# Mixture Density Network¶

The Mixture Density Network (MDN) [BISHOP1994] combines a conventional neural network (in our implementation specified as $$estimator$$) with a mixture density model for modeling conditional probability distributions $$p(t|x)$$. Given a sufficiently flexible network and considering a parametric mixture model, the parameters of the distribution $$t$$ can be determined by the outputs of the neural network provided the input to the network is $$x$$ (in our implementation specified as X_ph (placeholder) and X). This approach therefore constitutes as a framework capable of approximating arbitrary conditional distributions.

The following example develops a model for Gaussian components with isotropic component covariances, while $$K$$ being the number of components of a single mixture (our model currently allows to choose an arbitrary number of (global) mixture components, see parameter L below) and $$\pi(x)$$ denoting the mixing coefficients:

$p(t|x) = \sum_{k=1}^K \pi_{k}(x) \mathcal{N}(t|\mu_{k}(x), \sigma_{k}^2(x))$

It is both feasible to replace the components by components of other distributions and extending the MDN to arbitrary covariance matrices. Although the later is generally much more difficult, it has been shown by [TANSEY2016] that, for example one can have the MDN output the lower triangular entries in the Cholesky decomposition.

Using $$x$$ as input, the mixing coefficients $$\pi_{k}(x)$$, the means $$\mu_{k}(x)$$, and the variances $$\sigma_{k}^2(x)$$ can be governed by the outputs of neural network. Assuming the mixture model has L mixture components (in our implementation specified as n_centers), the total number of network outputs is given by $$(K+2)L$$.

The mixing coefficients are computed as a set of $$L$$ softmax outputs, where $$a_k^{\pi}$$ determine the mixing coefficients emitted by the network:

$\pi_k(x) = \frac{exp(a_k^{\pi})}{\sum_{l=1}^K exp(a_k^{\pi})}$

ensuring the constraint that $$\pi_k(x)$$ over $$K$$ must sum to 1. Similarly, the variances must me larger or equal to zero. Due to isotropy we have $$L$$ kernel widths $$\sigma_k(x)$$ which are determined by the network output $$a_k^{\sigma}$$ and can be represented as exponentials:

$\sigma_k(x) = exp(a_k^{\sigma})$

For the $$K \times L$$ means we directly use the network outputs: $$\mu_k(x) = a_{kj}^{\sigma}$$.

The weights and biases $$w$$ of the neural network are learned by minimizing the negative logarithm of the likelihood (maximum likelihood) over $$N$$ data points:

$E(w) = - \sum_{n=1}^N \ln \bigg\{\sum_{k=1}^k \pi_k(x_n, w) \mathcal{N} (t_n|\mu_k(x_n, w), \sigma_k^2(x_n,w)) \bigg\}$

This can be executed via the standard backpropagation algorithm, given that suitable expressions for the derivations can be obtained.

class cde.density_estimator.MixtureDensityNetwork(name, ndim_x, ndim_y, n_centers=10, hidden_sizes=(16, 16), hidden_nonlinearity=<function tanh>, n_training_epochs=1000, x_noise_std=None, y_noise_std=None, entropy_reg_coef=0.0, weight_decay=0.0, weight_normalization=True, data_normalization=True, dropout=0.0, random_seed=None)[source]

Mixture Density Network Estimator

See “Mixture Density networks”, Bishop 1994

Parameters
• name – (str) name space of MDN (should be unique in code, otherwise tensorflow namespace collitions may arise)

• ndim_x – (int) dimensionality of x variable

• ndim_y – (int) dimensionality of y variable

• n_centers – Number of Gaussian mixture components

• hidden_sizes – (tuple of int) sizes of the hidden layers of the neural network

• hidden_nonlinearity – (tf function) nonlinearity of the hidden layers

• n_training_epochs – Number of epochs for training

• x_noise_std – (optional) standard deviation of Gaussian noise over the the training data X -> regularization through noise

• y_noise_std – (optional) standard deviation of Gaussian noise over the the training data Y -> regularization through noise

• entropy_reg_coef – (optional) scalar float coefficient for shannon entropy penalty on the mixture component weight distribution

• weight_decay – (float) the amount of decoupled (http://arxiv.org/abs/1711.05101) weight decay to apply

• weight_normalization – (boolean) whether weight normalization shall be used

• data_normalization – (boolean) whether to normalize the data (X and Y) to exhibit zero-mean and std

• dropout – (float) the probability of switching off nodes during training

• random_seed – (optional) seed (int) of the random number generators used

cdf(X, Y)

Predicts the conditional cumulative probability p(Y<=y|X=x). Requires the model to be fitted.

