bayesml.gaussianmixture package

bayesml.gaussianmixture package#

Module contents#

The Gaussian mixture model with the Gauss-Wishart prior distribution and the Dirichlet prior distribution.

The stochastic data generative model is as follows:

\(K \in \mathbb{N}\): number of latent classes
\(\boldsymbol{z} \in \{ 0, 1 \}^K\): a one-hot vector representing the latent class (latent variable)
\(\boldsymbol{\pi} \in [0, 1]^K\): a parameter for latent classes, (\(\sum_{k=1}^K \pi_k=1\))
\(D \in \mathbb{N}\): a dimension of data
\(\boldsymbol{x} \in \mathbb{R}^D\): a data point
\(\boldsymbol{\mu}_k \in \mathbb{R}^D\): a parameter
\(\boldsymbol{\mu} = \{ \boldsymbol{\mu}_k \}_{k=1}^K\)
\(\boldsymbol{\Lambda}_k \in \mathbb{R}^{D\times D}\) : a parameter (a positive definite matrix)
\(\boldsymbol{\Lambda} = \{ \boldsymbol{\Lambda}_k \}_{k=1}^K\)
\(| \boldsymbol{\Lambda}_k | \in \mathbb{R}\): the determinant of \(\boldsymbol{\Lambda}_k\)

\[\begin{split}p(\boldsymbol{z} | \boldsymbol{\pi}) &= \mathrm{Cat}(\boldsymbol{z}|\boldsymbol{\pi}) = \prod_{k=1}^K \pi_k^{z_k},\\ p(\boldsymbol{x} | \boldsymbol{\mu}, \boldsymbol{\Lambda}, \boldsymbol{z}) &= \prod_{k=1}^K \mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_k,\boldsymbol{\Lambda}_k^{-1})^{z_k} \\ &= \prod_{k=1}^K \left( \frac{| \boldsymbol{\Lambda}_k |^{1/2}}{(2\pi)^{D/2}} \exp \left\{ -\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu}_k)^\top \boldsymbol{\Lambda}_k (\boldsymbol{x}-\boldsymbol{\mu}_k) \right\} \right)^{z_k}.\end{split}\]

The prior distribution is as follows:

\(\boldsymbol{m}_0 \in \mathbb{R}^{D}\): a hyperparameter
\(\kappa_0 \in \mathbb{R}_{>0}\): a hyperparameter
\(\nu_0 \in \mathbb{R}\): a hyperparameter (\(\nu_0 > D-1\))
\(\boldsymbol{W}_0 \in \mathbb{R}^{D\times D}\): a hyperparameter (a positive definite matrix)
\(\boldsymbol{\alpha}_0 \in \mathbb{R}_{> 0}^K\): a hyperparameter
\(\mathrm{Tr} \{ \cdot \}\): a trace of a matrix
\(\Gamma (\cdot)\): the gamma function

\[\begin{split}p(\boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_0,(\kappa_0 \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_0, \nu_0) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\alpha}_0) \\ &= \Biggl[\, \prod_{k=1}^K \left( \frac{\kappa_0}{2\pi} \right)^{D/2} |\boldsymbol{\Lambda}_k|^{1/2} \exp \left\{ -\frac{\kappa_0}{2}(\boldsymbol{\mu}_k -\boldsymbol{m}_0)^\top \boldsymbol{\Lambda}_k (\boldsymbol{\mu}_k - \boldsymbol{m}_0) \right\} \notag \\ &\qquad \times B(\boldsymbol{W}_0, \nu_0) | \boldsymbol{\Lambda}_k |^{(\nu_0 - D - 1) / 2} \exp \left\{ -\frac{1}{2} \mathrm{Tr} \{ \boldsymbol{W}_0^{-1} \boldsymbol{\Lambda}_k \} \right\} \Biggr] \notag \\ &\qquad \times C(\boldsymbol{\alpha}_0)\prod_{k=1}^K \pi_k^{\alpha_{0,k}-1},\\\end{split}\]

