Tf, Ts, T, St, a, eta)).reshape(
len(t), 100, len(T)),
axis=1)
if np.any(model < 0):
return np.amin(model) * np.ones((len(nu), 100, len(Q)))
I = np.arange(len(nu) * len(Q) * len(t) * len(n))
B = binom.pmf(
nu[I // (len(Q) * len(t) * len(n))], (n[I % len(n)]).astype(int),
dS[(I // (len(n) * len(t))) % len(Q)] *
Q[(I // (len(n) * len(t))) % len(Q)] *
model[(I // len(n)) % len(t),
(I // (len(n) * len(t))) % len(Q)]).reshape(
len(nu), len(Q), len(t), len(n))
# P=np.matmul(B, pois)
if np.any(np.isnan(B)):
print('B is nan')
sys.exit()
if np.any(np.isinf(B)):
print('B is inf')
sys.exit()
P0 = np.vstack((np.ones(len(n)), np.cumprod(np.prod(B[0], axis=0),
axis=0)[:-1]))
P1 = (P0 * (1 - np.prod(B[0], axis=0)))
P = np.zeros((100, len(nu), len(Q)))
for i in range(len(Q)):
P2 = P0 * (1 - np.prod(B[0, np.delete(np.arange(len(Q)), i)], axis=0))
P[0, 0, i] = np.sum(np.sum(B[0, i] * P2 * pois, axis=1), axis=0)
P[0, 1:] = np.sum(np.sum(B[1:] * P0 * pois, axis=3), axis=2)
for i in range(1, 100):
P[i] = np.sum(np.sum(B[:, :, i:, :] * P1[:len(t) - i, :] * pois,
axis=3),
axis=2)
return np.transpose(P, (1, 0, 2))
def kmr_markov_matrix(p, N, epsilon=0, simultaneous=False):
"""
Generate the transition probability matrix for the KMR dynamics with
two acitons.
Parameters
----------
p : scalar(int)
Probability on mixed strategy nash equilibrium.
N : scalar(int)
Number of players.
epsilon : scalar(int), optional(default=0)
perturbations.
simultaneous : True or False
sequential or simultaneous
Returns
---------
P : ndarray(int, ndim=2)
The transition probability matrix for the KMR dynamics.
"""
P = np.zeros((N + 1, N + 1), dtype=float)
if simultaneous is False:
P[0][1] = (epsilon) * (1 / 2)
P[N][N - 1] = (epsilon) * (1 / 2)
P[0][0], P[N][N] = 1 - P[0][1], 1 - P[N][N - 1]
for i in range(1, N):
P[i][i - 1] = (i / N) * (epsilon * (1 / 2) + (1 - epsilon) *
(((i - 1) / (N - 1) < p) +
((i - 1) / (N - 1) == p) * (1 / 2)))
P[i][i + 1] = ((N - i) / N) * (epsilon * (1 / 2) + (1 - epsilon) *
((i / (N - 1) > p) +
(i / (N - 1) == p) * (1 / 2)))
P[i][i] = 1 - P[i][i - 1] - P[i][i + 1]
return P
else:
for i in range(0, N + 1):
P[i] = binom.pmf(range(N + 1), N,
(i / N < p) * epsilon / 2 + (i / N == p) / 2 +
(i / N > p) * (1 - epsilon / 2))
return P
def compare_aln(fg_aln, bg_aln):
"""Compare alignments using the ball-in-urn model.
Like CHAIN does.
