Deep Probabilistic Programming Library

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MindSpore deep probabilistic programming is to combine Bayesian learning with deep learning, including probability distribution, probability distribution mapping, deep probability network, probability inference algorithm, Bayesian layer, Bayesian conversion, and Bayesian toolkit. For professional Bayesian learning users, it provides probability sampling, inference algorithms, and model build libraries. On the other hand, advanced APIs are provided for users who are unfamiliar with Bayesian deep learning, so that they can use Bayesian models without changing the deep learning programming logic.

Probability Distribution

Probability distribution (mindspore.nn.probability.distribution) is the basis of probabilistic programming. The Distribution class provides various probability statistics APIs, such as pdf for probability density, cdf for cumulative density, kl_loss for divergence calculation, and sample for sampling. Existing probability distribution examples include Gaussian distribution, Bernoulli distribution, exponential distribution, geometric distribution, and uniform distribution.

Probability Distribution Class

  • Distribution: base class of all probability distributions.

  • Bernoulli: Bernoulli distribution, with a parameter indicating the number of experiment successes.

  • Exponential: exponential distribution, with a rate parameter.

  • Geometric: geometric distribution, with a parameter indicating the probability of initial experiment success.

  • Normal: normal distribution (Gaussian distribution), with two parameters indicating the average value and standard deviation.

  • Uniform: uniform distribution, with two parameters indicating the minimum and maximum values on the axis.

  • Categorical: categorical distribution, with one parameter indicating the probability of each category.

  • Cauchy: cauchy distribution, with two parameters indicating the location and scale.

  • LogNormal: lognormal distribution, with two parameters indicating the location and scale.

  • Logistic: logistic distribution, with two parameters indicating the location and scale.

  • Gumbel: gumbel distribution, with two parameters indicating the location and scale.

Distribution Base Class

Distribution is the base class for all probability distributions.

The Distribution class supports the following functions: prob, log_prob, cdf, log_cdf, survival_function, log_survival, mean, sd, var, entropy, kl_loss, cross_entropy, and sample. The input parameters vary according to the distribution. These functions can be used only in a derived class and their parameters are determined by the function implementation of the derived class.

  • prob: probability density function (PDF) or probability quality function (PMF)

  • log_prob: log-like function

  • cdf: cumulative distribution function (CDF)

  • log_cdf: log-cumulative distribution function

  • survival_function: survival function

  • log_survival: logarithmic survival function

  • mean: average value

  • sd: standard deviation

  • var: variance

  • entropy: entropy

  • kl_loss: Kullback-Leibler divergence

  • cross_entropy: cross entropy of two probability distributions

  • sample: random sampling of probability distribution

  • get_dist_args: returns the parameters of the distribution used in the network

  • get_dist_name: returns the type of the distribution

Bernoulli Distribution

Bernoulli distribution, inherited from the Distribution class.

Properties are described as follows:

  • Bernoulli.probs: returns the probability of success in the Bernoulli experiment as a Tensor.

The Distribution base class invokes the private API in the Bernoulli to implement the public APIs in the base class. Bernoulli supports the following public APIs:

  • mean,mode,var, and sd: The input parameter probs1 that indicates the probability of experiment success is optional.

  • entropy: The input parameter probs1 that indicates the probability of experiment success is optional.

  • cross_entropy and kl_loss: The input parameters dist and probs1_b are mandatory. dist indicates another distribution type. Currently, only ‘Bernoulli’ is supported. probs1_b is the experiment success probability of distribution b. Parameter probs1_a of distribution a is optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. The input parameter probs that indicates the probability of experiment success is optional.

  • sample: Optional input parameters include sample shape shape and experiment success probability probs1.

  • get_dist_args: The input parameter probs1 that indicates the probability of experiment success is optional. Return (probs1,) with type tuple.

  • get_dist_type: return ‘Bernoulli’.

Exponential Distribution

Exponential distribution, inherited from the Distribution class.

Properties are described as follows:

  • Exponential.rate: returns the rate parameter as a Tensor.

The Distribution base class invokes the Exponential private API to implement the public APIs in the base class. Exponential supports the following public APIs:

  • mean,mode,var, and sd: The input rate parameter rate is optional.

  • entropy: The input rate parameter rate is optional.

  • cross_entropy and kl_loss: The input parameters dist and rate_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Exponential’ is supported. rate_b is the rate parameter of distribution b. Parameter rate_a of distribution a is optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. The input rate parameter rateis optional.

  • sample: Optional input parameters include sample shape shape and rate parameter rate.

  • get_dist_args: The input rate parameter rate is optional. Return (rate,) with type tuple.

  • get_dist_type: returns ‘Exponential’.

Geometric Distribution

Geometric distribution, inherited from the Distribution class.

Properties are described as follows:

  • Geometric.probs: returns the probability of success in the Bernoulli experiment as a Tensor.

The Distribution base class invokes the private API in the Geometric to implement the public APIs in the base class. Geometric supports the following public APIs:

  • mean,mode,var, and sd: The input parameter probs1 that indicates the probability of experiment success is optional.

  • entropy: The input parameter probs1 that indicates the probability of experiment success is optional.

  • cross_entropy and kl_loss: The input parameters dist and probs1_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Geometric’ is supported. probs1_b is the experiment success probability of distribution b. Parameter probs1_a of distribution a is optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. The input parameter probs1 that indicates the probability of experiment success is optional.

  • sample: Optional input parameters include sample shape shape and experiment success probability probs1.

  • get_dist_args: The input parameter probs1 that indicates the probability of experiment success is optional. Return (probs1,) with type tuple.

  • get_dist_type: returns ‘Geometric’.

Normal Distribution

Normal distribution (also known as Gaussian distribution), inherited from the Distribution class.

The Distribution base class invokes the private API in the Normal to implement the public APIs in the base class. Normal supports the following public APIs:

  • mean,mode,var, and sd: Input parameters mean (for average value) and sd (for standard deviation) are optional.