where \(B(\boldsymbol{W}_0, \nu_0)\) and \(C(\boldsymbol{\alpha}_0)\) are defined as follows:

\[\begin{split}B(\boldsymbol{W}_0, \nu_0) &= | \boldsymbol{W}_0 |^{-\nu_0 / 2} \left( 2^{\nu_0 D / 2} \pi^{D(D-1)/4} \prod_{i=1}^D \Gamma \left( \frac{\nu_0 + 1 - i}{2} \right) \right)^{-1}, \\ C(\boldsymbol{\alpha}_0) &= \frac{\Gamma(\sum_{k=1}^K \alpha_{0,k})}{\Gamma(\alpha_{0,1})\cdots\Gamma(\alpha_{0,K})}.\end{split}\]

The apporoximate posterior distribution in the \(t\)-th iteration of a variational Bayesian method is as follows:

\(\boldsymbol{x}^n = (\boldsymbol{x}_1, \boldsymbol{x}_2, \dots , \boldsymbol{x}_n) \in \mathbb{R}^{D \times n}\): given data
\(\boldsymbol{z}^n = (\boldsymbol{z}_1, \boldsymbol{z}_2, \dots , \boldsymbol{z}_n) \in \{ 0, 1 \}^{K \times n}\): latent classes of given data
\(\boldsymbol{r}_i^{(t)} = (r_{i,1}^{(t)}, r_{i,2}^{(t)}, \dots , r_{i,K}^{(t)}) \in [0, 1]^K\): a parameter for the \(i\)-th latent class. (\(\sum_{k=1}^K r_{i, k}^{(t)} = 1\))
\(\boldsymbol{m}_{n,k}^{(t)} \in \mathbb{R}^{D}\): a hyperparameter
\(\kappa_{n,k}^{(t)} \in \mathbb{R}_{>0}\): a hyperparameter
\(\nu_{n,k}^{(t)} \in \mathbb{R}\): a hyperparameter \((\nu_n > D-1)\)
\(\boldsymbol{W}_{n,k}^{(t)} \in \mathbb{R}^{D\times D}\): a hyperparameter (a positive definite matrix)
\(\boldsymbol{\alpha}_n^{(t)} \in \mathbb{R}_{> 0}^K\): a hyperparameter

\[\begin{split}q(\boldsymbol{z}^n, \boldsymbol{\mu},\boldsymbol{\Lambda},\boldsymbol{\pi}) &= \left\{ \prod_{i=1}^n \mathrm{Cat} (\boldsymbol{z}_i | \boldsymbol{r}_i^{(t)}) \right\} \left\{ \prod_{k=1}^K \mathcal{N}(\boldsymbol{\mu}_k|\boldsymbol{m}_{n,k}^{(t)},(\kappa_{n,k}^{(t)} \boldsymbol{\Lambda}_k)^{-1})\mathcal{W}(\boldsymbol{\Lambda}_k|\boldsymbol{W}_{n,k}^{(t)}, \nu_{n,k}^{(t)}) \right\} \mathrm{Dir}(\boldsymbol{\pi}|\boldsymbol{\alpha}_n^{(t)}) \\ &= \Biggl[\, \prod_{i=1}^n \prod_{k=1}^K (r_{i,k}^{(t)})^{z_{i,k}} \Biggr] \Biggl[\, \prod_{k=1}^K \left( \frac{\kappa_{n,k}^{(t)}}{2\pi} \right)^{D/2} |\boldsymbol{\Lambda}_k|^{1/2} \exp \left\{ -\frac{\kappa_{n,k}^{(t)}}{2}(\boldsymbol{\mu}_k -\boldsymbol{m}_{n,k}^{(t)})^\top \boldsymbol{\Lambda}_k (\boldsymbol{\mu}_k - \boldsymbol{m}_{n,k}^{(t)}) \right\} \\ &\qquad \times B(\boldsymbol{W}_{n,k}^{(t)}, \nu_{n,k}^{(t)}) | \boldsymbol{\Lambda}_k |^{(\nu_{n,k}^{(t)} - D - 1) / 2} \exp \left\{ -\frac{1}{2} \mathrm{Tr} \{ ( \boldsymbol{W}_{n,k}^{(t)} )^{-1} \boldsymbol{\Lambda}_k \} \right\} \Biggr] \\ &\qquad \times C(\boldsymbol{\alpha}_n^{(t)})\prod_{k=1}^K \pi_k^{\alpha_{n,k}^{(t)}-1},\\\end{split}\]

where the updating rule of the hyperparameters is as follows.

