Gaussian Regression observation model: Variational Bayesian Methods

Generative model

The Gaussian regression observation model explains a observed collection of input/ouptut data pairs \(\{x_n, y_n\}_{n=1}^N\). Each input observation \(x_n\) is a vector of length D, while each output observation \(y_n\) is a scalar.

In this document, we assume that the observed input data \(x\) are fixed and focus on a generative model for the output data \(y\) which depends on \(x\). Various generative models, such as the diagonal-covariance Gaussian, are possible for the observed data \(x\). These can be straight-forwardly combined with the model here to produce a joint model for both \(x\) and \(y\).

Each cluster k produces the output values according to the following standard Bayesian linear regression model:

\[y_{n} \sim \mathcal{N} \left( b_k + \sum_{d=1}^D w_{kd} x_{nd}, \delta_{k}^{-1} \right)\]

Here, the cluster-specific parameters are a weight vector \(w_k\), an intercept weight \(b_k\), and a precision scalar \(\delta_k\). These are the global random variables of this observation model.

Alternatively, if we define an expanded input data vector \(\tilde{x}_n = [x_{n1} x_{n2} \ldots x_{nD} ~ 1]\), we can write the generative model more simply as:

\[y_{n} \sim \mathcal{N} \left( \sum_{d=1}^{D+1} w_{kd} \tilde{x}_{nd}, \delta_{k}^{-1} \right)\]

Global Random Variables

The global random variables are the cluster-specific weights and precisions. For each cluster k, we have

\[\begin{split}w_{k} &\in \mathbb{R}^D, \qquad w_k = [w_{k1}, w_{k2}, \ldots w_{kD} w_{kD+1} ] \\ \delta_k &\in (0, +\infty)\end{split}\]

For convenience, let \(E\) denote the size of this expanded representation, where \(E = D+1\).

Local Random Variables

Each dataset observation at index n has its own cluster assignment:

\[z_n \in \{1, 2, \ldots K \}\]

The generative model and approximate posterior for \(z_n\) is determined by an allocation model. For all computations needed by our current observation model, we’ll assume either a point estimate or an approximate posterior for \(z_n\) is known.

Normal-Wishart prior

We assume that the weights \(w_k\) and the precision \(\delta_k\) have a joint Normal-Wishart prior with hyperparameters:

  • \(\bar{\nu}\) : positive scalar

    count parameter of the Wishart prior on precision

  • \(\bar{\tau}\) : positive scalar

    location parameter of the Wishart prior on precision

  • \(\bar{w}\) : vector of size E

    mean value of the Normal prior on cluster weights

  • \(\bar{P}\) : positive definite E x E matrix

    precision matrix for the Normal prior on cluster weights

Mathematically, we have:

\[\begin{split}\delta_{k} &\sim \mathcal{W}_1(\bar{\nu}, \bar{\tau}) \\ w_{k} &\sim \mathcal{N}_E( \bar{w}, \delta_k^{-1} \bar{P}^{-1} )\end{split}\]

Under this prior, here are some useful expectations for the precision random variable:

\[\begin{split}\E_{\mbox{prior}}[ \delta_k ] &= \frac{\bar{\nu}}{\bar{\tau}} \\ \E_{\mbox{prior}}[ \delta_k^{-1} ] &= \frac{\bar{\tau}}{\bar{\nu} - 2} \\ \mbox{Var}_{\mbox{prior}}[ \delta_k ] &= \frac{\bar{\nu}}{\bar{\tau}^2}\end{split}\]

Likewise, here are some useful prior expectations for the weight vector random variable:

\[\begin{split}\E_{\mbox{prior}}[w_k] &= \bar{w} \\ \mbox{Cov}_{\mbox{prior}}[w_k] &= \frac{\bar{\tau}}{\bar{\nu} - 2} \bar{P}^{-1}\end{split}\]

And some useful joint expectations:

\[\begin{split}\E_{\mbox{prior}}[\delta_k w_k ] &= \frac{\bar{\nu}}{\bar{\tau}}\bar{w} \\ \E_{\mbox{prior}}[\delta_k w_k w_k^T ] &= \bar{P}^{-1} + \frac{\bar{\nu}}{\bar{\tau}} \bar{w} \bar{w}^{T}\end{split}\]

In our Python implementation of the GaussRegressYFromFixedXObsModel class, these quantities are represented by the following numpy array attributes of the Prior parameter bag:

  • pnu : float

    value of \(\bar{\nu}\)

  • ptau : float

    value of \(\bar{\tau}\)

  • w_E : 1D array, size E

    value of \(\bar{w}\)

  • P_EE : 2D array, size E x E

    value of \(\bar{P}\)

Several keyword arguments can be used to determine the values of the prior hyperparameters when calling bnpy.run

  • --pnu : float

    Sets value of \(\bar{\nu}\). Defaults to 1.