Parameters
• X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_samples, n_dim_y)

Returns

conditional cumulative probability p(Y<=y|X=x) - numpy array of shape (n_query_samples, )

conditional_value_at_risk(x_cond, alpha=0.01, n_samples=10000000)

Computes the Conditional Value-at-Risk (CVaR) / Expected Shortfall of a GMM. Only if ndim_y = 1

Based on formulas from section 2.3.2 in “Expected shortfall for distributions in finance”, Simon A. Broda, Marc S. Paolella, 2011

Parameters
• x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

• alpha – quantile percentage of the distribution

Returns

CVaR values for each x to condition on - numpy array of shape (n_values)

covariance(x_cond, n_samples=None)

Covariance of the fitted distribution conditioned on x_cond

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns

Covariances Cov[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y, ndim_y)

eval_by_cv(X, Y, n_splits=5, verbose=True)

Fits the conditional density model with cross-validation by using the score function of the BaseDensityEstimator for scoring the various splits.

Parameters
• X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_samples, n_dim_y)

• n_splits – number of cross-validation folds (positive integer)

• verbose – the verbosity level

fit(X, Y, random_seed=None, verbose=True, eval_set=None, **kwargs)[source]

Fits the conditional density model with provided data

Parameters
• X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_samples, n_dim_y)

• eval_set – (tuple) eval/test set - tuple (X_test, Y_test)

• verbose – (boolean) controls the verbosity (console output)

fit_by_cv(X, Y, n_folds=3, param_grid=None, random_state=None, verbose=True, n_jobs=-1)

Fits the conditional density model with hyperparameter search and cross-validation.

• Determines the best hyperparameter configuration from a pre-defined set using cross-validation. Thereby, the conditional log-likelihood is used for simulation_eval.

• Fits the model with the previously selected hyperparameter configuration

Parameters
• X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_samples, n_dim_y)

• n_folds – number of cross-validation folds (positive integer)

• param_grid

(optional) a dictionary with the hyperparameters of the model as key and and a list of respective parametrizations as value. The hyperparameter search is performed over the cartesian product of the provided lists. Example:

{"n_centers": [20, 50, 100, 200],
"center_sampling_method": ["agglomerative", "k_means", "random"],
"keep_edges": [True, False]
}


• random_state – (int) seed used by the random number generator for shuffeling the data

get_configuration(deep=True)

Get parameter configuration for this estimator.

Parameters

deep – boolean, optional If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params - mapping of string to any Parameter names mapped to their values.

get_params(deep=True)

Get parameters for this estimator.

Parameters

deep (boolean, optional) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params – Parameter names mapped to their values.

Return type

mapping of string to any

get_params_internal(**tags)

Internal method to be implemented which does not perform caching

kurtosis(x_cond, n_samples=1000000)

Kurtosis of the fitted distribution conditioned on x_cond

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns

Kurtosis Kurt[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y, ndim_y)

log_pdf(X, Y)

Predicts the conditional log-probability log p(y|x). Requires the model to be fitted.

Parameters
• X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_samples, n_dim_y)

Returns

onditional log-probability log p(y|x) - numpy array of shape (n_query_samples, )

mean_(x_cond, n_samples=None)

Mean of the fitted distribution conditioned on x_cond :param x_cond: different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns

Means E[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y)

mean_std(x_cond, n_samples=None)
Computes Mean and Covariance of the fitted distribution conditioned on x_cond.