\[\begin{split}N_k^{(t)} &= \sum_{i=1}^n r_{i,k}^{(t)}, \\ \bar{\boldsymbol{x}}_k^{(t)} &= \frac{1}{N_k^{(t)}} \sum_{i=1}^n r_{i,k}^{(t)} \boldsymbol{x}_i, \\ \boldsymbol{m}_{n,k}^{(t+1)} &= \frac{\kappa_0\boldsymbol{m}_0 + N_k^{(t)} \bar{\boldsymbol{x}}_k^{(t)}}{\kappa_0 + N_k^{(t)}}, \\ \kappa_{n,k}^{(t+1)} &= \kappa_0 + N_k^{(t)}, \\ (\boldsymbol{W}_{n,k}^{(t+1)})^{-1} &= \boldsymbol{W}_0^{-1} + \sum_{i=1}^{n} r_{i,k}^{(t)} (\boldsymbol{x}_i-\bar{\boldsymbol{x}}_k^{(t)})(\boldsymbol{x}_i-\bar{\boldsymbol{x}}_k^{(t)})^\top + \frac{\kappa_0 N_k^{(t)}}{\kappa_0 + N_k^{(t)}}(\bar{\boldsymbol{x}}_k^{(t)}-\boldsymbol{m}_0)(\bar{\boldsymbol{x}}_k^{(t)}-\boldsymbol{m}_0)^\top, \\ \nu_{n,k}^{(t+1)} &= \nu_0 + N_k^{(t)},\\ \alpha_{n,k}^{(t+1)} &= \alpha_{0,k} + N_k^{(t)}, \\ \ln \rho_{i,k}^{(t+1)} &= \psi (\alpha_{n,k}^{(t+1)}) - \psi ( {\textstyle \sum_{k=1}^K \alpha_{n,k}^{(t+1)}} ) \notag \\ &\qquad + \frac{1}{2} \Biggl[\, \sum_{d=1}^D \psi \left( \frac{\nu_{n,k}^{(t+1)} + 1 - d}{2} \right) + D \ln 2 + \ln | \boldsymbol{W}_{n,k}^{(t+1)} | \notag \\ &\qquad - D \ln (2 \pi ) - \frac{D}{\kappa_{n,k}^{(t+1)}} - \nu_{n,k}^{(t+1)} (\boldsymbol{x}_i - \boldsymbol{m}_{n,k}^{(t+1)})^\top \boldsymbol{W}_{n,k}^{(t+1)} (\boldsymbol{x}_i - \boldsymbol{m}_{n,k}^{(t+1)}) \Biggr], \\ r_{i,k}^{(t+1)} &= \frac{\rho_{i,k}^{(t+1)}}{\sum_{j=1}^K \rho_{i,j}^{(t+1)}}.\end{split}\]

The approximate predictive distribution is as follows:

\(\boldsymbol{x}_{n+1} \in \mathbb{R}^D\): a new data point
\(\boldsymbol{\mu}_{\mathrm{p},k} \in \mathbb{R}^D\): the parameter of the predictive distribution
\(\boldsymbol{\Lambda}_{\mathrm{p},k} \in \mathbb{R}^{D \times D}\): the parameter of the predictive distribution (a positive definite matrix)
\(\nu_{\mathrm{p},k} \in \mathbb{R}_{>0}\): the parameter of the predictive distribution