  • entropy: Input parameters mean (for average value) and sd (for standard deviation) are optional.

  • cross_entropy and kl_loss: The input parameters dist, mean_b, and sd_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Normal’ is supported. mean_b and sd_b indicate the mean value and standard deviation of distribution b, respectively. Input parameters mean value mean_a and standard deviation sd_a of distribution a are optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. Input parameters mean value mean_a and standard deviation sd_a are optional.

  • sample: Input parameters sample shape shape, average value mean_a, and standard deviation sd_a are optional.

  • get_dist_args: Input parameters mean value mean and standard deviation sd are optional. Return (mean, sd) with type tuple.

  • get_dist_type: returns ‘Normal’.

Uniform Distribution

Uniform distribution, inherited from the Distribution class.

Properties are described as follows:

  • Uniform.low: returns the minimum value as a Tensor.

  • Uniform.high: returns the maximum value as a Tensor.

The Distribution base class invokes Uniform to implement public APIs in the base class. Uniform supports the following public APIs:

  • mean,mode,var, and sd: Input parameters maximum value high and minimum value low are optional.

  • entropy: Input parameters maximum value high and minimum value low are optional.

  • cross_entropy and kl_loss: The input parameters dist, high_b, and low_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Uniform’ is supported. high_b and low_b are parameters of distribution b. Input parameters maximum value high and minimum value low of distribution a are optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. Input parameters maximum value high and minimum value low are optional.

  • sample: Input parameters shape, maximum value high, and minimum value low are optional.

  • get_dist_args: Input parameters maximum value high and minimum value low are optional. Return (low, high) with type tuple.

  • get_dist_type: returns ‘Uniform’.

Categorical Distribution

Categorical distribution, inherited from the Distribution class.

Properties are described as follows:

  • Categorical.probs: returns the probability of each category as a Tensor.

The Distribution base class invokes the private API in the Categorical to implement the public APIs in the base class. Categorical supports the following public APIs:

  • mean,mode,var, and sd: The input parameter probs that indicates the probability of each category is optional.

  • entropy: The input parameter probs that indicates the probability of each category is optional.

  • cross_entropy and kl_loss: The input parameters dist and probs_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Categorical’ is supported. probs_b is the categories’ probabilities of distribution b. Parameter probs_a of distribution a is optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. The input parameter probs that indicates the probability of each category is optional.

  • sample: Optional input parameters include sample shape shape and the categories’ probabilities probs.

  • get_dist_args: The input parameter probs that indicates the probability of each category is optional. Return (probs,) with type tuple.

  • get_dist_type: returns ‘Categorical’.

Cauchy Distribution

Cauchy distribution, inherited from the Distribution class.

Properties are described as follows:

  • Cauchy.loc: returns the location parameter as a Tensor.

  • Cauchy.scale: returns the scale parameter as a Tensor.

The Distribution base class invokes the private API in the Cauchy to implement the public APIs in the base class. Cauchy supports the following public APIs:

  • entropy: Input parameters loc (for location) and scale (for scale) are optional.

  • cross_entropy and kl_loss: The input parameters dist, loc_b, and scale_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Cauchy’ is supported. loc_b and scale_b indicate the location and scale of distribution b, respectively. Input parameters loc and scale of distribution a are optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. Input parameters location loc and scale scale are optional.

  • sample: Input parameters sample shape shape, location loc and scale scale are optional.

  • get_dist_args: Input parameters location loc and scale scale are optional. Return (loc, scale) with type tuple.

  • get_dist_type: returns ‘Cauchy’.

LogNormal Distribution

LogNormal distribution, inherited from the TransformedDistribution class, constructed by Exp Bijector and Normal Distribution.

Properties are described as follows:

  • LogNormal.loc: returns the location parameter as a Tensor.

  • LogNormal.scale: returns the scale parameter as a Tensor.

The Distribution base class invokes the private API in the LogNormal and TransformedDistribution to implement the public APIs in the base class. LogNormal supports the following public APIs:

  • mean,mode,var, and sd：Input parameters loc (for location) and scale (for scale) are optional.

  • entropy: Input parameters loc (for location) and scale (for scale) are optional.

  • cross_entropy and kl_loss: The input parameters dist, loc_b, and scale_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘LogNormal’ is supported. loc_b and scale_b indicate the location and scale of distribution b, respectively. Input parameters loc and scale of distribution a are optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. Input parameters location loc and scale scale are optional.

  • sample: Input parameters sample shape shape, location loc and scale scale are optional.

  • get_dist_args: Input parameters location loc and scale scale are optional. Return (loc, scale) with type tuple.

  • get_dist_type: returns ‘LogNormal’.

Gumbel Distribution

Gumbel distribution, inherited from the TransformedDistribution class, constructed by GumbelCDF Bijector and Uniform Distribution.

Properties are described as follows:

  • Gumbel.loc: returns the location parameter as a Tensor.

  • Gumbel.scale: returns the scale parameter as a Tensor.

The Distribution base class invokes the private API in the Gumbel and TransformedDistribution to implement the public APIs in the base class. Gumbel supports the following public APIs:

  • mean,mode,var, and sd：No parameter.

  • entropy: No parameter.

  • cross_entropy and kl_loss: The input parameters dist, loc_b, and scale_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Gumbel’ is supported. loc_b and scale_b indicate the location and scale of distribution b.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory.

  • sample: Input parameters sample shape shape is optional.

  • get_dist_args: Input parameters location loc and scale scale are optional. Return (loc, scale) with type tuple.

  • get_dist_type: returns ‘Gumbel’.

Logistic Distribution

Logistic distribution, inherited from the Distribution class.

Properties are described as follows:

  • Logistic.loc: returns the location parameter as a Tensor.

  • Logistic.scale: returns the scale parameter as a Tensor.