  • --ptau : float

    Sets value of \(\bar{\tau}\). Defaults to 1.

  • --w_E : float or 1D array

    Sets value of the vector \(\bar{w}\). If float is provided, the whole vector is filled with that value. Defaults to 0.

  • --P_diag_val : float or 1D array

    Sets \(\bar{P}\) to diagonal matrix with specified values. Defaults to 1e-6.

Approximate posterior

We assume the following factorized approximate posterior family for variational optimization:

\[q(z, w, \delta) = \prod_{n=1}^N q(z_n) \cdot \prod_{k=1}^K (w_k, \delta_k )\]

The specific forms of the global and local factors are given below.

Posterior for local assignments

For each observation vector at index n, we assume an independent approximate posterior over the assigned cluster indicator \(z_n \in \{1, 2, \ldots K \}\).

\[\begin{split}q( z ) &= \prod_{n=1}^N q(z_n | \hat{r}_n ) \\ &= \prod_{n=1}^N \mbox{Discrete}( z_n | \hat{r}_{n1}, \hat{r}_{n2}, \ldots \hat{r}_{nK})\end{split}\]

Thus, for this observation model the only local variational parameter is the assignment responsibility array \(\hat{r} = \{ \{ \hat{r}_{nk} \}_{k=1}^K \}_{n=1}^N\).

Inside the LP dict, this is represented by the resp numpy array:

  • resp : 2D array, size N x K

    Parameters of approximate posterior q(z) over cluster assignments. resp[n,k] = probability observation n is assigned to component k.

Remember, all computations required by our observation model assume that the resp array is given. The actual values of resp are updated by an allocation model.

Posterior for global parameters

The goal of variational optimization is to find the best approximate posterior distribution for the mean and precision parameters of each cluster k:

\[\begin{split}q( w, \delta ) &= \prod_{k=1}^K \mathcal{W}_1( \delta_{k} | \hat{\nu}_k, \hat{\tau}_{k} ) \mathcal{N}_E( w_{k} | \hat{w}_{k}, \delta_k^{-1} \hat{P}_k^{-1} )\end{split}\]

Within our Python implementation in the class GaussRegressYFromFixedXObsModel, this approximate posterior is represented within the Post attribute. This attribute is a ParamBag object containing the following numpy arrays:

  • K : int

    number of active clusters

  • pnu_K : 1D array, size K

    Defines \(\hat{\nu}_k\) for each cluster

  • ptau_K : 1D array, size K

    Defines \(\hat{\tau}_{k}\) for each cluster

  • w_KE : 2D array, size K x E

    Defines \(\hat{w}_{ke}\) for each cluster and expanded dimension

  • P_KEE : 2D array, size K x E x E

    Defines precision matrix \(\hat{P}_{k}\) for each cluster

Objective function

Variational optimization will find the approximate posterior parameters that maximize the following objective function, given a fixed observed dataset \(x = \{x_1, \ldots x_N \}\) and fixed prior hyparparameters \(\bar{\nu}, \bar{\tau}, \bar{w}, \bar{P}\).

\[\begin{split}\mathcal{L}^{\smalltext{Gaussian Regression}}(y, x, \hat{\nu}, \hat{\tau}, \hat{w}, \hat{P} ) &= -\frac{N}{2} \log 2\pi \\ & \quad + \sum_{k=1}^K \sum_{d=1}^D c^{\smalltext{NW}}_{1,E}( \hat{\nu}_k, \hat{\tau}_{k}, \hat{w}_{k}, \hat{P}_k) - c^{\smalltext{NW}}_{1,E}( \bar{\nu}, \bar{\tau}, \bar{w}, \bar{P}) \\ & \quad -\frac{1}{2} \sum_{k=1}^K \left( N_k(\hat{r}) + \bar{\nu} - \hat{\nu}_k \right) \E_q[ \log \delta_k ] \\ & \quad -\frac{1}{2} \sum_{k=1}^K \left( S_{k}^{yy}(y, \hat{r}) + \bar{\tau} + \bar{w}\bar{P}\bar{w} - \hat{\tau}_k - \hat{w}_k \hat{P}_k \hat{w}_k \right) \E_q[ \delta_k ] \\ & \quad + \sum_{k=1}^K \left( S_k^{yx}(x, y, \hat{r}) + \bar{P} \bar{w} - \hat{P}_k \hat{w}_k \right)^T \E_q[ \delta_k w_k ] \\ & \quad - \frac{1}{2} \sum_{k=1}^K \mbox{trace} \left( \left( S_k^{xx^T}(x, \hat{r}) + \bar{P} - \hat{P}_k \right) \E_q[ \delta_k w_k w_k^T] \right)\end{split}\]

This objective function is computed by calling the Python function calc_evidence.