\[\begin{split}&p(x_{n+1}|x^n) \\ &= \frac{1}{\sum_{k=1}^K \alpha_{n,k}^{(t)}} \sum_{k=1}^K \alpha_{n,k}^{(t)} \mathrm{St}(x_{n+1}|\boldsymbol{\mu}_{\mathrm{p},k},\boldsymbol{\Lambda}_{\mathrm{p},k}, \nu_{\mathrm{p},k}) \\ &= \frac{1}{\sum_{k=1}^K \alpha_{n,k}^{(t)}} \sum_{k=1}^K \alpha_{n,k}^{(t)} \Biggl[ \frac{\Gamma (\nu_{\mathrm{p},k} / 2 + D / 2)}{\Gamma (\nu_{\mathrm{p},k} / 2)} \frac{|\boldsymbol{\Lambda}_{\mathrm{p},k}|^{1/2}}{(\nu_{\mathrm{p},k} \pi)^{D/2}} \notag \\ &\qquad \qquad \qquad \qquad \qquad \times \left( 1 + \frac{1}{\nu_{\mathrm{p},k}} (\boldsymbol{x}_{n+1} - \boldsymbol{\mu}_{\mathrm{p},k})^\top \boldsymbol{\Lambda}_{\mathrm{p},k} (\boldsymbol{x}_{n+1} - \boldsymbol{\mu}_{\mathrm{p},k}) \right)^{-\nu_{\mathrm{p},k}/2 - D/2} \Biggr],\end{split}\]

where the parameters are obtained from the hyperparameters of the posterior distribution as follows:

\[\begin{split}\boldsymbol{\mu}_{\mathrm{p},k} &= \boldsymbol{m}_{n,k}^{(t)}, \\ \nu_{\mathrm{p},k} &= \nu_{n,k}^{(t)} - D + 1,\\ \boldsymbol{\Lambda}_{\mathrm{p},k} &= \frac{\kappa_{n,k}^{(t)} \nu_{\mathrm{p},k}}{\kappa_{n,k}^{(t)} + 1} \boldsymbol{W}_{n,k}^{(t)}.\end{split}\]

class bayesml.gaussianmixture.GenModel(c_num_classes, c_degree, pi_vec=None, mu_vecs=None, lambda_mats=None, h_alpha_vec=None, h_m_vecs=None, h_kappas=None, h_nus=None, h_w_mats=None, seed=None)#

Bases: Generative

The stochastic data generative model and the prior distribution

Parameters:

c_num_classesint: a positive integer
c_degreeint: a positive integer
pi_vecnumpy.ndarray, optional: a real vector in \([0, 1]^K\), by default [1/K, 1/K, … , 1/K]
mu_vecsnumpy.ndarray, optional: vectors of real numbers, by default zero vectors.
lambda_matsnumpy.ndarray, optional: positive definite symetric matrices, by default the identity matrices
h_alpha_vecnumpy.ndarray, optional: a vector of positive real numbers, by default [1/2, 1/2, … , 1/2]
h_m_vecsnumpy.ndarray, optional: vectors of real numbers, by default zero vectors
h_kappasfloat, optional: positive real numbers, by default [1.0, 1.0, … , 1.0]
h_nusfloat, optional: real numbers greater than c_degree-1, by default the value of c_degree
h_w_matsnumpy.ndarray, optional: positive definite symetric matrices, by default the identity matrices
seed{None, int}, optional: A seed to initialize numpy.random.default_rng(), by default None

Methods

`gen_params`()	Generate the parameter from the prior distribution.
`gen_sample`(sample_size)	Generate a sample from the stochastic data generative model.
`get_constants`()	Get constants of GenModel.
`get_h_params`()	Get the hyperparameters of the prior distribution.
`get_params`()	Get the parameter of the sthocastic data generative model.
`load_h_params`(filename)	Load the hyperparameters to h_params.
`load_params`(filename)	Load the parameters saved by `save_params`.
`save_h_params`(filename)	Save the hyperparameters using python `pickle` module.
`save_params`(filename)	Save the parameters using python `pickle` module.
`save_sample`(filename, sample_size)	Save the generated sample as NumPy `.npz` format.
`set_h_params`([h_alpha_vec, h_m_vecs, ...])	Set the hyperparameters of the prior distribution.
`set_params`([pi_vec, mu_vecs, lambda_mats])	Set the parameter of the sthocastic data generative model.
`visualize_model`([sample_size])	Visualize the stochastic data generative model and generated samples.

get_constants()#

Get constants of GenModel.

