如何解决如何创建 Hypsecant 分布以及与该分布一起使用的先验条件？
我一直在使用以下链接， Fitting empirical distribution to theoretical ones with Scipy。我一直在使用我的数据，发现常见的分布是 Hypsecant 分布。我在 pymc3 包中找不到发行版，因此，我决定用 scipy 看一下，以了解发行版是如何形成的。我在 pymc3 中为 Hypsecant 分布创建了一个自定义分布，但我有几个问题：
我的自定义发行版：
import numpy as np
import theano.tensor as tt
from scipy import stats
from scipy.special import hyp1f1,nctdtr
import math
import warnings
from pymc3.theanof import floatX
from pymc3.distributions.dist_math import bound,gammaln
from pymc3.distributions.continuous import assert_negative_support,get_tau_sigma
from pymc3.distributions.distribution import Continuous,draw_values,generate_samples
class Hypsecant(Continuous):
"""
Parameters
----------
mu: float
Location parameter.
sigma: float
Scale parameter (sigma > 0). Converges to the standard deviation as nu increases. (only required if lam is not specified)
lam: float
Scale parameter (lam > 0). Converges to the precision as nu increases. (only required if sigma is not specified)
"""
def __init__(self,mu=0,sigma=1,lam=None,sd=None,*args,**kwargs):
super().__init__(*args,**kwargs)
super(Hypsecant,self).__init__(*args,**kwargs)
if sd is not None:
sigma = sd
warnings.warn("sd is deprecated,use sigma instead",DeprecationWarning)
lam,sigma = get_tau_sigma(tau=lam,sigma=sigma)
self.lam = lam = tt.as_tensor_variable(lam)
self.sigma = self.sd = sigma = tt.as_tensor_variable(sigma)
self.mean = self.median = self.mode = self.mu = mu = tt.as_tensor_variable(mu)
self.variance = tt.as_tensor_variable(1)
assert_negative_support(lam,'lam (sigma)','Hypsecant')
def random(self,point=None,size=None):
"""
Draw random values from Hypsecant distribution.
Parameters
----------
point: dict,optional
Dict of variable values on which random values are to be
conditioned (uses default point if not specified).
size: int,optional
Desired size of random sample (returns one sample if not
specified).
Returns
-------
array
"""
mu,lam = draw_values([self.mu,self.lam],point=point,size=size)
return generate_samples(stats.hypsecant.rvs,loc=mu,scale=lam,dist_shape=self.shape,size=size)
def logp(self,value):
"""
Calculate log-probability of Hypsecant distribution at specified value.
Parameters
----------
value: numeric
Value(s) for which log-probability is calculated. If the log probabilities for multiple
values are desired the values must be provided in a numpy array or theano tensor
Returns
-------
TensorVariable
"""
mu = self.mu
lam = self.lam
return bound()
def logcdf(self,value):
"""
Compute the log of the cumulative distribution function for Hypsecant distribution
at the specified value.
Parameters
----------
value: numeric
Value(s) for which log CDF is calculated. If the log CDF for multiple
values are desired the values must be provided in a numpy array or theano tensor.
Returns
-------
TensorVariable
"""
mu = self.mu
lam = self.lam
return 2.0 / math.pi * tt.math.atan(tt.math.exp(value))
我的自定义模型：
with pm.Model() as model:
# Prior distributions for unkNown model parameters:
mu = pm.normal('mu',sigma=1)
sd = pm.Halfnormal('sd',sigma=1)
# Observed data is from a Likelihood distributions (Likelihood (sampling distribution) of observations):
observed_data = Hypsecant('observed_data',mu=mu,sd=sd,observed=data_list)
# draw 5000 posterior samples
trace_S = pm.sample(draws=5000,tune=2000,chains=3,cores=1)
# Obtaining Posterior Predictive Sampling:
post_pred_S = pm.sample_posterior_predictive(trace_S,samples=3000)
print(post_pred_S['observed_data'].shape)
print('\nSummary: ')
print(pm.stats.summary(data=trace_S))
print(pm.stats.summary(data=post_pred_S))
编辑 1：
class Hypsecant(Continuous):
"""
Parameters
----------
mu: float
Location parameter.
sigma: float
Scale parameter (sigma > 0). Converges to the standard deviation as nu increases. (only required if lam is not specified)
lam: float
Scale parameter (lam > 0). Converges to the precision as nu increases. (only required if sigma is not specified)
"""
def __init__(self,value):
"""
Calculate log-probability of Hypsecant distribution at specified value.
