Python 如何在PyMC3中定义一个模型，其中一个参数在多个条件下约束为相同的值

我想写一个模型，就像下面的那个。主要的想法是，我有几个条件（或治疗），所有的参数都是独立估计每个条件，除了kappa参数是相同的所有条件

with pm.Model() as model:
    trace_per_condition = []
    # define the kappa hyperparameter
    kappa = pm.Gamma('kappa', 1, 0.1)
    for condition in range(0, ncond):
        z_cond = z[condition]
        # define the mu hyperparameter
        mu = pm.Beta('mu', 1, 1)
        # define the prior
        theta = pm.Beta('theta', mu * kappa, (1 - mu) * kappa, shape=len(z_cond))
        # define the likelihood
        y = pm.Binomial('y', p=theta, n=trials, observed=z_cond)
    # Generate a MCMC chain
        start = pm.find_MAP()
        step1 = pm.Metropolis([theta, mu])
        step2 = pm.NUTS([kappa])
        trace = pm.sample(1000, [step1, step2], progressbar=False)
        trace_per_condition.append(trace)

当我运行模型时，我得到以下消息

/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/gradient.py:513:  UserWarning: grad method was asked to compute the gradient with respect to a variable that is not part of the computational graph of the cost, or is used only by a non-differentiable operator: mu handle_disconnected(elem)
/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/gradient.py:533: UserWarning: grad method was asked to compute the gradient with respect to a variable that is not part of the computational graph of the cost, or is used only by a non-differentiable operator: <DisconnectedType>
  handle_disconnected(rval[i])
/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/gradient.py:513: UserWarning: grad method was asked to compute the gradient with respect to a variable that is not part of the computational graph of the cost, or is used only by a non-differentiable operator: theta
  handle_disconnected(elem)
Traceback (most recent call last):
  File "<stdin>", line 46, in <module>
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/tuning/starting.py", line 80, in find_MAP
    start), fprime=grad_logp_o, disp=disp, *args, **kwargs)
  File "/usr/lib/python2.7/dist-packages/scipy/optimize/optimize.py", line 777, in fmin_bfgs
    res = _minimize_bfgs(f, x0, args, fprime, callback=callback, **opts)
  File "/usr/lib/python2.7/dist-packages/scipy/optimize/optimize.py", line 832, in _minimize_bfgs
    gfk = myfprime(x0)
  File "/usr/lib/python2.7/dist-packages/scipy/optimize/optimize.py", line 281, in function_wrapper
    return function(*(wrapper_args + args))
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/tuning/starting.py", line 75, in grad_logp_o
    return nan_to_num(-dlogp(point))
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/blocking.py", line 119, in __call__
    return self.fa(self.fb(x))
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/model.py", line 284, in __call__
    return self.f(**state)
  File "/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/compile/function_module.py", line 516, in __call__
    self[k] = arg
  File "/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/compile/function_module.py", line 452, in __setitem__
    self.value[item] = value
  File "/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/compile/function_module.py", line 413, in __setitem__
    "of the inputs of your function for duplicates." % str(item)) 
TypeError: Ambiguous name: mu - please check the names of the inputs of your function for duplicates.

我得到一个错误：

/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/gradient.py:513:
UserWarning: grad method was asked to compute the gradient with respect to a variable  
that  is not part of the computational graph of the cost, or is used only by a  
non-differentiable operator: mu_1
handle_disconnected(elem)

/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/gradient.py:533: 
UserWarning: grad method was asked to compute the gradient with respect to a variable 
that is not part of the computational graph of the cost, or is used only by a 
non-differentiable operator: <DisconnectedType>
handle_disconnected(rval[i])

/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/gradient.py:513: 
UserWarning: grad method was asked to compute the gradient with respect to a variable 
that is not part of the computational graph of the cost, or is used only by a 
non-differentiable operator: theta_1
handle_disconnected(elem)