"""
# BG seqs are weighted, FG seqs are not
bg_weights = alnutils.sequence_weights(bg_aln, 'none')
bg_size = sum(bg_weights)
bg_cons = consensus.consensus(bg_aln, weights=bg_weights)
# Height of the foreground alignment column
fg_size = len(fg_aln)
fg_cons = consensus.consensus(fg_aln)
fg_cols = zip(*fg_aln)
bg_cols = zip(*bg_aln)
fg_weights = [1] * fg_size
pseudocounts = combined_frequencies(fg_aln, fg_weights, bg_aln, bg_weights)
hits = []
for faa, baa, fg_col, bg_col in zip(fg_cons, bg_cons, fg_cols, bg_cols):
if faa == '-' or baa == '-':
# Ignore indel columns -- there are better ways to look at these
pvalue = 1.0
else:
# Cumulative binomial test
# Number of consensus-type residues in the foreground column
fg_counts = count_col(fg_col, fg_weights, pseudocounts)
fg_tot = fg_counts['S'] + fg_counts['T'] + fg_counts['Y']
# Consensus residue frequency in the combined alignment column
bg_counts = count_col(bg_col, bg_weights, pseudocounts)
p_j = (bg_counts['S'] + bg_counts['T'] + bg_counts['Y'] + fg_tot
) / (bg_size + fg_size + 2.0) # pseudocount size = 1.0
# Probability of fg col conservation vs. the combined/main set
# (P_j_LB in the publication)
# NB: Some tweaks for pseudocounts
pvalue = binom.pmf(range(int(math.ceil(fg_tot)), fg_size + 2),
fg_size + 1, p_j).sum()
if pvalue == 1.0:
logging.info("Meaningless p-value: p_j=%s, fg=%s vs. bg=%s",
p_j, fg_tot, bg_counts)
hits.append((faa, baa, pvalue))
return hits
def matrix_selection_more_contributors(n, N, s=0):
rocc = reduced_occupancy(N)
mtx = np.zeros((n + 1, n + 1))
# cache is (io, no, ip, nc)
cache = np.full((2*n+1, n+1, 2*n+1, n+1), np.nan)
for ip in range(0, n + 1):
for io in range(0, n + 1):
for nc in range(0, n+1):
for ic in range(0, nc+1):
p = binom.pmf(ic, nc, ip/n)
mtx[ip, io] += Qs(io, n, ic, nc, N, s, cache) * p
# for nc in range(0, n+1):
# for ic in range(0, nc+1):
# # p = binom.pmf(ic, nc, np.arange(0, n+1)/n)
# p = hypergeom.pmf(ic, n, np.arange(0, n+1), nc)
# for io in range(0, n + 1):
# mtx[:, io] += Qs(io, n, ic, nc, N, s, cache) * p
return mtx, cache
def UtilIG_bayes_risk(self):
"""
効用としてベイズリスクを計算する関数
推定1回目はランダムに実験を選ぶ
"""
if self.i != 0:
m = np.argmax(self.w)
L_w = np.ones(self.n_particles())
dU = np.zeros([self.n_exp(), 1])
for i in range(self.n_exp()):
L_w[m] = binom.pmf(self.num, n=self.d,
p=self.ptable_C[i]) #各パーティクルでの実験でms=0にいた確率
w_new = self.w * L_w.reshape([len(L_w), 1]) #重みの更新
x_infer = self.Mean(w_new, self.x)
dU[i] = np.trace(self.Q * np.dot(
(self.x - x_infer[0]).T,
(self.x - x_infer[0]))) #実験C[i]でのベイズリスク
self.U = dU * self.U
self.U = self.U / np.sum(self.U)
self.C_best_i = np.argmin(self.U)
self.C_best = self.C[self.C_best_i]
def EuropeanBinomialPricer(pricing_engine, option, data):
expiry = option.expiry
strike = option.strike
(spot, rate, volatility, dividend) = data.get_data()
steps = pricing_engine.steps
nodes = steps + 1
dt = expiry / steps
u = np.exp(((rate - dividend) * dt) + volatility * np.sqrt(dt))
d = np.exp(((rate - dividend) * dt) - volatility * np.sqrt(dt))
pu = (np.exp((rate - dividend) * dt) - d) / (u - d)
pd = 1 - pu
disc = np.exp(-rate * expiry)
spotT = 0.0
payoffT = 0.0
for i in range(nodes):
spotT = spot * (u ** (steps - i)) * (d ** (i))
payoffT += option.payoff(spotT) * binom.pmf(steps - i, steps, pu)
price = disc * payoffT
return price
def get_binominal_probabilities(n):
params = {}
params['4'] = [0.2, 0.4, 0.6, 0.8]
params['5'] = [0.1, 0.3, 0.5, 0.7, 0.9]
params['6'] = [0.1, 0.26, 0.42, 0.58, 0.74, 0.9]
params['8'] = [
0.1, 0.21428571, 0.32857143, 0.44285714, 0.55714286, 0.67142857,
0.78571429, 0.9
]
probs = []
for true_class in range(0, n):
probs.append(
binom.pmf(np.arange(0, n), n - 1, params[str(n)][true_class]))
return np.array(probs)
def UtilIG_bayes_risk(self):
"""
効用としてベイズリスクを計算する関数
重みの仮更新がこれで良いのか議論の余地あり
"""
m = np.argmax(self.w)
p_exp = self.Expsim(self.x[m], self.C_best) #真値におけるms0の確立
num = binomial(self.d, p_exp) #実験をd回行いｍs=0であった回数
L_w = np.ones(self.n_particles())
dU = np.zeros([self.n_exp(), 1])
for i in range(self.n_exp()):
L_w[m] = binom.pmf(num, n=self.d,
p=self.ptable_C[i]) #各パーティクルでの実験でms=0にいた確率
w_new = self.w * L_w.reshape([len(L_w), 1]) #重みの更新
x_infer = self.Mean(w_new, self.x)
dU[i] = np.trace(self.Q * np.dot((self.x - x_infer[0]).T,
(self.x - x_infer[0])))
self.U = dU * self.U