The Distribution base class invokes the private API in the Logistic and TransformedDistribution to implement the public APIs in the base class. Logistic supports the following public APIs:

  • mean,mode,var, and sd：Input parameters loc (for location) and scale (for scale) are optional.

  • entropy: Input parameters loc (for location) and scale (for scale) are optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. Input parameters location loc and scale scale are optional.

  • sample: Input parameters sample shape shape, location loc and scale scale are optional.

  • get_dist_args: Input parameters location loc and scale scale are optional. Return (loc, scale) with type tuple.

  • get_dist_type: returns ‘Logistic’.

Poisson Distribution

Poisson distribution, inherited from the Distribution class.

Properties are described as follows:

  • Poisson.rate: returns the rate as a Tensor.

The Distribution base class invokes the private API in the Poisson to implement the public APIs in the base class. Poisson supports the following public APIs:

  • mean,mode,var, and sd: The input parameter rate is optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. The input parameter rate* is optional.

  • sample: Optional input parameters include sample shape shape and the parameter rate.

  • get_dist_args: The input parameter rate is optional. Return (rate,) with type tuple.

  • get_dist_type: returns ‘Poisson’.

Gamma Distribution

Gamma distribution, inherited from the Distribution class.

Properties are described as follows:

  • Gamma.concentration: returns the concentration as a Tensor.

  • Gamma.rate: returns the rate as a Tensor.

The Distribution base class invokes the private API in the Gamma to implement the public APIs in the base class. Gamma supports the following public APIs:

  • mean,mode,var, and sd: The input parameters concentration and rate are optional.

  • entropy: The input parameters concentration and rate are optional.

  • cross_entropy and kl_loss: The input parameters dist, concentration_b and rate_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Gamma’ is supported. concentration_b and rate_b are the parameters of distribution b. The input parameters concentration_a and rate_a for distribution a are optional.

  • prob, log_prob, cdf, log_cdf, survival_function, and log_survival: The input parameter value is mandatory. The input parameters concentration and rate are optional.

  • sample: Optional input parameters include sample shape shape and parameters concentration and rate.

  • get_dist_args: The input parameters concentration and rate are optional. Return (concentration, rate) with type tuple.

  • get_dist_type: returns ‘Gamma’.

Beta Distribution

Beta distribution, inherited from the Distribution class.

Properties are described as follows:

  • Beta.concentration1: returns the rate as a Tensor.

  • Beta.concentration0: returns the rate as a Tensor.

The Distribution base class invokes the private API in the Beta to implement the public APIs in the base class. Beta supports the following public APIs:

  • mean,mode,var, and sd: The input parameters concentration1 and concentration0 are optional.

  • entropy: The input parameters concentration1 and concentration0 are optional.

  • cross_entropy and kl_loss: The input parameters dist, concentration1_b and rateconcentration0_b are mandatory. dist indicates the name of another distribution type. Currently, only ‘Beta’ is supported. concentration1_b and concentration0_b are the parameters of distribution b. The input parameters concentratio1n_a and concentration0_a for distribution a are optional.

  • prob and log_prob: The input parameter value is mandatory. The input parameters concentration1 and concentration0 are optional.

  • sample: Optional input parameters include sample shape shape and parameters concentration1 and concentration0.

  • get_dist_args: The input parameters concentration1 and concentration0 are optional. Return (concentration1, concentration0) with type tuple.

  • get_dist_type: returns ‘Beta’.

Probability Distribution Class Application in PyNative Mode

Distribution subclasses can be used in PyNative mode.

Use Normal as an example. Create a normal distribution whose average value is 0.0 and standard deviation is 1.0.

from mindspore import Tensor
from mindspore import dtype as mstype
import mindspore.context as context
import mindspore.nn.probability.distribution as msd
context.set_context(mode=context.PYNATIVE_MODE)

my_normal = msd.Normal(0.0, 1.0, dtype=mstype.float32)

mean = my_normal.mean()
var = my_normal.var()
entropy = my_normal.entropy()

value = Tensor([-0.5, 0.0, 0.5], dtype=mstype.float32)
prob = my_normal.prob(value)
cdf = my_normal.cdf(value)

mean_b = Tensor(1.0, dtype=mstype.float32)
sd_b = Tensor(2.0, dtype=mstype.float32)
kl = my_normal.kl_loss('Normal', mean_b, sd_b)

# get the distribution args as a tuple
dist_arg = my_normal.get_dist_args()

print("mean: ", mean)
print("var: ", var)
print("entropy: ", entropy)
print("prob: ", prob)
print("cdf: ", cdf)
print("kl: ", kl)
print("dist_arg: ", dist_arg)

The output is as follows:

mean:  0.0
var:  1.0
entropy:  1.4189385
prob:  [0.35206532 0.3989423  0.35206532]
cdf:  [0.30853754 0.5        0.69146246]
kl:  0.44314718
dist_arg: (Tensor(shape=[], dtype=Float32, value= 0), Tensor(shape=[], dtype=Float32, value= 1))

Probability Distribution Class Application in Graph Mode

In graph mode, Distribution subclasses can be used on the network.

import mindspore.nn as nn
from mindspore import Tensor
from mindspore import dtype as mstype
import mindspore.context as context
import mindspore.nn.probability.distribution as msd
context.set_context(mode=context.GRAPH_MODE)

class Net(nn.Cell):
    def __init__(self):
        super(Net, self).__init__()
        self.normal = msd.Normal(0.0, 1.0, dtype=mstype.float32)

    def construct(self, value, mean, sd):
        pdf = self.normal.prob(value)
        kl = self.normal.kl_loss("Normal", mean, sd)
        return pdf, kl

net = Net()
value = Tensor([-0.5, 0.0, 0.5], dtype=mstype.float32)
mean = Tensor(1.0, dtype=mstype.float32)
sd = Tensor(1.0, dtype=mstype.float32)
pdf, kl = net(value, mean, sd)
print("pdf: ", pdf)
print("kl: ", kl)

The output is as follows:

pdf:  [0.35206532 0.3989423  0.35206532]
kl:  0.5

TransformedDistribution Class API Design

TransformedDistribution, inherited from Distribution, is a base class for mathematical distribution that can be obtained by mapping f(x) changes. The APIs are as follows:

  1. Properties

    • bijector: returns the distribution transformation method.