We can directly interpret this function as a lower bound on the marginal evidence:

\[\log p(y | x, \bar{\nu}, \bar{\tau}, \bar{w}, \bar{P}) \geq \mathcal{L}^{\smalltext{Gaussian Regression}} (y, x, \hat{\nu}, \hat{\tau}, \hat{w}, \hat{P} )\]

Sufficient statistics

The sufficient statistics of this observation model are functions of the local parameters \(\hat{r}\), the observed input data \(x\), and the observed output data \(y\).

\[\begin{split}N_{k}(\hat{r}) &= \sum_{n=1}^N \hat{r}_{nk} \\ S^{y^2}_{k}(y, \hat{r}) &= \sum_{n=1}^N \hat{r}_{nk} y_n^2 \\ S^{yx}_{k}(x, y, \hat{r}) &= \sum_{n=1}^N \hat{r}_{nk} y_n x_{n} \\ S^{xx^T}_{k}(x, \hat{r}) &= \sum_{n=1}^N \hat{r}_{nk} x_{n} x_{n}^T\end{split}\]

These fields are stored within the sufficient statistics parameter bag SS as the following fields:

  • SS.N : 1D array, size K

    SS.N[k] = \(N_k\)

  • SS.yy_K : 1D array, size K

    SS.yy[k] = \(S^{y^2}_{k}(y, \hat{r})\)

  • SS.yx : 2D array, size K x E

    SS.yx[k] = \(S^{yx}_{k}(x, y, \hat{r})\)

  • SS.xxT : 3D array, size K x E x E

    SS.xxT[k] = \(S^{xx^T}_{k}(x, \hat{r})\)

Cumulant function

The cumulant function of the Normal-Wishart produces a scalar output from 4 input arguments:

\[\begin{split}c^{\smalltext{NW}}_{1,E}(\nu, \tau, w, P) &= \frac{E}{2} \log 2 \pi - \frac{1}{2} \log |P| - \frac{\nu}{2} \log \frac{\tau}{2} + \log \Gamma \left( \frac{\nu}{2} \right)\end{split}\]

where \(\Gamma(\cdot)\) is the gamma function, and \(\log |P|\) is the log determinant of the E x E matrix \(P\).

Coordinate Ascent Updates

Local step update

As with all observation models, the local step computes the expected log conditional probability of assigning each observation to each cluster:

\[\E_q[ \log p( y_n | x_n, w_k, \delta_k) ] = - \frac{1}{2} \log 2 \pi + \frac{1}{2} \E[ \log \delta_{k} ] - \frac{1}{2} \E[ \delta_{k} (y_{n} - w_k ^T \tilde{x}_n)^2 ]\]

where the elementary expectations required are:

\[\begin{split}\E_q [ \log \delta_{k} ] &= - \log \frac{\hat{\tau}_k}{2} + \psi \left( \frac{\hat{\nu}_k}{2} \right) \\ \E_q \left[ \delta_{k} \left( y_{n} - w_k^T \tilde{x}_n \right)^2 \right] &= \tilde{x}_n^T \hat{P}_k^{-1} \tilde{x}_n + \frac{ \hat{\nu}_k }{ \hat{\tau}_{k} } (y_{n} - \bar{w}_{k}^T \tilde{x}_n)^2\end{split}\]

The above operations can be efficiently computed via smart vectorized calculations on modern cpus.

In our implementation, this is done via the function calc_local_params, which computes the following arrays and places them inside the local parameter dict LP.

  • E_log_soft_ev : 2D array, N x K

    log probability of assigning each observation n to each cluster k

Global step update

The global step update produces an updated approximate posterior over the global random variables. Concretely, this means updated values for each of the four parameters which define each cluster-specific Normal-Wishart:

\[\begin{split}\hat{\nu}_k &\gets N_k(\hat{r}) + \bar{\nu} \\ \hat{P}_{k} &\gets \bar{P}_k + S^{xx^T}_k(x, \hat{r}) \\ \hat{w}_{k} &\gets \hat{P}_k^{-1} \left( \bar{P} \bar{w} + S^{yx}_k(x, y, \hat{r}) \right) \\ \hat{\tau}_k &\gets \bar{\tau} + S^{y^2}_k(y, \hat{r}) + \bar{w}^T \bar{P} \bar{w} - \hat{w}_k^T \hat{P}_k \hat{w}_k\end{split}\]

Our implementation performs this update when calling the function update_global_params.

Initialization

Initialization creates valid values of the parameters which define the approximate posterior over the global random variables. Concretely, this means it creates a valid setting of the Post attribute of the GaussRegressYFromFixedXObsModel object.

TODO