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varbvs.m
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%--------------------------------------------------------------------------
% varbvs.m: Fit variable selection model using variational approximation.
%--------------------------------------------------------------------------
%
% DESCRIPTION:
% Compute fully-factorized variational approximation for Bayesian variable
% selection in linear (family = 'gaussian') or logistic regression (family
% = 'binomial'). More precisely, find the "best" fully-factorized
% approximation to the posterior distribution of the coefficients, with
% spike-and-slab priors on the coefficients. By "best", we mean the
% approximating distribution that locally minimizes the Kullback-Leibler
% divergence between the approximating distribution and the exact
% posterior.
%
% USAGE:
% fit = varbvs(X, Z, y)
% fit = varbvs(X, Z, y, labels, family, options)
% (Use empty matrix [] to apply the default value)
%
% INPUT ARGUMENTS:
% X n x p input matrix, where n is the number of samples,
% and p is the number of variables. X cannot be sparse.
%
% Z n x m covariate data matrix, where m is the number of
% covariates. Do not supply an intercept as a covariate (i.e., a
% column of ones), because an intercept is automatically included
% in the regression model. For no covariates, set Z to the empty
% matrix []. The covariates are assigned an improper, uniform
% prior. Although improper priors are generally not advisable
% because they can result in improper posteriors and Bayes factors,
% this choice allows us to easily integrate out these covariates.
%
% y Vector of length n containing observations of binary
% (family = 'binomial') or continuous (family = 'gaussian')
% outcome. For binary outcomes, all entries of y must be
% 0 or 1.
%
% labels Cell array with p entries containing variable labels. By
% default, it is set to the empty matrix [].
%
% family Outcome type, either 'gaussian' or 'binomial'. Default
% is 'gaussian'.
%
% options A structure (type 'help struct') containing additional
% model parameters and optimization settings. More details about
% these options are given below. Fields, with their default
% settings given, include:
%
% options.tol = 1e-4 (convergence tolerance for inner loop)
% options.maxiter = 1e4 (maximum number of inner loop iterations)
% options.verbose = true (print progress of algorithm to console)
% options.sa = 1 (prior variance parameter settings)
% options.logodds = linspace(-log10(p),-1,20) (log-odds settings)
% options.update_sa (fit model parameter sa to data)
% options.sa0 = 1 (scale parameter for prior on sa)
% options.n0 = 10 (degrees of freedom for prior on sa)
% options.alpha (initial estimate of variational parameter alpha)
% options.mu (initial estimate of variational parameter mu)
% options.initialize_params (see below)
% options.nr = 100 (samples of PVE to draw from posterior)
%
% For family = 'gaussian' only:
% options.sigma = var(y) (residual variance parameter settings)
% options.update_sigma (fit model parameter sigma to data)
%
% For family = 'binomial' only:
% options.eta (initial estimate of variational parameter eta)
% options.optimize_eta (optimize parameter eta)
%
% OUTPUT ARGUMENTS:
% fit A structure (type 'help struct').
% fit.family Either 'gaussian' or 'binomial'.
% fit.n Number of training samples.
% fit.labels Variable names.
% fit.sigma settings for sigma (family = 'gaussian' only).
% fit.sa settings for prior variance parameter, sa.
% fit.logodds prior log-odds settings.
% fit.prior_same true if prior is identical for all variables.
% fit.sa0 scale parameter for prior on sa.
% fit.n0 degrees of freedom for prior on sa.
% fit.update_sigma whether sigma was fit to data (family = 'gaussian' only).
% fit.update_sa whether hyperparameter sa was fit to data.
% fit.logw approximate marginal log-likelihood for each
% setting of hyperparameters.
% fit.w normalized weights compute from logw.
% fit.alpha variational estimates of posterior inclusion probs.
% fit.mu variational estimates of posterior mean coefficients.
% fit.s variational estimates of posterior variances.
% fit.pip "Averaged" posterior inclusion probabilities.
