在PyMC3中使用BetaBinomial

Question

我有一张二进制结果计数表，我想拟合一个β二项分布来估计$ \\ alpha $和$ \\ beta $参数，但是当我尝试以我的方式拟合/抽样模型分布时，我遇到了错误针对其他情况：

import pymc3 as pm
import pandas as pd

df = pd.read_csv('~/data.csv', low_memory=False)
df = df[df.Clicks >= 0]

C0=df.C.values
I0=df.N.values
N0 = C0 + I0

with pm.Model() as model:
    C=pm.constant(C0)
    I=pm.constant(I0)
    C1=pm.constant(C0 + 1)
    I1=pm.constant(I0 + 1)
    N=pm.constant(N0)
    alpha = pm.Exponential('alpha', 1/(C0.sum()+1))
    beta = pm.Exponential('beta', 1/(I0.sum()+1))
    obs = pm.BetaBinomial('obs', alpha, beta, N, observed=C0)


with model:
    advi_fit = pm.variational.advi(n=int(1e4))
    trace1 = pm.variational.sample_vp(advi_fit, draws=int(1e4))

pm.traceplot(trace1[::10])

with model:
    step = pm.NUTS()
    #step = pm.Metropolis() # <== same problem
    trace2 = pm.sample(int(1e3), step)

pm.traceplot(trace2[::10])

在这两种情况下，采样均失败：

MissingInputError: ("An input of the graph, used to compute Elemwise{neg,no_inplace}(P_logodds_), was not provided and not given a value.Use the Theano flag exception_verbosity='high',for more information on this error.", P_logodds

在advi情况下，完整堆栈跟踪为：

---

MissingInputError                         Traceback (most recent call last)
<ipython-input-46-8947c7c798e5> in <module>()
----> 1 import codecs, os;__pyfile = codecs.open('''/tmp/py7996Jip''', encoding='''utf-8''');__code = __pyfile.read().encode('''utf-8''');__pyfile.close();os.remove('''/tmp/py7996Jip''');exec(compile(__code, '''/home/dmahler/Scripts/adops-bayes2.py''', 'exec'));

/home/dmahler/Scripts/adops-bayes2.py in <module>()
     59     advi_fit = pm.variational.advi(n=int(J*6.4e4), learning_rate=1e-3/J, epsilon=1e-8, accurate_elbo=False)
     60     #advi_fit = pm.variational.advi_minibatch(minibatch_RVs=[alpha, beta, p], minibatch_tensors=[C,I,N])
---> 61     trace = pm.variational.sample_vp(advi_fit, draws=int(2e4))
     62 
     63 pm.traceplot(trace[::10])

/home/dmahler/Anaconda/lib/python2.7/site-packages/pymc3/variational/advi.pyc in sample_vp(vparams, draws, model, local_RVs, random_seed, hide_transformed)
    317 
    318     varnames = [str(var) for var in model.unobserved_RVs]
--> 319     trace = NDArray(model=model, vars=vars_sampled)
    320     trace.setup(draws=draws, chain=0)
    321 

/home/dmahler/Anaconda/lib/python2.7/site-packages/pymc3/backends/ndarray.pyc in __init__(self, name, model, vars)
     21     """
     22     def __init__(self, name=None, model=None, vars=None):
---> 23         super(NDArray, self).__init__(name, model, vars)
     24         self.draw_idx = 0
     25         self.draws = None

/home/dmahler/Anaconda/lib/python2.7/site-packages/pymc3/backends/base.pyc in __init__(self, name, model, vars)
     34         self.vars = vars
     35         self.varnames = [var.name for var in vars]
---> 36         self.fn = model.fastfn(vars)
     37 
     38 

/home/dmahler/Anaconda/lib/python2.7/site-packages/pymc3/model.pyc in fastfn(self, outs, mode, *args, **kwargs)
    374         Compiled Theano function as point function.
    375         """
--> 376         f = self.makefn(outs, mode, *args, **kwargs)
    377         return FastPointFunc(f)
    378 

/home/dmahler/Anaconda/lib/python2.7/site-packages/pymc3/memoize.pyc in memoizer(*args, **kwargs)
     12 
     13         if key not in cache:
---> 14             cache[key] = obj(*args, **kwargs)
     15 
     16         return cache[key]

/home/dmahler/Anaconda/lib/python2.7/site-packages/pymc3/model.pyc in makefn(self, outs, mode, *args, **kwargs)
    344                                on_unused_input='ignore',
    345                                accept_inplace=True,
--> 346                                mode=mode, *args, **kwargs)
    347 
    348     def fn(self, outs, mode=None, *args, **kwargs):

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/compile/function.pyc in function(inputs, outputs, mode, updates, givens, no_default_updates, accept_inplace, name, rebuild_strict, allow_input_downcast, profile, on_unused_input)
    318                    on_unused_input=on_unused_input,
    319                    profile=profile,
--> 320                    output_keys=output_keys)
    321     # We need to add the flag check_aliased inputs if we have any mutable or
    322     # borrowed used defined inputs

