Bayesian Neural Network (VI) for regression

Zhenwen Dai (2018-8-21)

import mxfusion as mf
import mxnet as mx
import numpy as np
import mxnet.gluon.nn as nn
import mxfusion.components
import mxfusion.inference

/Users/zhenwend/anaconda3/lib/python3.6/site-packages/h5py/__init__.py:36: FutureWarning: Conversion of the second argument of issubdtype from `float` to `np.floating` is deprecated. In future, it will be treated as `np.float64 == np.dtype(float).type`.
  from ._conv import register_converters as _register_converters

Generate Synthetic Data

import GPy
%matplotlib inline
from pylab import *

k = GPy.kern.RBF(1, lengthscale=0.1)
x = np.random.rand(1000,1)
y = np.random.multivariate_normal(mean=np.zeros((1000,)), cov=k.K(x), size=(1,)).T
plot(x[:,0], y[:,0], '.')

[<matplotlib.lines.Line2D at 0x1097ed358>]

Model definition

D = 50
net = nn.HybridSequential(prefix='nn_')
with net.name_scope():
    net.add(nn.Dense(D, activation="tanh"))
    net.add(nn.Dense(D, activation="tanh"))
    net.add(nn.Dense(1, flatten=True))
net.initialize(mx.init.Xavier(magnitude=3))
_=net(mx.nd.array(x))

from mxfusion.components.var_trans import PositiveTransformation
from mxfusion.inference import VariationalPosteriorForwardSampling

m = mf.core.Model()
m.N = mf.components.Variable()
m.f = mf.components.functions.MXFusionGluonFunction(net, nOutputs=1,broadcastable=False)
m.x = mf.components.Variable(shape=(m.N,1))
m.v = mf.components.Variable(shape=(1,), transformation=PositiveTransformation(), initial_value=mx.nd.array([0.01]))
#m.prior_variance = mf.core.Variable(shape=(1,), transformation=PositiveTransformation())
m.r = m.f(m.x)
for _,v in m.r.factor.block_variables:
    v.set_prior(mf.components.distributions.Normal(mean=mx.nd.array([0]),variance=mx.nd.array([1.])))
m.y = mf.components.distributions.Normal.define_variable(mean=m.r, variance=m.v, shape=(m.N,1))
m.show()

Variable(24825) ~ Normal(mean=Variable(3706e), variance=Variable(78c3f))
Variable(1f4c5) ~ Normal(mean=Variable(d8e05), variance=Variable(b73d6))
Variable(1dbe9) ~ Normal(mean=Variable(6e428), variance=Variable(ee4e8))
Variable(9abb8) ~ Normal(mean=Variable(f8cb1), variance=Variable(73d44))
Variable(cdf2a) ~ Normal(mean=Variable(b4c6f), variance=Variable(5f593))
Variable(e28f4) ~ Normal(mean=Variable(528e0), variance=Variable(c17f7))
r = GluonFunctionEvaluation(nn_dense0_weight=Variable(e28f4), nn_dense0_bias=Variable(cdf2a), nn_dense1_weight=Variable(9abb8), nn_dense1_bias=Variable(1dbe9), nn_dense2_weight=Variable(1f4c5), nn_dense2_bias=Variable(24825), nn_input_0=x)
y ~ Normal(mean=r, variance=v)

Inference with Meanfield

from mxfusion.inference import BatchInferenceLoop, create_Gaussian_meanfield, GradBasedInference, StochasticVariationalInference

observed = [m.y, m.x]
q = create_Gaussian_meanfield(model=m, observed=observed)
alg = StochasticVariationalInference(num_samples=3, model=m, posterior=q, observed=observed)
infr = GradBasedInference(inference_algorithm=alg, grad_loop=BatchInferenceLoop())

infr.initialize(y=mx.nd.array(y), x=mx.nd.array(x))

 /Users/zhenwend/mxfusion/src/MXFusion/mxfusion/inference/inference_parameters.py:52: UserWarning:InferenceParameters has already been initialized.  The existing one will be overwritten.

for v_name, v in m.r.factor.block_variables:
    uuid = v.uuid
    loc_uuid = infr.inference_algorithm.posterior[uuid].factor.variance.uuid
    a = infr.params.param_dict[loc_uuid].data().asnumpy()
    a[:] = 1e-6
    infr.params[infr.inference_algorithm.posterior[uuid].factor.mean] = net.collect_params()[v_name].data()
    infr.params[infr.inference_algorithm.posterior[uuid].factor.variance] = mx.nd.array(a)

infr.run(max_iter=2000, learning_rate=1e-2, y=mx.nd.array(y), x=mx.nd.array(x), verbose=True)

 /Users/zhenwend/mxfusion/src/MXFusion/mxfusion/inference/inference.py:111: UserWarning:Trying to initialize the inference twice, skipping.

