如何在 PyMC3 中定义一个模型，其中一个参数在多个条件下限制为相同的值

Question

我想写一个模型，如下所示。主要思想是我有几个条件（或治疗），所有参数都是独立估计每个条件的，除了对所有条件都相同的 kappa 参数。

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，我收到以下消息。

/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.

编辑 按照 chris-fonnesbeck 的回答，我尝试了以下方法：

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_%i' % condition, 1, 1)
        # define the prior
        theta = pm.Beta('theta_%i' % condition, mu * kappa, (1 - mu) * kappa, shape=len(z_cond))
        # define the likelihood
        y = pm.Binomial('y_%i' % condition, 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(10000, [step1, step2], start=start, progressbar=False)
        trace_per_condition.append(trace)

我得到错误：

/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

UserWarning 与起点的优化有关，如果我不使用 pm.find_MAP()，则将其删除。其余的错误仍然存​​在。

Answer 1

我注意到的一件事是，您每次添加条件时都会进行采样，我认为您可能希望将其拉出循环。

此外，您无需为每个条件的每个 mu、theta、y 定义单独的变量。 例如，如果您的数据在列中的data 中，您应该能够执行类似的操作

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)

Answer 2

如果您在循环中定义 PyMC 对象，则必须在每次迭代时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)

这应该可以消除您遇到的错误。