Returns:

constantsdict of {str: int, numpy.ndarray}

"c_num_classes" : the value of self.c_num_classes
"c_degree" : the value of self.c_degree

set_h_params(h_alpha_vec=None, h_m_vecs=None, h_kappas=None, h_nus=None, h_w_mats=None)#

Set the hyperparameters of the prior distribution.

Parameters:

h_alpha_vecnumpy.ndarray, optional: a vector of positive real numbers, by default None
h_m_vecsnumpy.ndarray, optional: vectors of real numbers, by default None
h_kappasfloat, optional: positive real numbers, by default None
h_nusfloat, optional: real numbers greater than c_degree-1, by default None
h_w_matsnumpy.ndarray, optional: positive definite symetric matrices, by default None

get_h_params()#

Get the hyperparameters of the prior distribution.

Returns:

h_params{str:float, np.ndarray}

"h_alpha_vec" : The value of self.h_alpha_vec
"h_m_vecs" : The value of self.h_m_vecs
"h_kappas" : The value of self.h_kappas
"h_nus" : The value of self.h_nus
"h_w_mats" : The value of self.h_w_mats

gen_params()#

Generate the parameter from the prior distribution.

The generated vaule is set at self.pi_vec, self.mu_vecs and self.lambda_mats.

set_params(pi_vec=None, mu_vecs=None, lambda_mats=None)#

Set the parameter of the sthocastic data generative model.

Parameters:

pi_vecnumpy.ndarray: a real vector in \([0, 1]^K\). The sum of its elements must be 1.
mu_vecsnumpy.ndarray: vectors of real numbers
lambda_matsnumpy.ndarray: positive definite symetric matrices

get_params()#

Get the parameter of the sthocastic data generative model.

Returns:

params{str:float, numpy.ndarray}

"pi_vec" : The value of self.pi_vec
"mu_vecs" : The value of self.mu_vecs
"lambda_mats" : The value of self.lambda_mats

gen_sample(sample_size)#

Generate a sample from the stochastic data generative model.

Parameters:

sample_sizeint: A positive integer

Returns:

xnumpy ndarray: 2-dimensional array whose shape is (sammple_size,c_degree) and its elements are real numbers.
znumpy ndarray: 2-dimensional array whose shape is (sample_size,c_num_classes) whose rows are one-hot vectors.

save_sample(filename, sample_size)#

Save the generated sample as NumPy .npz format.

It is saved as a NpzFile with keyword: “x”, “z”.

Parameters:

filenamestr: The filename to which the sample is saved. .npz will be appended if it isn’t there.
sample_sizeint: A positive integer

See also

numpy.savez_compressed

visualize_model(sample_size=100)#

Visualize the stochastic data generative model and generated samples.

Parameters:

sample_sizeint, optional: A positive integer, by default 100

Examples

>>> from bayesml import gaussianmixture
>>> import numpy as np
>>> model = gaussianmixture.GenModel(
>>>             c_num_classes=3,
>>>             c_degree=1
>>>             pi_vec=np.array([0.444,0.444,0.112]),
>>>             mu_vecs=np.array([[-2.8],[-0.8],[2]]),
>>>             lambda_mats=np.array([[[6.25]],[[6.25]],[[100]]])
>>>             )
>>> model.visualize_model()
pi_vec:
 [0.444 0.444 0.112]
mu_vecs:
 [[-2.8]
 [-0.8]
 [ 2. ]]
lambda_mats:
 [[[  6.25]]
 [[  6.25]]
 [[100.  ]]]

class bayesml.gaussianmixture.LearnModel(c_num_classes, c_degree, h0_alpha_vec=None, h0_m_vecs=None, h0_kappas=None, h0_nus=None, h0_w_mats=None, seed=None)#

Bases: Posterior, PredictiveMixin

The posterior distribution and the predictive distribution.