Computationally more efficient than calling mean and covariance computatio separately

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns

Means E[y|x] and Covariances Cov[y|x]

pdf(X, Y)

Predicts the conditional probability p(y|x). Requires the model to be fitted.

Parameters
• X – numpy array to be conditioned on - shape: (n_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_samples, n_dim_y)

Returns

conditional probability p(y|x) - numpy array of shape (n_query_samples, )

plot2d(x_cond=[0, 1, 2], ylim=(-8, 8), resolution=100, mode='pdf', show=True, prefix='', numpyfig=False)

Generates a 3d surface plot of the fitted conditional distribution if x and y are 1-dimensional each

Parameters
• xlim – 2-tuple specifying the x axis limits

• ylim – 2-tuple specifying the y axis limits

• resolution – integer specifying the resolution of plot

plot3d(xlim=(-5, 5), ylim=(-8, 8), resolution=100, show=False, numpyfig=False)

Generates a 3d surface plot of the fitted conditional distribution if x and y are 1-dimensional each

Parameters
• xlim – 2-tuple specifying the x axis limits

• ylim – 2-tuple specifying the y axis limits

• resolution – integer specifying the resolution of plot

predict_density(X, Y=None, resolution=100)

Computes conditional density p(y|x) over a predefined grid of y target values

Parameters
• X – values/vectors to be conditioned on - shape: (n_instances, n_dim_x)

• Y – (optional) y values to be evaluated from p(y|x) - if not set, Y will be a grid with with specified resolution

• resolution

integer specifying the resolution of simulation_eval grid

Returns: tuple (P, Y)
• P - density p(y|x) - shape (n_instances, resolution**n_dim_y)

• Y - grid with with specified resolution - shape (resolution**n_dim_y, n_dim_y) or a copy of Y in case it was provided as argument

reset_fit()

resets all tensorflow objects and :return:

sample(X)

sample from the conditional mixture distributions - requires the model to be fitted

Parameters

X – values to be conditioned on when sampling - numpy array of shape (n_instances, n_dim_x)

Returns: tuple (X, Y)
• X - the values to conditioned on that were provided as argument - numpy array of shape (n_samples, ndim_x)

• Y - conditional samples from the model p(y|x) - numpy array of shape (n_samples, ndim_y)

score(X, Y)

Computes the mean conditional log-likelihood of the provided data (X, Y)

Parameters
• X – numpy array to be conditioned on - shape: (n_query_samples, n_dim_x)

• Y – numpy array of y targets - shape: (n_query_samples, n_dim_y)

Returns

average log likelihood of data

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Returns

Return type

self

skewness(x_cond, n_samples=1000000)

Skewness of the fitted distribution conditioned on x_cond

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns

Skewness Skew[y|x] corresponding to x_cond - numpy array of shape (n_values, ndim_y, ndim_y)

std_(x_cond, n_samples=1000000)

Standard deviation of the fitted distribution conditioned on x_cond

Parameters

x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

Returns

Standard deviations sqrt(Var[y|x]) corresponding to x_cond - numpy array of shape (n_values, ndim_y)

tail_risk_measures(x_cond, alpha=0.01, n_samples=10000000)

Computes the Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)

Parameters
• x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

• alpha – quantile percentage of the distribution

• n_samples – number of samples for monte carlo model_fitting

Returns

• VaR values for each x to condition on - numpy array of shape (n_values)

• CVaR values for each x to condition on - numpy array of shape (n_values)

value_at_risk(x_cond, alpha=0.01, n_samples=1000000)

Computes the Value-at-Risk (VaR) of the fitted distribution. Only if ndim_y = 1

Parameters
• x_cond – different x values to condition on - numpy array of shape (n_values, ndim_x)

• alpha – quantile percentage of the distribution

Returns

VaR values for each x to condition on - numpy array of shape (n_values)

BISHOP1994

Bishop (1994). Mixture Density Networks, Technical Report, Aston University (http://publications.aston.ac.uk/373/)

TANSEY2016

Tansey et al. (2016). Better Conditional Density Estimation for Neural Networks (https://arxiv.org/abs/1606.02321)