Parameters
----------
value: numeric
Value(s) for which log-probability is calculated. If the log probabilities for multiple
values are desired the values must be provided in a numpy array or theano tensor
Returns
-------
TensorVariable
"""
mu = self.mu
lam = self.lam
Px = 1.0/(math.pi * tt.math.cosh(value))
return bound(Px,mu,lam)
def logcdf(self,value):
"""
Compute the log of the cumulative distribution function for Hypsecant distribution
at the specified value.
Parameters
----------
value: numeric
Value(s) for which log CDF is calculated. If the log CDF for multiple
values are desired the values must be provided in a numpy array or theano tensor.
Returns
-------
TensorVariable
"""
mu = self.mu
lam = self.lam
return 2.0 / math.pi * tt.math.atan(tt.math.exp(value))
型号：
with pm.Model() as model:
# Prior distributions for unkNown model parameters:
mu = pm.normal('mu',sigma=1)
sd = pm.normal('sd',sigma=1)
# Custom distribution:
# observed_data = pm.Densitydist('observed_data',NonCentralStudentT,observed=data_list)
# Observed data is from a Likelihood distributions (Likelihood (sampling distribution) of observations):
observed_data = Hypsecant('observed_data',observed=data)
# draw 5000 posterior samples
trace = pm.sample(draws=5000,tune=3000,chains=2,cores=1)
# Obtaining Posterior Predictive Sampling:
post_pred = pm.sample_posterior_predictive(trace,samples=3000)
print(post_pred['observed_data'].shape)
print('\nSummary: ')
print(pm.stats.summary(data=trace))
print(pm.stats.summary(data=post_pred))
编辑 2：
class Hypsecant(Continuous):
"""
Parameters
----------
mu: float
Location parameter.
sigma: float
Scale parameter (sigma > 0). Converges to the standard deviation as nu increases. (only required if lam is not specified)
lam: float
Scale parameter (lam > 0). Converges to the precision as nu increases. (only required if sigma is not specified)
"""
def __init__(self,mu=None,sigma=None,**kwargs):
# super().__init__(*args,**kwargs)
# super(Hypsecant,sigma=sigma)
self.lam = lam = tt.as_tensor_variable(lam)
self.sigma = self.sd = sigma = tt.as_tensor_variable(sigma)
self.mean = self.median = self.mode = self.mu = mu = tt.as_tensor_variable(mu)
self.variance = tt.as_tensor_variable(1)
assert_negative_support(mu,'mu','Hypsecant')
assert_negative_support(sigma,'sigma (lam)','Hypsecant')
# return super(Hypsecant,self).__init__(shape=[mu,sigma],**kwargs)
return super().__init__(*args,**kwargs)
def random(self,optional
Desired size of random sample (returns one sample if not
specified).
Returns
-------
array
"""
# mu = self.mu
# lam,sigma=sigma)
mu,value):
"""
Calculate log-probability of Hypsecant distribution at specified value.
Parameters
----------
value: numeric
Value(s) for which log-probability is calculated. If the log probabilities for multiple
values are desired the values must be provided in a numpy array or theano tensor
Returns
-------
TensorVariable
"""
# mu = self.mu
# lam = self.lam
# sigma = self.sigma
# # sd = self.sd
# mu = self.mu
# lam,sigma = get_tau_sigma(sigma=sigma)
Px = pm.math.log(1.0/(math.pi * pm.math.cosh(value)))
return bound(Px)
# return bound(Px,lam,sigma)
def logcdf(self,value):
"""
Compute the log of the cumulative distribution function for Hypsecant distribution
at the specified value.
Parameters
----------
value: numeric
Value(s) for which log CDF is calculated. If the log CDF for multiple
values are desired the values must be provided in a numpy array or theano tensor.
Returns
-------
TensorVariable
"""
# # mu = self.mu
# # lam = self.lam
# # # self.sigma = self.sd = sigma = tt.as_tensor_variable(sigma)
# mu = self.mu
# lam = self.lam
# sigma = self.sigma
# # sd = self.sd
# mu = self.mu
# lam,sigma = get_tau_sigma(sigma=sigma)
# return bound(pm.math.log(2.0 / math.pi * tt.math.atan(tt.math.exp(value))),sigma)
return bound(pm.math.log(2.0 / math.pi * tt.math.atan(tt.math.exp(value))))
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