self.U = self.U / np.sum(self.U)
self.C_best_i = np.argmin(self.U)
self.C_best = self.C[self.C_best_i]
def Update(self): #ベイズ推定を行う関数 引数(self,Ci)
"""
パーティクルの重みを更新する関数
"""
#ラビ振動とラムゼー干渉の実験を行う
exp_flag_best=self.exp_flag
self.mode=1
self.p_exp=[]
self.exp_flag="rabi"
self.p_exp.append(self.Expsim(self.x0,self.C_best[0]))#真値におけるms0の確立
self.exp_flag="ramsey"
self.p_exp.append(self.Expsim(self.x0,self.C_best[1]))#真値におけるms0の確立
self.num=binomial(self.d, self.p_exp)#実験をd回行いｍs=0であった回数
self.exp_flag=exp_flag_best
if exp_flag_best=="rabi":
_num=self.num[0]
elif exp_flag_best=="ramsey":
_num=self.num[1]
temp=binom.pmf(_num,n=self.d,p=self.ptable_x)#各パーティクルでの実験でms=0にいた確率
self.w=self.w*temp.reshape([len(temp),1]) #重みの更新
self.w=self.w/np.sum(self.w) #重みの規格化
def get_series_prob(series_games, series_wins, series_losses, team_winProb):
team_probs = []
if series_wins == series_games / 2 + 1:
team_probs.append(1)
total_games = series_wins + series_losses
if series_losses == series_games / 2 + 1:
team_probs.append(0)
if (series_wins != series_games / 2 + 1
and series_losses != series_games / 2 + 1):
for end_game in range(series_games / 2 + 1,
series_games + 1 - series_losses):
team_in_N = BinomDist.pmf(n=end_game - 1 - series_wins,
k=(series_games / 2 - series_wins),
p=team_winProb) * team_winProb
team_probs.append(team_in_N)
return float(sum(team_probs))
def compare_cols(fg_col, fg_cons, fg_size, fg_weights, bg_col, bg_cons,
bg_size, bg_weights, aa_freqs, pseudo_size):
"Compare alignments using the ball-in-urn model (cumulative binomial test)"
# Number of consensus-type residues in the foreground column
fg_counts = count_col(fg_col, fg_weights, aa_freqs, pseudo_size)
fg_tot = fg_counts['S'] + fg_counts['T'] + fg_counts['Y']
# Consensus residue frequency in the combined alignment column
bg_counts = count_col(bg_col, bg_weights, aa_freqs, pseudo_size)
p_j = (bg_counts['S'] + bg_counts['T'] + bg_counts['Y'] + fg_tot) / (
bg_size + fg_size + 2.0) # pseudocount size = 1.0
# Probability of fg col conservation vs. the combined/main set
# (P_j_LB in the publication)
# NB: Some tweaks for pseudocounts
pvalue = binom.pmf(range(int(math.ceil(fg_tot)), fg_size + 2), fg_size + 1,
p_j).sum()
if pvalue == 1.0:
logging.info("Meaningless p-value: p_j=%s, fg=%s vs. bg=%s", p_j,
fg_tot, bg_counts)
return pvalue
def show_my_result(name_list, correct_list, num_questions=30, ax1=None):
n_correct = np.arange(num_questions + 1)
if ax1 is None:
fig, ax1 = plt.subplots(1, 1, figsize=(10, 5), dpi=250)
binom_pmf = binom.pmf(n_correct, num_questions, 0.5)
binom_cdf = np.cumsum(binom_pmf)
ax1.bar(n_correct, binom_pmf / np.max(binom_pmf), color='k', alpha=0.5)
ax1.plot(n_correct, binom_cdf, 'k-', label='Cumulative')
prop_cycle = plt.rcParams['axes.prop_cycle']
color_cycle = cycle(prop_cycle.by_key()['color'])
y_pos_list = np.linspace(0.2, 0.8, len(name_list))
for y_pos, name, correct, color in zip(y_pos_list, name_list, correct_list,
color_cycle):
ax1.axvline(correct,
label='{}\nTop {:2.1f}%'.format(name,
100 * binom_cdf[correct]),
color=color)
ax1.text(correct, y_pos, name)
ax1.legend()
return ax1
def model(args):
"""
%prog model erate
Model kmer distribution given error rate. See derivation in FIONA paper:
<http://bioinformatics.oxfordjournals.org/content/30/17/i356.full>
"""
from scipy.stats import binom, poisson
p = OptionParser(model.__doc__)
p.add_option("-k", default=23, type="int", help="Kmer size")
p.add_option("--cov", default=50, type="int", help="Expected coverage")
opts, args = p.parse_args(args)
if len(args) != 1:
sys.exit(not p.print_help())
(erate, ) = args
erate = float(erate)
cov = opts.cov
k = opts.k
xy = []
# Range include c although it is unclear what it means to have c=0
for c in range(0, cov * 2 + 1):
Prob_Yk = 0
for i in range(k + 1):
# Probability of having exactly i errors
pi_i = binom.pmf(i, k, erate)
# Expected coverage of kmer with exactly i errors
mu_i = cov * (erate / 3)**i * (1 - erate)**(k - i)
# Probability of seeing coverage of c
Prob_Yk_i = poisson.pmf(c, mu_i)
# Sum i over 0, 1, ... up to k errors
Prob_Yk += pi_i * Prob_Yk_i
xy.append((c, Prob_Yk))
x, y = zip(*xy)
asciiplot(x, y, title="Model")
def single_gen_matrix(Ne= 1092, ploidy= 2,precision= 1092,s=0):
'''
define transition probabilities for alleles at any given frequency from 1 to Ne.
