bayesml.linearregression package

bayesml.linearregression package#

Module contents#

The Baysian Linear Regression.

The stochastic data generative model is as follows:

\(d \in \mathbb N\): a dimension
\(\boldsymbol{x} = [x_1, x_2, \dots , x_d] \in \mathbb{R}^d\): an explanatory variable. If you consider an intercept term, it should be included as one of the elements of \(\boldsymbol{x}\).
\(y\in\mathbb{R}\): an objective variable
\(\tau \in\mathbb{R}_{>0}\): a parameter
\(\boldsymbol{\theta}\in\mathbb{R}^{d}\): a parameter

\[\begin{split}p(y|\boldsymbol{x},\boldsymbol{\theta},\tau) &= \mathcal N (y| \boldsymbol{\theta}^{\top} \boldsymbol{x},\tau^{-1}) \\ &= \sqrt{\frac{\tau}{2 \pi}} \exp \left\{ -\frac{\tau}{2} (y - \boldsymbol{\theta}^\top \boldsymbol{x})^2 \right\}.\end{split}\]

\[\begin{split}&\mathbb{E}[ y | \boldsymbol{x},\boldsymbol{\theta},\tau] = \boldsymbol{\theta}^{\top} \boldsymbol{x}, \\ &\mathbb{V}[ y | \boldsymbol{x},\boldsymbol{\theta},\tau ] = \tau^{-1}.\end{split}\]

The prior distribution is as follows:

\(\boldsymbol{\mu_0} \in \mathbb{R}^d\): a hyperparameter
\(\boldsymbol{\Lambda_0} \in \mathbb{R}^{d\times d}\): a hyperparameter (a positive definite matrix)
\(\alpha_0\in \mathbb{R}_{>0}\): a hyperparameter
\(\beta_0\in \mathbb{R}_{>0}\): a hyperparameter

\[\begin{split}p(\boldsymbol{\theta}, \tau) &= \mathcal{N}(\boldsymbol{\theta}|\boldsymbol{\mu}_0, (\tau \boldsymbol{\Lambda}_0)^{-1}) \mathrm{Gam}(\tau|\alpha_0,\beta_0)\\ &= \frac{|\tau \boldsymbol{\Lambda}_0|^{1/2}}{(2 \pi)^{d/2}} \exp \left\{ -\frac{\tau}{2} (\boldsymbol{\theta} - \boldsymbol{\mu}_0)^\top \boldsymbol{\Lambda}_0 (\boldsymbol{\theta} - \boldsymbol{\mu}_0) \right\} \frac{\beta_0^{\alpha_0}}{\Gamma (\alpha_0)} \tau^{\alpha_0 - 1} \exp \{ -\beta_0 \tau \} .\end{split}\]

\[\begin{split}\mathbb{E}[\boldsymbol{\theta}] &= \boldsymbol{\mu}_0 & \left( \alpha_0 > \frac{1}{2} \right), \\ \mathrm{Cov}[\boldsymbol{\theta}] &= \frac{\beta_0}{\alpha_0 - 1} \boldsymbol{\Lambda}_0^{-1} & (\alpha_0 > 1), \\ \mathbb{E}[\tau] &= \frac{\alpha_0}{\beta_0}, \\ \mathbb{V}[\tau] &= \frac{\alpha_0}{\beta_0^2}.\end{split}\]

The posterior distribution is as follows:

\(n \in \mathbb N\): a sample size
\(\boldsymbol{X} = [\boldsymbol{x}_1, \boldsymbol{x}_2, \dots , \boldsymbol{x}_n]^\top \in \mathbb{R}^{n \times d}\)
\(\boldsymbol{y} = [y_1, y_2, \dots , y_n]^\top \in \mathbb{R}^n\)
\(\boldsymbol{\mu}_n\in \mathbb{R}^d\): a hyperparameter
\(\boldsymbol{\Lambda_n} \in \mathbb{R}^{d\times d}\): a hyperparameter (a positive definite matrix)
\(\alpha_n\in \mathbb{R}_{>0}\): a hyperparameter
\(\beta_n\in \mathbb{R}_{>0}\): a hyperparameter