Parameters:

c_num_classesint: a positive integer
c_degreeint: a positive integer
h0_alpha_vecnumpy.ndarray, optional: a vector of positive real numbers, by default [1/2, 1/2, … , 1/2]
h0_m_vecsnumpy.ndarray, optional: vectors of real numbers, by default zero vectors
h0_kappas{float, numpy.ndarray}, optional: positive real numbers, by default [1.0, 1.0, … , 1.0]
h0_nus{float, numpy.ndarray}, optional: real numbers greater than c_degree-1, by default the value of c_degree
h0_w_matsnumpy.ndarray, optional: positive definite symetric matrices, by default the identity matrices
seed{None, int}, optional: A seed to initialize numpy.random.default_rng(), by default None

Attributes:

h0_w_mats_invnumpy.ndarray: the inverse matrices of h0_w_mats
hn_alpha_vecnumpy.ndarray: a vector of positive real numbers
hn_m_vecsnumpy.ndarray: vectors of real numbers
hn_kappasnumpy.ndarray: positive real numbers
hn_nusnumpy.ndarray: real numbers greater than c_degree-1
hn_w_matsnumpy.ndarray: positive definite symetric matrices
hn_w_mats_invnumpy.ndarray: the inverse matrices of hn_w_mats
r_vecsnumpy.ndarray: vectors of real numbers. The sum of its elenemts is 1.
nsnumpy.ndarray: positive real numbers
s_matsnumpy.ndarray: positive difinite symmetric matrices
p_mu_vecsnumpy.ndarray: vectors of real numbers
p_nusnumpy.ndarray: positive real numbers
p_lambda_matsnumpy.ndarray: positive definite symetric matrices

Methods

`calc_pred_dist`()	Calculate the parameters of the predictive distribution.
`estimate_latent_vars`(x[, loss])	Estimate latent variables corresponding to x under the given criterion.
`estimate_latent_vars_and_update`(x[, loss, ...])	Estimate latent variables and update the posterior sequentially.
`estimate_params`([loss])	Estimate the parameter of the stochastic data generative model under the given criterion.
`get_constants`()	Get constants of LearnModel.
`get_h0_params`()	Get the initial values of the hyperparameters of the posterior distribution.
`get_hn_params`()	Get the hyperparameters of the posterior distribution.
`get_p_params`()	Get the parameters of the predictive distribution.
`load_h0_params`(filename)	Load the hyperparameters to h0_params.
`load_hn_params`(filename)	Load the hyperparameters to hn_params.
`make_prediction`([loss])	Predict a new data point under the given criterion.
`overwrite_h0_params`()	Overwrite the initial values of the hyperparameters of the posterior distribution by the learned values.
`pred_and_update`(x[, loss, max_itr, ...])	Predict a new data point and update the posterior sequentially.
`reset_hn_params`()	Reset the hyperparameters of the posterior distribution to their initial values.
`save_h0_params`(filename)	Save the hyperparameters using python `pickle` module.
`save_hn_params`(filename)	Save the hyperparameters using python `pickle` module.
`set_h0_params`([h0_alpha_vec, h0_m_vecs, ...])	Set the hyperparameters of the prior distribution.
`set_hn_params`([hn_alpha_vec, hn_m_vecs, ...])	Set updated values of the hyperparameter of the posterior distribution.
`update_posterior`(x[, max_itr, num_init, ...])	Update the hyperparameters of the posterior distribution using traning data.
`visualize_posterior`()	Visualize the posterior distribution for the parameter.

get_constants()#

Get constants of LearnModel.