% fit.mu_cov posterior estimates of coefficients for covariates.
% fit.eta variational parameters for family = 'binomial' only.
% fit.optimize_eta whether eta was fit to data (family = 'binomial' only).
% fit.pve estimated PVE per variable (family = 'gaussian' only).
% fit.model_pve posterior samples of proportion of variance in Y
% explained by variable selection model (only for
% family = 'gaussian').
%
% REGRESSION MODELS:
% Two types of outcomes Y are modeled: (1) a continuous outcome, which
% is also referred to as a "quantitative trait" in the genetics
% literature; or (2) a binary outcome with possible values 0 and 1. Y is
% modeled as a continuous outcome by setting family = 'gaussian'. In this
% case, Y is i.i.d. normal with mean u0 + Z*u + X*b and variance sigma,
% in which u and b are vectors of regresion coefficients, and u0 is the
% intercept. In the second case, we use logistic regression to model Y,
% in which the probability that Y = 1 is given by
%
% Pr(Y = 1) = sigmoid(u0 + Z*u + X*b).
%
% See 'help sigmoid' for a description of the sigmoid function. Note
% that the regression always includes an intercept term (u0).
%
% For both regression models, the fitting procedure consists of an inner
% loop and an outer loop. The outer loop iterates over each of the
% hyperparameter settings. The hyperparameters sa, sigma and logodds are
% specified by three arrays with the same number of elements, in which
% options.sa(i), options.sigma(i) and options.logodds(i) specify the ith
% hyperparameter setting. (The exception to this is when logodds is a
% matrix.) Note that sigma is only used for the linear regression model,
% and will generate an error if family = 'binomial'.
%
% HYPERPARAMETERS:
% Hyperparameter sa is the prior variance of regression coefficients for
% variables that are included in the model. This prior variance is always
% scaled by sigma (for logistic regression, we take sigma = 1). Scaling
% the variance of the coefficients in this way is necessary to ensure
% that this prior is invariant to measurement scale (e.g., switching from
% grams to kilograms). Hyperparameter logodds is the prior log-odds that
% a variable is included in the regression model; it is defined as
% logodds = log10(q/(1-q)), where q is the prior probability that a
% variable is included in the regression model. Note that we use the
% base-10 logarithm instead of the natural logarithm because it is
% usually more natural to specify prior log-odds settings in this way.
%
% The prior log-odds may also be specified separately for each variable,
% which is useful is there is prior information about which variables are
% most relevant to the outcome Y. This is accomplished by setting
% options.logodds to a p x ns matrix, where p is the number of
% variables, and ns is the number of hyperparameter settings. In this
% case, fit.prior_same = false.
%
% CO-ORDINATE ASCENT OPTIMIZATION:
% Given a setting of the hyperparameters, options.sa(i), options.sigma(i)
% and options.logodds(:,i), the inner loop cycles through coordinate
% ascent updates to tighten the lower bound on the marginal likelihood,
%
% Pr(Y | X, sigma, sa, logodds).
%
% The inner loop coordinate ascent updates terminate when either (1) the
% maximum number of inner loop iterations is reached, as specified by
% options.maxiter, or (2) the maximum difference between the estimated
% posterior inclusion probabilities (see below) is less than options.tol.
%
% To provide a more accurate variational approximation of the posterior
% distribution, by default the fitting procedure has two stages. In the
% first stage, the entire fitting procedure is run to completion, and the
% variational parameters (alpha, mu, s, eta) corresponding to the maximum
% lower bound are then used to initialize the coordinate ascent updates
% in a second stage. Although this has the effect of doubling the
% computation time (in the worst case), the final posterior estimates
% tend to be more accurate with this two-stage fitting procedure. The
% initial stage will be automatically skipped if initial estimates of the
% variational parameters are provided in options.alpha, options.mu and/or
% options.s, unless options.initialize_params = true.