    • distribution: returns the original distribution.

    • is_linear_transformation: returns the linear transformation flag.

  2. API functions (The parameters of the following APIs are the same as those of the corresponding APIs of distribution in the constructor function.)

    • cdf: cumulative distribution function (CDF)

    • log_cdf: log-cumulative distribution function

    • survival_function: survival function

    • log_survival: logarithmic survival function

    • prob: probability density function (PDF) or probability quality function (PMF)

    • log_prob: log-like function

    • sample: random sampling

    • mean: a non-parametric function, which can be invoked only when Bijector.is_constant_jacobian=true is invoked.

Invoking a TransformedDistribution Instance in PyNative Mode

The TransformedDistribution subclass can be used in PyNative mode.

import numpy as np
import mindspore.nn as nn
import mindspore.nn.probability.bijector as msb
import mindspore.nn.probability.distribution as msd
import mindspore.context as context
from mindspore import Tensor, dtype

context.set_context(mode=context.PYNATIVE_MODE)

normal = msd.Normal(0.0, 1.0, dtype=dtype.float32)
exp = msb.Exp()
LogNormal = msd.TransformedDistribution(exp, normal, seed=0, name="LogNormal")

# compute cumulative distribution function
x = np.array([2.0, 5.0, 10.0], dtype=np.float32)
tx = Tensor(x, dtype=dtype.float32)
cdf = LogNormal.cdf(tx)

# generate samples from the distribution
shape = (3, 2)
sample = LogNormal.sample(shape)

# get information of the distribution
print(LogNormal)
# get information of the underlying distribution and the bijector separately
print("underlying distribution:\n", LogNormal.distribution)
print("bijector:\n", LogNormal.bijector)
# get the computation results
print("cdf:\n", cdf)
print("sample shape:\n", sample.shape)

The output is as follows:

TransformedDistribution<
  (_bijector): Exp<exp>
  (_distribution): Normal<mean = 0.0, standard deviation = 1.0>
  >
underlying distribution:
 Normal<mean = 0.0, standard deviation = 1.0>
bijector:
 Exp<exp>
cdf:
 [0.7558914 0.9462397 0.9893489]
sample shape:
(3, 2)

When the TransformedDistribution is constructed to map the transformed is_constant_jacobian = true (for example, ScalarAffine), the constructed TransformedDistribution instance can use the mean API to calculate the average value. For example:

normal = msd.Normal(0.0, 1.0, dtype=dtype.float32)
scalaraffine = msb.ScalarAffine(1.0, 2.0)
trans_dist = msd.TransformedDistribution(scalaraffine, normal, seed=0)
mean = trans_dist.mean()
print(mean)

The output is as follows:

2.0

Invoking a TransformedDistribution Instance in Graph Mode

In graph mode, the TransformedDistribution class can be used on the network.

import numpy as np
import mindspore.nn as nn
from mindspore import Tensor, dtype
import mindspore.context as context
import mindspore.nn.probability.bijector as msb
import mindspore.nn.probability.distribution as msd
context.set_context(mode=context.GRAPH_MODE)

class Net(nn.Cell):
    def __init__(self, shape, dtype=dtype.float32, seed=0, name='transformed_distribution'):
        super(Net, self).__init__()
        # create TransformedDistribution distribution
        self.exp = msb.Exp()
        self.normal = msd.Normal(0.0, 1.0, dtype=dtype)
        self.lognormal = msd.TransformedDistribution(self.exp, self.normal, seed=seed, name=name)
        self.shape = shape

    def construct(self, value):
        cdf = self.lognormal.cdf(value)
        sample = self.lognormal.sample(self.shape)
        return cdf, sample

shape = (2, 3)
net = Net(shape=shape, name="LogNormal")
x = np.array([2.0, 3.0, 4.0, 5.0]).astype(np.float32)
tx = Tensor(x, dtype=dtype.float32)
cdf, sample = net(tx)
print("cdf: ", cdf)
print("sample shape: ", sample.shape)

The output is as follows:

cdf:  [0.7558914  0.86403143 0.9171715  0.9462397 ]
sample shape:  (2, 3)

Probability Distribution Mapping

Bijector (mindspore.nn.probability.bijector) is a basic component of probability programming. Bijector describes a random variable transformation method, and a new random variable \(Y = f(x)\) may be generated by using an existing random variable X and a mapping function f. Bijector provides four mapping-related transformation methods. It can be directly used as an operator, or used to generate a Distribution class instance of a new random variable on an existing Distribution class instance.

Bijector API Design

Bijector Base Class

The Bijector class is the base class for all probability distribution mappings. The APIs are as follows:

  1. Properties

    • name: returns the value of name.

    • dtype: returns the value of dtype.

    • parameters: returns the value of parameter.

    • is_constant_jacobian: returns the value of is_constant_jacobian.

    • is_injective: returns the value of is_injective.

  2. Mapping functions

    • forward: forward mapping, whose parameter is determined by _forward of the derived class.

    • inverse: backward mapping, whose parameter is determined by_inverse of the derived class.

    • forward_log_jacobian: logarithm of the derivative of the forward mapping, whose parameter is determined by _forward_log_jacobian of the derived class.

    • inverse_log_jacobian: logarithm of the derivative of the backward mapping, whose parameter is determined by _inverse_log_jacobian of the derived class.

When Bijector is invoked as a function: The input is a Distribution class and a TransformedDistribution is generated (cannot be invoked in a graph).

PowerTransform

PowerTransform implements variable transformation with \(Y = g(X) = {(1 + X * c)}^{1 / c}\). The APIs are as follows:

  1. Properties

    • power: returns the value of power as a Tensor.