Traceback (most recent call last):
  File "<stdin>", line 43, in <module>
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/tuning/starting.py", line 80, in find_MAP
    start), fprime=grad_logp_o, disp=disp, *args, **kwargs)
  File "/usr/lib/python2.7/dist-packages/scipy/optimize/optimize.py", line 777, in fmin_bfgs
    res = _minimize_bfgs(f, x0, args, fprime, callback=callback, **opts)
  File "/usr/lib/python2.7/dist-packages/scipy/optimize/optimize.py", line 837, in _minimize_bfgs
    old_fval = f(x0)
  File "/usr/lib/python2.7/dist-packages/scipy/optimize/optimize.py", line 281, in function_wrapper
    return function(*(wrapper_args + args))
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/tuning/starting.py", line 72, in logp_o
    return nan_to_high(-logp(point))
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/blocking.py", line 119, in __call__
    return self.fa(self.fb(x))
  File "/usr/local/lib/python2.7/dist-packages/pymc-3.0-py2.7.egg/pymc/model.py", line 283, in __call__
    return self.f(**state)
  File "/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/theano/compile/function_module.py", line 482, in __call__
    raise TypeError("Too many parameter passed to theano function")
TypeError: Too many parameter passed to theano function

/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/Theano/gradient.py:513:
用户警告：要求grad方法计算变量的梯度
这不是成本计算图的一部分，或仅由
不可微算子：mu_1
断开手柄（elem）
/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/Theano/gradient.py:533:
用户警告：要求grad方法计算变量的梯度
这不是成本计算图的一部分，或仅由
不可微算子：
句柄\u已断开（rval[i]）
/usr/local/lib/python2.7/dist-packages/Theano-0.6.0-py2.7.egg/Theano/gradient.py:513:
用户警告：要求grad方法计算变量的梯度
这不是成本计算图的一部分，或仅由
不可微算子：θ_1
断开手柄（elem）
回溯（最近一次呼叫最后一次）：
文件“”，第43行，在
文件“/usr/local/lib/python2.7/dist packages/pymc-3.0-py2.7.egg/pymc/tuning/starting.py”，第80行，在find_映射中
开始），fprime=grad\u logp\u o，disp=disp，*args，**kwargs）
文件“/usr/lib/python2.7/dist packages/scipy/optimize/optimize.py”，第777行，在fmin_bfgs中
res=\u最小化\u bfgs（f，x0，args，fprime，callback=callback，**选项）
文件“/usr/lib/python2.7/dist packages/scipy/optimize/optimize.py”，第837行，在
旧的fval=f（x0）
文件“/usr/lib/python2.7/dist packages/scipy/optimize/optimize.py”，第281行，在函数包装器中
返回函数（*（包装器参数+参数））
文件“/usr/local/lib/python2.7/dist packages/pymc-3.0-py2.7.egg/pymc/tuning/starting.py”，第72行，在logp_o中
将nan_返回至_高（-logp（点））
文件“/usr/local/lib/python2.7/dist packages/pymc-3.0-py2.7.egg/pymc/blocking.py”，第119行，在调用__
返回self.fa（self.fb（x））
文件“/usr/local/lib/python2.7/dist packages/pymc-3.0-py2.7.egg/pymc/model.py”，第283行，在调用中__
返回self.f（**状态）
文件“/usr/local/lib/python2.7/dist packages/Theano-0.6.0-py2.7.egg/Theano/compile/function_module.py”，第482行，在调用中__
raise TypeError（“传递给THANO函数的参数太多”）
TypeError:传递给theano函数的参数太多

---

UserWarning与起始点的优化有关，如果我不使用pm.find_MAP（），它将被删除。其余的错误仍然存在。

如果在循环中定义PyMC对象，则必须在每次迭代中为它们指定不同的名称。例如，您可以定义：

mu = pm.Beta('mu_%i' % condition, 1, 1)

---

这应该可以消除您得到的错误。

我注意到的一件事是，您每次添加条件时都在采样，我认为您可能希望将其从循环中拉出来

此外，您不需要为每个条件的每个mu、θ、y定义单独的变量。 例如，如果您的数据位于列中的

data

，那么您应该能够执行以下操作

with pm.Model() as model:

    kappa = pm.Gamma('kappa', 1, 0.1)

    mu = pm.Beta('mu', 1, 1, shape=ncond)

    mu_c = mu[data.condition]
    theta = pm.Beta('theta', mu_c * kappa, (1 - mu_c) * kappa, shape=len(data))

    y = pm.Binomial('y', p=theta, n=data.trials, observed=data.z_cond)

---

谢谢，这很有道理，但现在我又犯了一个错误。我编辑问题以添加新错误。

with pm.Model() as model:

    kappa = pm.Gamma('kappa', 1, 0.1)

    mu = pm.Beta('mu', 1, 1, shape=ncond)

    mu_c = mu[data.condition]
    theta = pm.Beta('theta', mu_c * kappa, (1 - mu_c) * kappa, shape=len(data))

    y = pm.Binomial('y', p=theta, n=data.trials, observed=data.z_cond)