\[\begin{split}p(\boldsymbol{\theta}, \tau | \boldsymbol{X}, \boldsymbol{y}) &= \mathcal{N}(\boldsymbol{\theta}|\boldsymbol{\mu}_n, (\tau \boldsymbol{\Lambda}_n)^{-1}) \mathrm{Gam}(\tau|\alpha_n,\beta_n)\\ &= \frac{|\tau \boldsymbol{\Lambda}_n|^{1/2}}{(2 \pi)^{d/2}} \exp \left\{ -\frac{\tau}{2} (\boldsymbol{\theta} - \boldsymbol{\mu}_n)^\top \boldsymbol{\Lambda}_n (\boldsymbol{\theta} - \boldsymbol{\mu}_n) \right\} \frac{\beta_n^{\alpha_n}}{\Gamma (\alpha_n)} \tau^{\alpha_n - 1} \exp \{ -\beta_n \tau \} .\end{split}\]

\[\begin{split}\mathbb{E}[\boldsymbol{\theta} | \boldsymbol{X}, \boldsymbol{y}] &= \boldsymbol{\mu}_n & \left( \alpha_n > \frac{1}{2} \right), \\ \mathrm{Cov}[\boldsymbol{\theta} | \boldsymbol{X}, \boldsymbol{y}] &= \frac{\beta_n}{\alpha_n - 1} \boldsymbol{\Lambda}_n^{-1} & (\alpha_n > 1), \\ \mathbb{E}[\tau | \boldsymbol{X}, \boldsymbol{y}] &= \frac{\alpha_n}{\beta_n}, \\ \mathbb{V}[\tau | \boldsymbol{X}, \boldsymbol{y}] &= \frac{\alpha_n}{\beta_n^2},\end{split}\]

where the updating rules of the hyperparameters are

\[\begin{split}\boldsymbol{\Lambda}_n &= \boldsymbol{\Lambda}_0 + \boldsymbol{X}^\top \boldsymbol{X},\\ \boldsymbol{\mu}_n &= \boldsymbol{\Lambda}_n^{-1} (\boldsymbol{\Lambda}_0 \boldsymbol{\mu}_0 + \boldsymbol{X}^\top \boldsymbol{y}),\\ \alpha_n &= \alpha_0 + \frac{n}{2},\\ \beta_n &= \beta_0 + \frac{1}{2} \left( -\boldsymbol{\mu}_n^\top \boldsymbol{\Lambda}_n \boldsymbol{\mu}_n + \boldsymbol{y}^\top \boldsymbol{y} + \boldsymbol{\mu}_0^\top \boldsymbol{\Lambda}_0 \boldsymbol{\mu}_0 \right).\end{split}\]

The predictive distribution is as follows:

\(\boldsymbol{x}_{n+1}\in \mathbb{R}^d\): a new data point
\(y_{n+1}\in \mathbb{R}\): a new objective variable
\(m_\mathrm{p}\in \mathbb{R}\): a parameter
\(\lambda_\mathrm{p}\in \mathbb{R}\): a parameter
\(\nu_\mathrm{p}\in \mathbb{R}\): a parameter

\[\begin{split}p(y_{n+1} | \boldsymbol{X}, \boldsymbol{y}, \boldsymbol{x}_{n+1} ) &= \mathrm{St}\left(y_{n+1} \mid m_\mathrm{p}, \lambda_\mathrm{p}, \nu_\mathrm{p}\right) \\ &= \frac{\Gamma (\nu_\mathrm{p} / 2 + 1/2 )}{\Gamma (\nu_\mathrm{p} / 2)} \left( \frac{\lambda_\mathrm{p}}{\pi \nu_\mathrm{p}} \right)^{1/2} \left( 1 + \frac{\lambda_\mathrm{p} (y_{n+1} - m_\mathrm{p})^2}{\nu_\mathrm{p}} \right)^{-\nu_\mathrm{p}/2 - 1/2},\end{split}\]

\[\begin{split}\mathbb{E}[y_{n+1} | \boldsymbol{X}, \boldsymbol{y}, \boldsymbol{x}_{n+1}] &= m_\mathrm{p} & (\nu_\mathrm{p} > 1), \\ \mathbb{V}[y_{n+1} | \boldsymbol{X}, \boldsymbol{y}, \boldsymbol{x}_{n+1}] &= \frac{1}{\lambda_\mathrm{p}} \frac{\nu_\mathrm{p}}{\nu_\mathrm{p}-2} & (\nu_\mathrm{p} > 2),\end{split}\]

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

\[\begin{split}m_\mathrm{p} &= \boldsymbol{x}_{n+1}^{\top} \boldsymbol{\mu}_{n}, \\ \lambda_\mathrm{p} &= \frac{\alpha_{n}}{\beta_{n}}\left(1+\boldsymbol{x}_{n+1}^{\top} \boldsymbol{\Lambda}_{n} \boldsymbol{x}_{n+1}\right)^{-1}, \\ \nu_\mathrm{p} &= 2 \alpha_{n}.\end{split}\]

class bayesml.linearregression.GenModel(c_degree, theta_vec=None, tau=1.0, h_mu_vec=None, h_lambda_mat=None, h_alpha=1.0, h_beta=1.0, seed=None)#

Bases: Generative

The stochastic data generative model and the prior distribution.