  2. Mapping functions

    • forward: forward mapping, with an input parameter Tensor.

    • inverse: backward mapping, with an input parameter Tensor.

    • forward_log_jacobian: logarithm of the derivative of the forward mapping, with an input parameter Tensor.

    • inverse_log_jacobian: logarithm of the derivative of the backward mapping, with an input parameter Tensor.

Exp

Exp implements variable transformation with \(Y = g(X)= exp(X)\). The APIs are as follows:

Mapping functions

  • forward: forward mapping, with an input parameter Tensor.

  • inverse: backward mapping, with an input parameter Tensor.

  • forward_log_jacobian: logarithm of the derivative of the forward mapping, with an input parameter Tensor.

  • inverse_log_jacobian: logarithm of the derivative of the backward mapping, with an input parameter Tensor.

ScalarAffine

ScalarAffine implements variable transformation with Y = g(X) = a * X + b. The APIs are as follows:

  1. Properties

    • scale: returns the value of scale as a Tensor.

    • shift: returns the value of shift as a Tensor.

  2. Mapping functions

    • forward: forward mapping, with an input parameter Tensor.

    • inverse: backward mapping, with an input parameter Tensor.

    • forward_log_jacobian: logarithm of the derivative of the forward mapping, with an input parameter Tensor.

    • inverse_log_jacobian: logarithm of the derivative of the backward mapping, with an input parameter Tensor.

Softplus

Softplus implements variable transformation with \(Y = g(X) = log(1 + e ^ {kX}) / k \). The APIs are as follows:

  1. Properties

    • sharpness: returns the value of sharpness as a Tensor.

  2. Mapping functions

    • forward: forward mapping, with an input parameter Tensor.

    • inverse: backward mapping, with an input parameter Tensor.

    • forward_log_jacobian: logarithm of the derivative of the forward mapping, with an input parameter Tensor.

    • inverse_log_jacobian: logarithm of the derivative of the backward mapping, with an input parameter Tensor.

GumbelCDF

GumbelCDF implements variable transformation with \(Y = g(X) = \exp(-\exp(-\frac{X - loc}{scale}))\). The APIs are as follows:

  1. Properties

    • loc: returns the value of loc as a Tensor.

    • scale: returns the value of scale as a Tensor.

  2. Mapping functions

    • forward: forward mapping, with an input parameter Tensor.

    • inverse: backward mapping, with an input parameter Tensor.

    • forward_log_jacobian: logarithm of the derivative of the forward mapping, with an input parameter Tensor.

    • inverse_log_jacobian: logarithm of the derivative of the backward mapping, with an input parameter Tensor.

Invert

Invert implements the inverse of another bijector. The APIs are as follows:

  1. Properties

    • bijector: returns the Bijector used during initialization with type msb.Bijector.

  2. Mapping functions

    • forward: forward mapping, with an input parameter Tensor.

    • inverse: backward mapping, with an input parameter Tensor.

    • forward_log_jacobian: logarithm of the derivative of the forward mapping, with an input parameter Tensor.

    • inverse_log_jacobian: logarithm of the derivative of the backward mapping, with an input parameter Tensor.

Invoking the Bijector Instance in PyNative Mode

Before the execution, import the required library file package. The main library of the Bijector class is mindspore.nn.probability.bijector. After the library is imported, msb is used as the abbreviation of the library for invoking.

The following uses PowerTransform as an example. Create a PowerTransform object whose power is 2.

import numpy as np
import mindspore.nn as nn
import mindspore.nn.probability.bijector as msb
import mindspore.context as context
from mindspore import Tensor, dtype

context.set_context(mode=context.PYNATIVE_MODE)

powertransform = msb.PowerTransform(power=2.)

x = np.array([2.0, 3.0, 4.0, 5.0], dtype=np.float32)
tx = Tensor(x, dtype=dtype.float32)
forward = powertransform.forward(tx)
inverse = powertransform.inverse(tx)
forward_log_jaco = powertransform.forward_log_jacobian(tx)
inverse_log_jaco = powertransform.inverse_log_jacobian(tx)

print(powertransform)
print("forward: ", forward)
print("inverse: ", inverse)
print("forward_log_jacobian: ", forward_log_jaco)
print("inverse_log_jacobian: ", inverse_log_jaco)

The output is as follows:

PowerTransform<power = 2.0>
forward:  [2.236068  2.6457515 3.        3.3166249]
inverse:  [ 1.5       4.        7.5      12.000001]
forward_log_jacobian:  [-0.804719  -0.9729551 -1.0986123 -1.1989477]
inverse_log_jacobian:  [0.6931472 1.0986123 1.3862944 1.609438 ]

Invoking a Bijector Instance in Graph Mode

In graph mode, the Bijector subclass can be used on the network.

import numpy as np
import mindspore.nn as nn
from mindspore import Tensor
from mindspore import dtype as mstype
import mindspore.context as context
import mindspore.nn.probability.bijector as msb
context.set_context(mode=context.GRAPH_MODE)

class Net(nn.Cell):
    def __init__(self):
        super(Net, self).__init__()
        # create a PowerTransform bijector
        self.powertransform = msb.PowerTransform(power=2.)

    def construct(self, value):
        forward = self.powertransform.forward(value)
        inverse = self.powertransform.inverse(value)
        forward_log_jaco = self.powertransform.forward_log_jacobian(value)
        inverse_log_jaco = self.powertransform.inverse_log_jacobian(value)
        return forward, inverse, forward_log_jaco, inverse_log_jaco

net = Net()
x = np.array([2.0, 3.0, 4.0, 5.0]).astype(np.float32)
tx = Tensor(x, dtype=mstype.float32)
forward, inverse, forward_log_jaco, inverse_log_jaco = net(tx)
print("forward: ", forward)
print("inverse: ", inverse)
print("forward_log_jaco: ", forward_log_jaco)
print("inverse_log_jaco: ", inverse_log_jaco)

