def model_best(y1, y2, samples=1000):
"""Bayesian Estimation Supersedes the T-Test
This model runs a Bayesian hypothesis comparing if y1 and y2 come
from the same distribution. Returns are assumed to be T-distributed.
In addition, computes annual volatility and Sharpe of in and
out-of-sample periods.
This model replicates the example used in:
Kruschke, John. (2012) Bayesian estimation supersedes the t
test. Journal of Experimental Psychology: General.
Parameters
----------
y1 : array-like
Array of returns (e.g. in-sample)
y2 : array-like
Array of returns (e.g. out-of-sample)
samples : int, optional
Number of posterior samples to draw.
Returns
-------
pymc3.sampling.BaseTrace object
A PyMC3 trace object that contains samples for each parameter
of the posterior.
See Also
--------
plot_stoch_vol : plotting of tochastic volatility model
"""
y = np.concatenate((y1, y2))
mu_m = np.mean(y)
mu_p = 0.000001 * 1 / np.std(y)**2
sigma_low = np.std(y)/1000
sigma_high = np.std(y)*1000
with pm.Model():
group1_mean = pm.Normal('group1_mean', mu=mu_m, tau=mu_p,
testval=y1.mean())
group2_mean = pm.Normal('group2_mean', mu=mu_m, tau=mu_p,
testval=y2.mean())
group1_std = pm.Uniform('group1_std', lower=sigma_low,
upper=sigma_high, testval=y1.std())
group2_std = pm.Uniform('group2_std', lower=sigma_low,
upper=sigma_high, testval=y2.std())
nu = pm.Exponential('nu_minus_two', 1/29., testval=4.) + 2.
returns_group1 = pm.T('group1', nu=nu, mu=group1_mean,
lam=group1_std**-2, observed=y1)
returns_group2 = pm.T('group2', nu=nu, mu=group2_mean,
lam=group2_std**-2, observed=y2)
diff_of_means = pm.Deterministic('difference of means',
group2_mean - group1_mean)
pm.Deterministic('difference of stds',
group2_std - group1_std)
pm.Deterministic('effect size', diff_of_means /
pm.sqrt((group1_std**2 +
group2_std**2) / 2))
pm.Deterministic('group1_annual_volatility',
returns_group1.distribution.variance**.5 *
np.sqrt(252))
pm.Deterministic('group2_annual_volatility',
returns_group2.distribution.variance**.5 *
np.sqrt(252))
pm.Deterministic('group1_sharpe', returns_group1.distribution.mean /
returns_group1.distribution.variance**.5 *
np.sqrt(252))
pm.Deterministic('group2_sharpe', returns_group2.distribution.mean /
returns_group2.distribution.variance**.5 *
np.sqrt(252))
step = pm.NUTS()
trace = pm.sample(samples, step)
return trace
def model_best(y1, y2, samples=1000):
"""Bayesian Estimation Supersedes the T-Test
This model runs a Bayesian hypothesis comparing if y1 and y2 come
from the same distribution. Returns are assumed to be T-distributed.
In addition, computes annual volatility and Sharpe of in and
out-of-sample periods.
This model replicates the example used in:
Kruschke, John. (2012) Bayesian estimation supersedes the t
test. Journal of Experimental Psychology: General.
Parameters
----------
y1 : array-like
Array of returns (e.g. in-sample)
y2 : array-like
Array of returns (e.g. out-of-sample)
samples : int, optional
Number of posterior samples to draw.
Returns
-------
model : pymc.Model object
PyMC3 model containing all random variables.
trace : pymc3.sampling.BaseTrace object
A PyMC3 trace object that contains samples for each parameter
of the posterior.