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Iteration 1853 logL: -1407.312255859375
Iteration 1854 logL: -1477.843505859375
Iteration 1855 logL: -1237.176025390625
Iteration 1856 logL: -1005.63037109375
Iteration 1857 logL: -1027.980224609375
Iteration 1858 logL: -1343.48583984375
Iteration 1859 logL: -1054.12548828125
Iteration 1860 logL: -1100.2314453125
Iteration 1861 logL: -940.71484375
Iteration 1862 logL: -1200.455322265625
Iteration 1863 logL: -1142.964111328125
Iteration 1864 logL: -905.65185546875
Iteration 1865 logL: -1021.826416015625
Iteration 1866 logL: -1014.808349609375
Iteration 1867 logL: -1071.56103515625
Iteration 1868 logL: -1333.482177734375
Iteration 1869 logL: -963.418701171875
Iteration 1870 logL: -1064.89697265625
Iteration 1871 logL: -1108.72900390625
Iteration 1872 logL: -1067.771484375
Iteration 1873 logL: -1127.8828125
Iteration 1874 logL: -1081.60302734375
Iteration 1875 logL: -965.2626953125
Iteration 1876 logL: -988.87255859375
Iteration 1877 logL: -1065.111083984375
Iteration 1878 logL: -998.36376953125
Iteration 1879 logL: -987.1142578125
Iteration 1880 logL: -1086.422119140625
Iteration 1881 logL: -1011.31591796875
Iteration 1882 logL: -1096.4521484375
Iteration 1883 logL: -1229.218994140625
Iteration 1884 logL: -1258.369873046875
Iteration 1885 logL: -1353.3291015625
Iteration 1886 logL: -857.21875
Iteration 1887 logL: -1257.5390625
Iteration 1888 logL: -983.54736328125
Iteration 1889 logL: -1463.050048828125
Iteration 1890 logL: -902.812255859375
Iteration 1891 logL: -1015.60205078125
Iteration 1892 logL: -937.58642578125
Iteration 1893 logL: -973.899169921875
Iteration 1894 logL: -1120.499755859375
Iteration 1895 logL: -1576.31640625
Iteration 1896 logL: -1102.259521484375
Iteration 1897 logL: -1324.40478515625
Iteration 1898 logL: -1041.040771484375
Iteration 1899 logL: -1416.749267578125
Iteration 1900 logL: -1493.891357421875
Iteration 1901 logL: -1039.11279296875
Iteration 1902 logL: -1012.100341796875
Iteration 1903 logL: -1212.315673828125
Iteration 1904 logL: -1539.478515625
Iteration 1905 logL: -1077.4384765625
Iteration 1906 logL: -992.618408203125
Iteration 1907 logL: -979.22802734375
Iteration 1908 logL: -1281.15771484375
Iteration 1909 logL: -1554.8798828125
Iteration 1910 logL: -1005.364990234375
Iteration 1911 logL: -899.75537109375
Iteration 1912 logL: -1055.173095703125
Iteration 1913 logL: -1163.792724609375
Iteration 1914 logL: -1157.046142578125
Iteration 1915 logL: -1776.942626953125
Iteration 1916 logL: -1032.344970703125
Iteration 1917 logL: -1102.173095703125
Iteration 1918 logL: -1496.106201171875
Iteration 1919 logL: -1098.893310546875
Iteration 1920 logL: -1040.317138671875
Iteration 1921 logL: -1673.8388671875
Iteration 1922 logL: -1684.88623046875
Iteration 1923 logL: -1116.3251953125
Iteration 1924 logL: -1416.927001953125
Iteration 1925 logL: -987.152587890625
Iteration 1926 logL: -1066.673095703125
Iteration 1927 logL: -1040.917724609375
Iteration 1928 logL: -1074.169921875
Iteration 1929 logL: -1119.952880859375
Iteration 1930 logL: -1414.101806640625
Iteration 1931 logL: -876.994384765625
Iteration 1932 logL: -971.498046875
Iteration 1933 logL: -1136.347412109375
Iteration 1934 logL: -1456.970703125
Iteration 1935 logL: -1219.20166015625
Iteration 1936 logL: -991.05908203125
Iteration 1937 logL: -1131.446044921875
Iteration 1938 logL: -944.871337890625
Iteration 1939 logL: -915.4677734375
Iteration 1940 logL: -1117.62548828125
Iteration 1941 logL: -1233.37841796875
Iteration 1942 logL: -1017.934814453125
Iteration 1943 logL: -832.17529296875
Iteration 1944 logL: -989.371826171875
Iteration 1945 logL: -1345.60302734375
Iteration 1946 logL: -1143.80224609375
Iteration 1947 logL: -1175.467041015625
Iteration 1948 logL: -1102.8701171875
Iteration 1949 logL: -1056.500244140625
Iteration 1950 logL: -951.403564453125
Iteration 1951 logL: -856.87353515625
Iteration 1952 logL: -1007.120849609375
Iteration 1953 logL: -896.989013671875
Iteration 1954 logL: -1194.041259765625
Iteration 1955 logL: -1053.199462890625
Iteration 1956 logL: -865.03466796875
Iteration 1957 logL: -1246.1025390625
Iteration 1958 logL: -1341.726318359375
Iteration 1959 logL: -889.03857421875
Iteration 1960 logL: -1292.580322265625
Iteration 1961 logL: -921.563232421875
Iteration 1962 logL: -1567.103759765625
Iteration 1963 logL: -1208.059814453125
Iteration 1964 logL: -921.08056640625
Iteration 1965 logL: -1033.3154296875
Iteration 1966 logL: -1243.2275390625
Iteration 1967 logL: -1147.265380859375
Iteration 1968 logL: -1195.745361328125
Iteration 1969 logL: -827.04443359375
Iteration 1970 logL: -1138.609375
Iteration 1971 logL: -975.8818359375
Iteration 1972 logL: -1208.3505859375
Iteration 1973 logL: -855.1357421875
Iteration 1974 logL: -957.8818359375
Iteration 1975 logL: -963.307373046875
Iteration 1976 logL: -1086.295166015625
Iteration 1977 logL: -902.845703125
Iteration 1978 logL: -1262.313232421875
Iteration 1979 logL: -1039.027587890625
Iteration 1980 logL: -892.798095703125
Iteration 1981 logL: -943.1220703125
Iteration 1982 logL: -1156.06591796875
Iteration 1983 logL: -1353.580078125
Iteration 1984 logL: -856.92529296875
Iteration 1985 logL: -1020.078857421875
Iteration 1986 logL: -1095.52685546875
Iteration 1987 logL: -1237.697265625
Iteration 1988 logL: -998.10693359375
Iteration 1989 logL: -1152.889892578125
Iteration 1990 logL: -963.375732421875
Iteration 1991 logL: -925.7431640625
Iteration 1992 logL: -906.114013671875
Iteration 1993 logL: -1085.943359375
Iteration 1994 logL: -1247.585693359375
Iteration 1995 logL: -931.400146484375
Iteration 1996 logL: -993.2607421875
Iteration 1997 logL: -1423.531005859375
Iteration 1998 logL: -968.134521484375
Iteration 1999 logL: -927.2412109375
Iteration 2000 logL: -987.966064453125