The output is as follows:

forward:  [2.236068  2.6457515 3.        3.3166249]
inverse:  [ 1.5       4.        7.5      12.000001]
forward_log_jacobian:  [-0.804719  -0.9729551 -1.0986123 -1.1989477]
inverse_log_jacobian:  [0.6931472 1.0986123 1.3862944 1.609438 ]

Deep Probabilistic Network

It is especially easy to use the MindSpore deep probabilistic programming library (mindspore.nn.probability.dpn) to construct a variational auto-encoder (VAE) for inference. You only need to define the encoder and decoder (a DNN model), invoke the VAE or conditional VAE (CVAE) API to form a derived network, invoke the ELBO API for optimization, and use the SVI API for variational inference. The advantage is that users who are not familiar with variational inference can build a probability model in the same way as they build a DNN model, and those who are familiar with variational inference can invoke these APIs to build a more complex probability model. VAE APIs are defined in mindspore.nn.probability.dpn, where dpn represents the deep probabilistic network. mindspore.nn.probability.dpn provides some basic APIs of the deep probabilistic network, for example, VAE.

VAE

First, we need to define the encoder and decoder and invoke the mindspore.nn.probability.dpn.VAE API to construct the VAE network. In addition to the encoder and decoder, we need to input the hidden size of the encoder output variable and the latent size of the VAE network storage potential variable. Generally, the latent size is less than the hidden size.

import mindspore.nn as nn
import mindspore.ops as ops
from mindspore.nn.probability.dpn import VAE

IMAGE_SHAPE = (-1, 1, 32, 32)


class Encoder(nn.Cell):
    def __init__(self):
        super(Encoder, self).__init__()
        self.fc1 = nn.Dense(1024, 800)
        self.fc2 = nn.Dense(800, 400)
        self.relu = nn.ReLU()
        self.flatten = nn.Flatten()

    def construct(self, x):
        x = self.flatten(x)
        x = self.fc1(x)
        x = self.relu(x)
        x = self.fc2(x)
        x = self.relu(x)
        return x


class Decoder(nn.Cell):
    def __init__(self):
        super(Decoder, self).__init__()
        self.fc1 = nn.Dense(400, 1024)
        self.sigmoid = nn.Sigmoid()
        self.reshape = ops.Reshape()

    def construct(self, z):
        z = self.fc1(z)
        z = self.reshape(z, IMAGE_SHAPE)
        z = self.sigmoid(z)
        return z


encoder = Encoder()
decoder = Decoder()
vae = VAE(encoder, decoder, hidden_size=400, latent_size=20)

ConditionalVAE

Similarly, the usage of CVAE is similar to that of VAE. The difference is that CVAE uses the label information of datasets. It is a supervised learning algorithm, and has a better generation effect than VAE.

First, define the encoder and decoder and invoke the mindspore.nn.probability.dpn.ConditionalVAE API to construct the CVAE network. The encoder here is different from that of the VAE because the label information of datasets needs to be input. The decoder is the same as that of the VAE. For the CVAE API, the number of dataset label categories also needs to be input. Other input parameters are the same as those of the VAE API.

import mindspore.nn as nn
import mindspore.ops as ops
from mindspore.nn.probability.dpn import ConditionalVAE

IMAGE_SHAPE = (-1, 1, 32, 32)


class Encoder(nn.Cell):
    def __init__(self, num_classes):
        super(Encoder, self).__init__()
        self.fc1 = nn.Dense(1024 + num_classes, 400)
        self.relu = nn.ReLU()
        self.flatten = nn.Flatten()
        self.concat = ops.Concat(axis=1)
        self.one_hot = nn.OneHot(depth=num_classes)

    def construct(self, x, y):
        x = self.flatten(x)
        y = self.one_hot(y)
        input_x = self.concat((x, y))
        input_x = self.fc1(input_x)
        input_x = self.relu(input_x)
        return input_x


class Decoder(nn.Cell):
    def __init__(self):
        super(Decoder, self).__init__()
        self.fc1 = nn.Dense(400, 1024)
        self.sigmoid = nn.Sigmoid()
        self.reshape = ops.Reshape()

    def construct(self, z):
        z = self.fc1(z)
        z = self.reshape(z, IMAGE_SHAPE)
        z = self.sigmoid(z)
        return z


encoder = Encoder(num_classes=10)
decoder = Decoder()
cvae = ConditionalVAE(encoder, decoder, hidden_size=400, latent_size=20, num_classes=10)

Load a dataset, for example, Mnist. For details about the data loading and preprocessing process, see Quick Start for Beginners. The create_dataset function is used to create a data iterator.

ds_train = create_dataset(image_path, 128, 1)

Next, use the infer API to perform variational inference on the VAE network.

Probability Inference Algorithm

Invoke the mindspore.nn.probability.infer.ELBO API to define the loss function of the VAE network, invoke WithLossCell to encapsulate the VAE network and loss function, define the optimizer, and transfer them to the mindspore.nn.probability.infer.SVI API. The run function of the SVI API can be understood to trigger training of the VAE network. You can specify the epochs of the training, so that a trained network is returned. If you specify the get_train_loss function, the loss of the trained model will be returned.

from mindspore.nn.probability.infer import ELBO, SVI

net_loss = ELBO(latent_prior='Normal', output_prior='Normal')
net_with_loss = nn.WithLossCell(vae, net_loss)
optimizer = nn.Adam(params=vae.trainable_params(), learning_rate=0.001)

vi = SVI(net_with_loss=net_with_loss, optimizer=optimizer)
vae = vi.run(train_dataset=ds_train, epochs=10)
trained_loss = vi.get_train_loss()

After obtaining the trained VAE network, use vae.generate_sample to generate a new sample. You need to specify the number of samples to be generated and the shape of the generated samples. The shape must be the same as that of the samples in the original dataset. You can also use vae.reconstruct_sample to reconstruct samples in the original dataset to test the reconstruction capability of the VAE network.