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/compile/pfunc.pyc in pfunc(params, outputs, mode, updates, givens, no_default_updates, accept_inplace, name, rebuild_strict, allow_input_downcast, profile, on_unused_input, output_keys)
    477                          accept_inplace=accept_inplace, name=name,
    478                          profile=profile, on_unused_input=on_unused_input,
--> 479                          output_keys=output_keys)
    480 
    481 

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/compile/function_module.pyc in orig_function(inputs, outputs, mode, accept_inplace, name, profile, on_unused_input, output_keys)
   1774                    profile=profile,
   1775                    on_unused_input=on_unused_input,
-> 1776                    output_keys=output_keys).create(
   1777             defaults)
   1778 

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/compile/function_module.pyc in __init__(self, inputs, outputs, mode, accept_inplace, function_builder, profile, on_unused_input, fgraph, output_keys)
   1426             # OUTPUT VARIABLES)
   1427             fgraph, additional_outputs = std_fgraph(inputs, outputs,
-> 1428                                                     accept_inplace)
   1429             fgraph.profile = profile
   1430         else:

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/compile/function_module.pyc in std_fgraph(input_specs, output_specs, accept_inplace)
    175 
    176     fgraph = gof.fg.FunctionGraph(orig_inputs, orig_outputs,
--> 177                                   update_mapping=update_mapping)
    178 
    179     for node in fgraph.apply_nodes:

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/gof/fg.pyc in __init__(self, inputs, outputs, features, clone, update_mapping)
    169 
    170         for output in outputs:
--> 171             self.__import_r__(output, reason="init")
    172         for i, output in enumerate(outputs):
    173             output.clients.append(('output', i))

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/gof/fg.pyc in __import_r__(self, variable, reason)
    358         # Imports the owners of the variables
    359         if variable.owner and variable.owner not in self.apply_nodes:
--> 360                 self.__import__(variable.owner, reason=reason)
    361         if (variable.owner is None and
    362                 not isinstance(variable, graph.Constant) and

/home/dmahler/Anaconda/lib/python2.7/site-packages/theano/gof/fg.pyc in __import__(self, apply_node, check, reason)
    472                             "for more information on this error."
    473                             % str(node)),
--> 474                             r)
    475 
    476         for node in new_nodes:

MissingInputError: ("An input of the graph, used to compute Elemwise{neg,no_inplace}(P_logodds_), was not provided and not given a value.Use the Theano flag exception_verbosity='high',for more information on this error.", P_logodds_)
> /home/dmahler/Anaconda/lib/python2.7/site-packages/theano/gof/fg.py(474)__import__()
    472                             "for more information on this error."
    473                             % str(node)),
--> 474                             r)
    475 
    476         for node in new_nodes:

在意识到pymc3.BetaBinomial之前，我已经尝试使用单独的Beta和Binomial实例来实现相同的结果：

with pm.Model() as model:
    C=pm.constant(C0)
    I=pm.constant(I0)
    C1=pm.constant(C0 + 1)
    I1=pm.constant(I0 + 1)
    N=pm.constant(N0)
    alpha = pm.Exponential('alpha', 1/(C0.sum()+1))
    beta = pm.Exponential('beta', 1/(I0.sum()+1))
    p = pm.Beta('P', alpha, beta,  shape=K)
    b = pm.Binomial('B', N, p, observed=C0)

这样可以成功完成，但是不同的方法会产生不同的结果。 我认为这可能部分是由于先验之间的间接性更高，并且观察结果使搜索空间更大。 当我遇到BetaBinomial时，我发现它不仅使搜索变得正确，而且使它成为正确的^ TM 。 否则，我认为to模型在逻辑上应该等效。 不幸的是，我无法弄清楚如何使batebinomial工作，并且无法在BetaBinomial上找到使用BetaBinomial任何示例。

• 如何使BetaBinomial修饰符起作用？
• 这些模型在逻辑上真的等效吗？
• 有人对初始分层版本的数字问题的原因有更好的猜测吗？
  • 我该如何解决？

Answer 1

您的模型应该运行，您可以编写此代码

with pm.Model() as model:
    alpha = pm.Exponential('alpha', 1/(C0.sum()+1))
    beta = pm.Exponential('beta', 1/(I0.sum()+1))
    obs = pm.BetaBinomial('obs', alpha, beta, N0, observed=C0)

也就是说，（C，I C1，I1）在您的模型中定义但未被使用。 无论如何，这不是问题。 您收到的错误是因为PyMC3期望变量P （就像您在第二个模型中一样），但是未定义该变量。 可能是您正在使用Jupyter笔记本，并且删除/注释了theano变量。 尝试再次运行笔记本电脑。

从理论上讲，使用Beta和二项式或BetaBinomial可以得出相同的结果。 从实际的角度来看。 从BetaBinomial进行采样应该比从beta和二项式采样更快，因为部分工作已经进行了分析！

假设采样正确，两个模型应提供相同的结果。 要检查两个结果是否相等，请尝试增加样本数量（并避免稀化 ）。 还要比较模型之间和模型内部的结果（差异应该大致相同）。 如果您不需要估计P变量（β分布），请使用BetaBinomial。