Use prediction to visualize the resulting BNN

xt = np.linspace(0,1,100)[:,None]

infr2 = VariationalPosteriorForwardSampling(10, [m.x], infr, [m.r])
res = infr2.run(x=mx.nd.array(xt))

 /Users/zhenwend/mxfusion/src/MXFusion/mxfusion/core/factor_graph.py:65: UserWarning:The value N has already been assigned in the model.
 /Users/zhenwend/mxfusion/src/MXFusion/mxfusion/core/factor_graph.py:65: UserWarning:The value y has already been assigned in the model.
 /Users/zhenwend/mxfusion/src/MXFusion/mxfusion/inference/inference_parameters.py:52: UserWarning:InferenceParameters has already been initialized.  The existing one will be overwritten.

yt = res[m.r].asnumpy()

# plot(xt[:,0],yt[:,0])
yt_mean = yt.mean(0)
yt_std = yt.std(0)
#plot(xt[:,0], yt.mean(0)[:,0])
#errorbar(xt[:,0],y=yt_mean[:,0],yerr=yt_std[:,0]*2)
for i in range(yt.shape[0]):
    plot(xt[:,0],yt[i,:,0],'k',alpha=0.2)
plot(x[:,0],y[:,0],'.')

[<matplotlib.lines.Line2D at 0x1a26059b00>]