generated_sample = vae.generate_sample(64, IMAGE_SHAPE)
for sample in ds_train.create_dict_iterator():
    sample_x = Tensor(sample['image'], dtype=mstype.float32)
    reconstructed_sample = vae.reconstruct_sample(sample_x)
print('The shape of the generated sample is ', generated_sample.shape)

The shape of the newly generated sample is as follows:

The shape of the generated sample is (64, 1, 32, 32)

The CVAE training process is similar to the VAE training process. However, when a trained CVAE network is used to generate a new sample and rebuild a new sample, label information needs to be input. For example, the generated new sample is 64 digits ranging from 0 to 7.

sample_label = Tensor([i for i in range(0, 8)] * 8, dtype=mstype.int32)
generated_sample = cvae.generate_sample(sample_label, 64, IMAGE_SHAPE)
for sample in ds_train.create_dict_iterator():
    sample_x = Tensor(sample['image'], dtype=mstype.float32)
    sample_y = Tensor(sample['label'], dtype=mstype.int32)
    reconstructed_sample = cvae.reconstruct_sample(sample_x, sample_y)
print('The shape of the generated sample is ', generated_sample.shape)

Check the shape of the newly generated sample:

The shape of the generated sample is  (64, 1, 32, 32)

If you want the generated sample to be better and clearer, you can define a more complex encoder and decoder. The example uses only two layers of full-connected layers.

Bayesian Layer

The following uses the APIs in nn.probability.bnn_layers of MindSpore to implement the BNN image classification model. The APIs in nn.probability.bnn_layers of MindSpore include NormalPrior, NormalPosterior, ConvReparam, DenseReparam, DenseLocalReparam and WithBNNLossCell. The biggest difference between BNN and DNN is that the weight and bias of the BNN layer are not fixed values, but follow a distribution. NormalPrior and NormalPosterior are respectively used to generate a prior distribution and a posterior distribution that follow a normal distribution. ConvReparam and DenseReparam are the Bayesian convolutional layer and fully connected layers implemented by using the reparameterization method, respectively. DenseLocalReparam is the Bayesian fully connected layers implemented by using the local reparameterization method. WithBNNLossCell is used to encapsulate the BNN and loss function.

For details about how to use the APIs in nn.probability.bnn_layers to build a Bayesian neural network and classify images, see Applying the Bayesian Network.

Bayesian Conversion

For researchers who are unfamiliar with the Bayesian model, the MDP provides the mindspore.nn.probability.transform API to convert the DNN model into the BNN model by one click.

The __init__ function of the model conversion API TransformToBNN is defined as follows:

class TransformToBNN:
    def __init__(self, trainable_dnn, dnn_factor=1, bnn_factor=1):
        net_with_loss = trainable_dnn.network
        self.optimizer = trainable_dnn.optimizer
        self.backbone = net_with_loss.backbone_network
        self.loss_fn = getattr(net_with_loss, "_loss_fn")
        self.dnn_factor = dnn_factor
        self.bnn_factor = bnn_factor
        self.bnn_loss_file = None

The trainable_bnn parameter is a trainable DNN model packaged by TrainOneStepCell, dnn_factor and bnn_factor are the coefficient of the overall network loss calculated by the loss function and the coefficient of the KL divergence of each Bayesian layer, respectively. TransformToBNN implements the following functions:

  • Function 1: Convert the entire model.

    The transform_to_bnn_model method can convert the entire DNN model into a BNN model. The definition is as follows:

      def transform_to_bnn_model(self,
                             get_dense_args=lambda dp: {"in_channels": dp.in_channels, "has_bias": dp.has_bias,
                                                        "out_channels": dp.out_channels, "activation": dp.activation},
                             get_conv_args=lambda dp: {"in_channels": dp.in_channels, "out_channels": dp.out_channels,
                                                       "pad_mode": dp.pad_mode, "kernel_size": dp.kernel_size,
                                                       "stride": dp.stride, "has_bias": dp.has_bias,
                                                       "padding": dp.padding, "dilation": dp.dilation,
                                                       "group": dp.group},
                             add_dense_args=None,
                             add_conv_args=None):
      r"""
      Transform the whole DNN model to BNN model, and wrap BNN model by TrainOneStepCell.

      Args:
          get_dense_args (function): The arguments gotten from the DNN full connection layer. Default: lambda dp:
              {"in_channels": dp.in_channels, "out_channels": dp.out_channels, "has_bias": dp.has_bias}.
          get_conv_args (function): The arguments gotten from the DNN convolutional layer. Default: lambda dp:
              {"in_channels": dp.in_channels, "out_channels": dp.out_channels, "pad_mode": dp.pad_mode,
              "kernel_size": dp.kernel_size, "stride": dp.stride, "has_bias": dp.has_bias}.
          add_dense_args (dict): The new arguments added to BNN full connection layer. Default: {}.
          add_conv_args (dict): The new arguments added to BNN convolutional layer. Default: {}.

      Returns:
          Cell, a trainable BNN model wrapped by TrainOneStepCell.
     """

    get_dense_args specifies the parameters to be obtained from the fully connected layer of the DNN model. The default value is the common parameters of the fully connected layers of the DNN and BNN models. For details about the parameters, see mindspore API. get_conv_args specifies the parameters to be obtained from the convolutional layer of the DNN model. The default value is the common parameters of the convolutional layers of the DNN and BNN models. For details about the parameters, see MindSpore API. add_dense_args and add_conv_args specify the new parameter values to be specified for the BNN layer. Note that the parameters in add_dense_args cannot be the same as those in get_dense_args. The same rule applies to add_conv_args and get_conv_args.

  • Function 2: Convert a specific layer.

    The transform_to_bnn_layer method can convert a specific layer (nn.Dense or nn.Conv2d) in the DNN model into a corresponding Bayesian layer. The definition is as follows:

     def transform_to_bnn_layer(self, dnn_layer, bnn_layer, get_args=None, add_args=None):
      r"""
      Transform a specific type of layers in DNN model to corresponding BNN layer.