Parameters:

c_degreeint: a positive integer.
theta_vecnumpy ndarray, optional: a vector of real numbers, by default [0.0, 0.0, … , 0.0]
taufloat, optional: a positive real number, by default 1.0
h_mu_vecnumpy ndarray, optional: a vector of real numbers, by default [0.0, 0.0, … , 0.0]
h_lambda_matnumpy ndarray, optional: a positive definate matrix, by default the identity matrix
h_alphafloat, optional: a positive real number, by default 1.0
h_betafloat, optional: a positive real number, by default 1.0
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, x, constant])	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, x, constant])	Save the generated sample as NumPy `.npz` format.
`set_h_params`([h_mu_vec, h_lambda_mat, ...])	Set the hyperparameters of the prior distribution.
`set_params`([theta_vec, tau])	Set the parameter of the sthocastic data generative model.
`visualize_model`([sample_size, constant])	Visualize the stochastic data generative model and generated samples.

get_constants()#

Get constants of GenModel.

Returns:

constantsdict of {str: int}

"c_degree" : the value of self.c_degree

set_h_params(h_mu_vec=None, h_lambda_mat=None, h_alpha=None, h_beta=None)#

Set the hyperparameters of the prior distribution.

Parameters:

h_mu_vecnumpy ndarray, optional: a vector of real numbers, by default None.
h_lambda_matnumpy ndarray, optional: a positive definate matrix, by default None.
h_alphafloat, optional: a positive real number, by default None.
h_betafloat, optional: a positive real number, by default None.

get_h_params()#

Get the hyperparameters of the prior distribution.

Returns:

h_paramsdict of {str: float or numpy ndarray}

"h_mu_vec" : The value of self.h_mu_vec
"h_lambda_mat" : The value of self.h_lambda_mat
"h_alpha" : The value of self.h_alpha
"h_beta" : The value of self.h_beta

gen_params()#

Generate the parameter from the prior distribution.

The generated vaule is set at self.theta_vec and ``self.tau.

set_params(theta_vec=None, tau=None)#

Set the parameter of the sthocastic data generative model.

Parameters:

theta_vecnumpy ndarray, optional: a vector of real numbers, by default None
taufloat, optional, optional: a positive real number, by default None

get_params()#

Get the parameter of the sthocastic data generative model.

Returns:

paramsdict of {str: float or numpy ndarray}

"theta_vec" : The value of self.theta_vec.
"tau" : The value of self.tau.

gen_sample(sample_size=None, x=None, constant=True)#

Generate a sample from the stochastic data generative model.

If x is given, it will be used for explanatory variables as it is (independent of the other options: sample_size and constant).

If x is not given, it will be generated from i.i.d. standard normal distribution. The size of the generated sample is defined by sample_size. If constant is True, the last element of the generated explanatory variables will be overwritten by 1.0.

Parameters:

sample_sizeint, optional: A positive integer, by default None.
xnumpy ndarray, optional: float array whose shape is (sammple_length,c_degree), by default None.
constantbool, optional: A boolean value, by default True.

Returns:

xnumpy ndarray: float array whose shape is (sammple_length,c_degree).
ynumpy ndarray: 1 dimensional float array whose size is sammple_length.

save_sample(filename, sample_size=None, x=None, constant=True)#

Save the generated sample as NumPy .npz format.

If x is given, it will be used for explanatory variables as it is (independent of the other options: sample_size and constant).

The generated sample is saved as a NpzFile with keyword: “x”, “y”.

Parameters:

filenamestr: The filename to which the sample is saved. .npz will be appended if it isn’t there.
xnumpy ndarray, optional: float array whose shape is (sammple_length,c_degree), by default None.
sample_sizeint, optional: A positive integer, by default None.
constantbool, optional: A boolean value, by default True.

See also

numpy.savez_compressed

visualize_model(sample_size=100, constant=True)#

Visualize the stochastic data generative model and generated samples.