Returns:

constantsdict of {str: int, numpy.ndarray}

"c_num_classes" : the value of self.c_num_classes
"c_degree" : the value of self.c_degree

set_h0_params(h0_alpha_vec=None, h0_m_vecs=None, h0_kappas=None, h0_nus=None, h0_w_mats=None)#

Set the hyperparameters of the prior distribution.

Parameters:

h0_alpha_vecnumpy.ndarray, optional: a vector of positive real numbers, by default None
h0_m_vecsnumpy.ndarray, optional: vectors of real numbers, by default None
h0_kappas{float, numpy.ndarray}, optional: positive real numbers, by default None
h0_nus{float, numpy.ndarray}, optional: real numbers greater than c_degree-1, by default None
h0_w_matsnumpy.ndarray, optional: positive definite symetric matrices, by default None

get_h0_params()#

Get the initial values of the hyperparameters of the posterior distribution.

Returns:

h0_paramsdict of {str: numpy.ndarray}

"h0_alpha_vec" : The value of self.h0_alpha_vec
"h0_m_vecs" : The value of self.h0_m_vecs
"h0_kappas" : The value of self.h0_kappas
"h0_nus" : The value of self.h0_nus
"h0_w_mats" : The value of self.h0_w_mats

set_hn_params(hn_alpha_vec=None, hn_m_vecs=None, hn_kappas=None, hn_nus=None, hn_w_mats=None)#

Set updated values of the hyperparameter of the posterior distribution.

Parameters:

hn_alpha_vecnumpy.ndarray, optional: a vector of positive real numbers, by default None
hn_m_vecsnumpy.ndarray, optional: vectors of real numbers, by default None
hn_kappas{float, numpy.ndarray}, optional: positive real numbers, by default None
hn_nus{float, numpy.ndarray}, optional: real numbers greater than c_degree-1, by default None
hn_w_matsnumpy.ndarray, optional: positive definite symetric matrices, by default None

get_hn_params()#

Get the hyperparameters of the posterior distribution.

Returns:

hn_paramsdict of {str: numpy.ndarray}

"hn_alpha_vec" : The value of self.hn_alpha_vec
"hn_m_vecs" : The value of self.hn_m_vecs
"hn_kappas" : The value of self.hn_kappas
"hn_nus" : The value of self.hn_nus
"hn_w_mats" : The value of self.hn_w_mats

update_posterior(x, max_itr=100, num_init=10, tolerance=1e-08, init_type='subsampling')#

Update the hyperparameters of the posterior distribution using traning data.

Parameters:

xnumpy.ndarray

(sample_size,c_degree)-dimensional ndarray. All the elements must be real number.

max_itrint, optional

maximum number of iterations, by default 100

num_initint, optional

number of initializations, by default 10

tolerancefloat, optional

convergence croterion of variational lower bound, by default 1.0E-8

init_typestr, optional

'subsampling': for each latent class, extract a subsample whose size is int(np.sqrt(x.shape[0])). and use its mean and covariance matrix as an initial values of hn_m_vecs and hn_lambda_mats.
'random_responsibility': randomly assign responsibility to r_vecs

Type of initialization, by default 'subsampling'

estimate_params(loss='squared')#

Estimate the parameter of the stochastic data generative model under the given criterion.

Note that the criterion is applied to estimating pi_vec, mu_vecs and lambda_mats independently. Therefore, a tuple of the dirichlet distribution, the student’s t-distributions and the wishart distributions will be returned when loss=”KL”

Parameters:

lossstr, optional: Loss function underlying the Bayes risk function, by default “squared”. This function supports “squared”, “0-1”, and “KL”.

Returns:

Estimatesa tuple of {numpy ndarray, float, None, or rv_frozen}

pi_vec_hat : the estimate for pi_vec
mu_vecs_hat : the estimate for mu_vecs
lambda_mats_hat : the estimate for lambda_mats

The estimated values under the given loss function. If it is not exist, np.nan will be returned. If the loss function is “KL”, the posterior distribution itself will be returned as rv_frozen object of scipy.stats.

bayesml.gaussianmixture package

Contents

bayesml.gaussianmixture package#

Module contents#