      Args:
          dnn_layer_type (Cell): The type of DNN layer to be transformed to BNN layer. The optional values are
          nn.Dense, nn.Conv2d.
          bnn_layer_type (Cell): The type of BNN layer to be transformed to. The optional values are
              DenseReparameterization, ConvReparameterization.
          get_args (dict): The arguments gotten from the DNN layer. Default: None.
          add_args (dict): The new arguments added to BNN layer. Default: None.

      Returns:
          Cell, a trainable model wrapped by TrainOneStepCell, whose sprcific type of layer is transformed to the corresponding bayesian layer.
      """

    Dnn_layer specifies a DNN layer to be converted into a BNN layer, and bnn_layer specifies a BNN layer to be converted into a DNN layer, and get_args and add_args specify the parameters obtained from the DNN layer and the parameters to be re-assigned to the BNN layer, respectively.

For details about how to use TransformToBNN in MindSpore, see DNN-to-BNN Conversion with One Click.

Bayesian Toolbox

Uncertainty Estimation

One of the advantages of the BNN is that uncertainty can be obtained. MDP provides a toolbox (mindspore.nn.probability.toolbox) for uncertainty estimation at the upper layer. You can easily use the toolbox to calculate uncertainty. Uncertainty means the uncertainty of the prediction result of the deep learning model. Currently, most deep learning algorithms can only provide high-confidence prediction results, but cannot determine the certainty of the prediction results. There are two types of uncertainty: aleatoric uncertainty and epistemic uncertainty.

  • Aleatoric uncertainty: describes the internal noise of data, that is, the unavoidable error. This phenomenon cannot be weakened by adding sampling data.

  • Epistemic uncertainty: describes the estimation inaccuracy of input data incurred due to reasons such as poor training or insufficient training data. This may be alleviated by adding training data.

The APIs of the uncertainty estimation toolbox are as follows:

  • model: trained model whose uncertainty is to be estimated.

  • train_dataset: dataset used for training, which is of the iterator type.

  • task_type: model type. The value is a character string. Enter regression or classification.

  • num_classes: For a classification model, you need to specify the number of labels of the classification.

  • epochs: number of epochs for training an uncertain model.

  • epi_uncer_model_path: path for storing or loading models that compute cognitive uncertainty.

  • ale_uncer_model_path: path used to store or load models that calculate epistemic uncertainty.

  • save_model: whether to store the model, which is of the Boolean type.

Before using the model, you need to train the model. The following uses LeNet5 as an example:

from mindspore.nn.probability.toolbox import UncertaintyEvaluation
from mindspore import load_checkpoint, load_param_into_net

if __name__ == '__main__':
    # get trained model
    network = LeNet5()
    param_dict = load_checkpoint('checkpoint_lenet.ckpt')
    load_param_into_net(network, param_dict)
    # get train and eval dataset
    ds_train = create_dataset('workspace/mnist/train')
    ds_eval = create_dataset('workspace/mnist/test')
    evaluation = UncertaintyEvaluation(model=network,
                                       train_dataset=ds_train,
                                       task_type='classification',
                                       num_classes=10,
                                       epochs=1,
                                       epi_uncer_model_path=None,
                                       ale_uncer_model_path=None,
                                       save_model=False)
    for eval_data in ds_eval.create_dict_iterator():
        eval_data = Tensor(eval_data['image'], mstype.float32)
        epistemic_uncertainty = evaluation.eval_epistemic_uncertainty(eval_data)
        aleatoric_uncertainty = evaluation.eval_aleatoric_uncertainty(eval_data)
    print('The shape of epistemic uncertainty is ', epistemic_uncertainty.shape)
    print('The shape of epistemic uncertainty is ', aleatoric_uncertainty.shape)

eval_epistemic_uncertainty calculates epistemic uncertainty, which is also called model uncertainty. Each estimation label of every sample has an uncertain value. eval_aleatoric_uncertainty calculates aleatoric uncertainty, which is also called data uncertainty. Each sample has an uncertain value. The output is as follows:

The shape of epistemic uncertainty is (32, 10)
The shape of epistemic uncertainty is (32,)

The value of uncertainty is greater than or equal to zero. A larger value indicates higher uncertainty.

Anomaly Detection

Anomaly Detection can find outliers that are “different from the main data distribution”. For example, finding outliers in data preprocessing can help improve the model’s fitting ability.

MDP provides anomaly detection toolbox (VAEAnomalyDetection) based on the variational autoencoder (VAE) in the upper layer. Similar to the use of VAE, we only need to customize the encoder and decoder (DNN model), initialize the relevant parameters, then you can use the toolbox to detect abnormal points.

The interface of the VAE-based anomaly detection toolbox is as follows:

  • encoder：Encoder(Cell)

  • decoder：Decoder(Cell)

  • hidden_size：The size of encoder’s output tensor

  • latent_size：The size of the latent space

Use Encoder and Decoder, set hidden_size and latent_size, initialize the class, and then pass the dataset to detect abnormal points.

from mindspore.nn.probability.toolbox import VAEAnomalyDetection

if __name__ == '__main__':
    encoder = Encoder()
    decoder = Decoder()
    ood = VAEAnomalyDetection(encoder=encoder, decoder=decoder,
                              hidden_size=400, latent_size=20)
    ds_train = create_dataset('workspace/mnist/train')
    ds_eval = create_dataset('workspace/mnist/test')
    model = ood.train(ds_train)
    for sample in ds_eval.create_dict_iterator(output_numpy=True, num_epochs=1):
        sample_x = Tensor(sample['image'], dtype=mstype.float32)
        score = ood.predict_outlier_score(sample_x)
        outlier = ood.predict_outlier(sample_x)
        print(score, outlier)

The output of score is the anomaly score of the sample. outlier is a Boolean type, True represents an abnormal point, and False represents a normal point.