在 PyMC3 中使用 BetaBinomial

Question

我有一个 table 的二元结果计数，我想拟合一个 beta 二项分布来估计 $\alpha$ 和 $\beta$ 参数，但是当我尝试时出现错误fit/sample 我在其他情况下的模型分布方式：

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，采样都失败了：

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 的情况下，完整的堆栈跟踪是：

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:

在我意识到 pymc3.BetaBinomial 之前，我正在尝试使用单独的 Beta 和 Binomial 实例来获得相同的结果：

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)

这已成功完成，但不同的方法会产生截然不同的结果。我认为这可能部分是由于先验和观察之间的额外间接级别使搜索 space 更大。当我遇到 BetaBinomial 时，我认为它会使搜索更容易，同时也是 正确的事情 ^TM。否则我相信 to 模型在逻辑上应该是等价的。不幸的是，我无法弄清楚如何使 batebinomial 工作，并且我无法在互联网上找到任何使用 BetaBinomial 的示例。

如何使 BetaBinomial modwel 工作？
这些模型在逻辑上真的是等价的吗？
有没有人能更好地猜测初始分层版本的数值问题的原因？
- 我该如何修复它们？

Answer 1

你的模型应该运行，你可以这样写

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) 已在您的模型中定义但未使用。无论如何，这不是问题。您得到的错误是因为 PyMC3 需要一个变量 P （就像在您拥有的第二个模型中一样）但未定义该变量。可能你正在使用 Jupyter 笔记本并且你 delete/comment 一个 theano 变量。再次尝试运行ning Notebook。

理论上使用 Beta 和 Binomial 或 BetaBinomial 应该得到相同的结果。从实用的角度来看。从 BetaBinomial 中抽样应该比从 Beta 和二项式中抽样更快，因为部分工作已经通过分析完成！

假设正确采样，两个模型应该提供相同的结果。要检查两个结果是否相同，请尝试增加样本数量（并避免 thinning）。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（beta 分布），则使用 BetaBinomial。

在 PyMC3 中使用 BetaBinomial

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