'''
pop_size= Ne * ploidy
space= np.linspace(0,pop_size,precision,dtype=int)
t= np.tile(np.array(space),(precision,1))
probs= [x / (pop_size) for x in space]
##
sadjust= [x * (1+s) for x in probs]
scomp= [(1-x) for x in probs]
new_probs= [sadjust[x] / (scomp[x]+sadjust[x]) for x in range(precision)]
new_probs= np.nan_to_num(new_probs)
probs= new_probs
freq_matrix= binom.pmf(t.T,pop_size,probs)
return freq_matrix
def sample_c_i(i):
global c, clusters
#print_cluster(clusters)
xi = x[i]
ci = c[i]
phi_i_j = {}
clusters_cp = deepcopy(clusters)
clusters_cp[ci].member.remove(xi)
for j in clusters_cp:
n_j_i = clusters_cp[j].size()
if n_j_i > 0:
theta_j = beta.rvs(clusters_cp[j].a1,clusters_cp[j].b1)
phi_i_j[j] = binom.pmf(xi,20,theta_j) * n_j_i / (alpha + N - 1)
j_new = max(clusters) + 1
phi_i_j[j_new] = p_new[xi]
# generate j1
c_phi = phi_i_j.keys()
p_phi = phi_i_j.values() / sum(phi_i_j.values())
#j1 = stats.rv_discrete(name='custm',values=(c_phi,p_phi))
j1 = np.random.choice(c_phi,p = p_phi)
#j1 = max(phi_i_j, key = phi_i_j.get)
#print "debug:x",i,"ci",ci, "j1",j1
if j1 == j_new:
c[i] = j1
clusters[j1] = Cluster(j1)
clusters[j1].member.append(xi)
clusters[ci].member.remove(xi)
if clusters[ci].size() == 0:
del clusters[ci]
else:
c[i] = j1
clusters[j1].member.append(xi)
#print xi, ci, clusters[ci].member
#print "ci",ci,"member",clusters[ci].member,"xi",xi
clusters[ci].member.remove(xi)
if clusters[ci].size() == 0:
del clusters[ci]
def _left_extreme(n_instances, n_total, prob):
"""Used for a binomial problem. Calculates the exact likelihood of finding observations as and less extreme
than our observed value
Parameters
----------
n_instances: int
The number of observed successes in our binomial problem
n_total: int
The total number of observations
prob: float, 0<=p<=1
The expected probability of success
Returns
-------
p: float
The exact likelihood that we would find observations less extreme than our observed number of
success
"""
counter = np.arange(n_instances + 1)
p = np.sum(binom.pmf(counter, n_total, prob))
return p
def part_a2(fname=fname):
fname = fname + '_a2'
m = 5
theta_set = np.linspace(1 / float(m), 1 - 1 / float(m), m)
print(theta_set)
n = 34
x = np.arange(n + 1)
fig = plt.figure()
hh = []
for i, theta in enumerate(theta_set):
h, = plt.plot(x,
binom.pmf(x, n, theta),
lw=1.5,
label=r'$\theta=' + '{:.2f}'.format(theta) + '$')
hh.append(h)
plt.xlabel(r'$y$', fontsize=labelFontSize)
plt.ylabel(r'$p(y \mid \theta)$', fontsize=labelFontSize)
plt.title('3.4a' + r' $p(y \mid \theta)$', fontsize=titleFontSize)
plt.xticks(fontsize=tickFontSize)
plt.yticks(fontsize=tickFontSize)
plt.legend()
plt.savefig(fname + '.' + imgFmt, format=imgFmt)