Parameters:

sample_sizeint, optional: A positive integer, by default 50
constantbool, optional

Examples

>>> import numpy as np
>>> from bayesml import linearregression
>>> model = linearregression.GenModel(c_degree=2,theta_vec=np.array([2,1]))
>>> model.visualize_model()

class bayesml.linearregression.LearnModel(c_degree, h0_mu_vec=None, h0_lambda_mat=None, h0_alpha=1.0, h0_beta=1.0)#

Bases: Posterior, PredictiveMixin

The posterior distribution and the predictive distribution.

Parameters:

c_degreeint: a positive integer.
h0_mu_vecnumpy ndarray, optional: a vector of real numbers, by default [0.0, 0.0, … , 0.0]
h0_lambda_matnumpy ndarray, optional: a positive definate matrix, by default the identity matrix
h0_alphafloat, optional: a positive real number, by default 1.0
h0_betafloat, optional: a positive real number, by default 1.0

Attributes:

hn_mu_vecnumpy ndarray: a vector of real numbers
hn_lambda_matnumpy ndarray: a positive definate matrix
hn_alphafloat: a positive real number
hn_betafloat: a positive real number
p_mfloat: a positive real number
p_lambdafloat: a positive real number
p_nufloat: a positive real number

Methods

`calc_log_marginal_likelihood`()	Calculate log marginal likelihood
`calc_pred_dist`(x)	Calculate the parameters of the predictive distribution.
`estimate_params`([loss, dict_out])	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, y[, loss])	Predict a new data 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_mu_vec, h0_lambda_mat, ...])	Set initial values of the hyperparameter of the posterior distribution.
`set_hn_params`([hn_mu_vec, hn_lambda_mat, ...])	Set updated values of the hyperparameter of the posterior distribution.
`update_posterior`(x, y)	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}

"c_degree" : the value of self.c_degree

set_h0_params(h0_mu_vec=None, h0_lambda_mat=None, h0_alpha=None, h0_beta=None)#

Set initial values of the hyperparameter of the posterior distribution.

Note that the parameters of the predictive distribution are also calculated from self.h0_mu_vec, slef.h0_lambda_mat, self.h0_alpha and self.h0_beta.

Parameters:

h0_mu_vecnumpy ndarray, optional: a vector of real numbers, by default None.
h0_lambda_matnumpy ndarray, optional: a positive definate matrix, by default None.
h0_alphafloat, optional: a positive real number, by default None.
h0_betafloat, optional: a positive real number, by default None.

get_h0_params()#

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

Returns:

h0_paramsdict of {str: float or numpy ndarray}

"h0_mu_vec" : The value of self.h0_mu_vec
"h0_lambda_mat" : The value of self.h0_lambda_mat
"h0_alpha" : The value of self.h0_alpha
"h0_beta" : The value of self.h0_beta

set_hn_params(hn_mu_vec=None, hn_lambda_mat=None, hn_alpha=None, hn_beta=None)#

Set updated values of the hyperparameter of the posterior distribution.

Note that the parameters of the predictive distribution are also calculated from self.hn_mu_vec, slef.hn_lambda_mat, self.hn_alpha and self.hn_beta.

Parameters:

hn_mu_vecnumpy ndarray, optional: a vector of real numbers, by default None.
hn_lambda_matnumpy ndarray, optional: a positive definate matrix, by default None.
hn_alphafloat, optional: a positive real number, by default None.
hn_betafloat, optional: a positive real number, by default None.

get_hn_params()#

Get the hyperparameters of the posterior distribution.

Returns:

hn_paramsdict of {str: float or numpy ndarray}

"hn_mu_vec" : The value of self.hn_mu_vec
"hn_lambda_mat" : The value of self.hn_lambda_mat
"hn_alpha" : The value of self.hn_alpha
"hn_beta" : The value of self.hn_beta

update_posterior(x, y)#

Update the hyperparameters of the posterior distribution using traning data.

Parameters:

xnumpy ndarray: float array. The size along the last dimension must conincides with the c_degree. If you want to use a constant term, it should be included in x.
ynumpy ndarray: float array.

estimate_params(loss='squared', dict_out=False)#

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

Note that the criterion is applied to estimating theta_vec and tau independently. Therefore, a tuple of the student’s t-distribution and the gamma distribution 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”, “abs”, and “KL”.
dict_outbool, optional: If True, output will be a dict, by default False.

Returns:

estimatestuple of {numpy ndarray, float, None, or rv_frozen}

theta_vec : the estimate for w
tau_hat : the estimate for tau

The estimated values under the given loss function. If it is not exist, None will be returned. If the loss function is “KL”, the posterior distribution itself will be returned as rv_frozen object of scipy.stats.

bayesml.linearregression package

Contents

bayesml.linearregression package#

Module contents#