See Also
--------
plot_stoch_vol : plotting of tochastic volatility model
"""
y = np.concatenate((y1, y2))
mu_m = np.mean(y)
mu_p = 0.000001 * 1 / np.std(y)**2
sigma_low = np.std(y) / 1000
sigma_high = np.std(y) * 1000
with pm.Model() as model:
group1_mean = pm.Normal('group1_mean',
mu=mu_m,
tau=mu_p,
testval=y1.mean())
group2_mean = pm.Normal('group2_mean',
mu=mu_m,
tau=mu_p,
testval=y2.mean())
group1_std = pm.Uniform('group1_std',
lower=sigma_low,
upper=sigma_high,
testval=y1.std())
group2_std = pm.Uniform('group2_std',
lower=sigma_low,
upper=sigma_high,
testval=y2.std())
nu = pm.Exponential('nu_minus_two', 1 / 29., testval=4.) + 2.
returns_group1 = pm.T('group1',
nu=nu,
mu=group1_mean,
lam=group1_std**-2,
observed=y1)
returns_group2 = pm.T('group2',
nu=nu,
mu=group2_mean,
lam=group2_std**-2,
observed=y2)
diff_of_means = pm.Deterministic('difference of means',
group2_mean - group1_mean)
pm.Deterministic('difference of stds', group2_std - group1_std)
pm.Deterministic(
'effect size', diff_of_means / pm.sqrt(
(group1_std**2 + group2_std**2) / 2))
pm.Deterministic(
'group1_annual_volatility',
returns_group1.distribution.variance**.5 * np.sqrt(252))
pm.Deterministic(
'group2_annual_volatility',
returns_group2.distribution.variance**.5 * np.sqrt(252))
pm.Deterministic(
'group1_sharpe', returns_group1.distribution.mean /
returns_group1.distribution.variance**.5 * np.sqrt(252))
pm.Deterministic(
'group2_sharpe', returns_group2.distribution.mean /
returns_group2.distribution.variance**.5 * np.sqrt(252))
step = pm.NUTS()
trace = pm.sample(samples, step)
return model, trace
lam1 = group1_std**-2
lam2 = group2_std**-2
group1 = pm.StudentT('drug', nu=nu, mu=group1_mean, lam=lam1, observed=y1)
group2 = pm.StudentT('placebo',
nu=nu,
mu=group2_mean,
lam=lam2,
observed=y2)
diff_of_means = pm.Deterministic('difference of means',
group1_mean - group2_mean)
diff_of_stds = pm.Deterministic('difference of stds',
group1_std - group2_std)
effect_size = pm.Deterministic(
'effect size', diff_of_means / pm.sqrt(
(group1_std**2 + group2_std**2) / 2))
step = pm.NUTS()
def run(n=3000):
if n == "short":
n = 500
with model:
trace = pm.sample(n, step)
burn = n / 10
pm.traceplot(trace[burn:])
pm.plots.summary(trace[burn:])
with pm.Model() as model:
group1_mean = pm.Normal('group1_mean', mu=mu_m, tau=mu_p, testval=y1.mean())
group2_mean = pm.Normal('group2_mean', mu=mu_m, tau=mu_p, testval=y2.mean())
group1_std = pm.Uniform('group1_std', lower=sigma_low, upper=sigma_high, testval=y1.std())
group2_std = pm.Uniform('group2_std', lower=sigma_low, upper=sigma_high, testval=y2.std())
nu = pm.Exponential('nu_minus_one', 1/29.) + 1
lam1 = group1_std**-2
lam2 = group2_std**-2
group1 = pm.T('drug', nu=nu, mu=group1_mean, lam=lam1, observed=y1)
group2 = pm.T('placebo', nu=nu, mu=group2_mean, lam=lam2, observed=y2)
diff_of_means = pm.Deterministic('difference of means', group1_mean - group2_mean)
diff_of_stds = pm.Deterministic('difference of stds', group1_std - group2_std)
effect_size = pm.Deterministic('effect size', diff_of_means / pm.sqrt((group1_std**2 + group2_std**2) / 2))
step = pm.NUTS()
def run(n=3000):
if n == "short":
n = 500
with model:
trace = pm.sample(n, step)
burn = n/10
pm.traceplot(trace[burn:]);
pm.plots.summary(trace[burn:])
if __name__ == '__main__':