%
% It is possible to optimize hyperparameter sa (and sigma for family =
% 'gaussian') as part of the inner loop fitting procedure. Parameters sa
% and sigma will automatically be fitted to the data, separately for each
% hyperparameter setting, when options.sa and options.sigma are not
% specified. Alternatively, this can be achieved by setting
% options.update_sa = true and options.update_sigma = true, in which case
% options.sa and options.sigma are treated as initial estimates of these
% parameters if they are provided. These parameters are fitted by
% computing approximate maximum-likelihood (ML) estimates. Optionally, an
% approximate maximum a posteriori (MAP) estimate of sa is computed by
% setting options.sa0 and options.n0 to positive scalars; these two
% numbers specify the scale parameter and number of degrees of freedom
% for a scaled inverse chi-square prior on sa. Large settings of n0
% provide greater stability of the parameter estimates for cases when the
% model is "sparse"; that is, when few variables are included in the
% model.
%
% Note it is not possible to fit the logodds parameter; if
% options.logodds is not provided, then it is set to the default value
% when options.sa and options.sigma are scalars, and otherwise an error
% is generated.
%
% VARIATIONAL APPROXIMATION:
% Outputs fit.alpha, fit.mu and fit.s specify the approximate posterior
% distribution of the regression coefficients. Each of these outputs is a
% p x ns matrix. For the ith hyperparameter setting, fit.alpha(:,i) is
% the variational estimate of the posterior inclusion probability (PIP)
% for each variable; fit.mu(:,i) is the variational estimate of the
% posterior mean coefficient given that it is included in the model; and
% fit.s(:,i) is the estimated posterior variance of the coefficient given
% that it is included in the model. These are also the quantities that
% are optimized as part of the inner loop coordinate ascent updates. From
% these quantities, we also provide fit.pve for family = 'gaussian'. This
% output provides, for each hyperparameter setting, the mean estimate of
% the proportion of variance in Y explained by each of the variables
% conditioned on being included in the model.
%
% An additional variational parameter, denoted by 'eta', is needed for
% fast computation with the logistic regression model (family =
% 'binomial'). The fitted value of eta is returned as an n x ns matrix
% fit.eta. If a good estimate of eta is already available (e.g., in a
% previous call to varbvs on the same data), provide this estimate in
% options.eta, in which case eta is not fitted to the data during the
% inner loop coordinate ascent updates (to override this behaviour, set
% options.optimize_eta = true, in which case options.eta is treated as an
% initial estimate).
%
% The variational estimates should be interpreted carefully, especially
% when variables are strongly correlated. For example, consider the
% simple scenario in which 2 candidate variables are closely correlated,
% and at least one of them explains the outcome with probability close to
% 1. Under the correct posterior distribution, we would expect that each
% variable is included with probability ~0.5. However, the variational
% approximation, due to the conditional independence assumption, will
% typically get this wrong, and concentrate most of the posterior weight
% on one variable (the actual variable that is chosen will depend on the
% starting conditions of the optimization). Although the individual PIPs
% are incorrect, a statistic summarizing the variable selection for both
% correlated variables (e.g., the total number of variables included in
% the model) should be reasonably accurate.
%
% More generally, if variables can be reasonably grouped together based
% on their correlations, we recommend interpreting the variable selection
% results at a group level. For example, in genome-wide association
% studies (see the vignettes) ,a SNP with a high PIP indicates that this
% SNP is probably associated with the trait, and one or more nearby SNPs
% within a chromosomal region, or ``locus,'' may be associated as
% well. Therefore, we interpreted the GWAS variable selection results at
% the level of loci, rather than at the level of individual SNPs.
%
% Also note that special care is required for interpreting the results of
% the variational approximation with the logistic regression model. In
% particular, interpretation of the individual estimates of the
% regression coefficients (e.g., the posterior mean estimates fit.mu) is
% not straightforward due to the additional approximation introduced on
% the individual nonlinear factors in the likelihood. As a general
% guideline, only the relative magnitudes of the coefficients are
% meaningful.
%
% AVERAGING OVER HYPERPARAMETER SETTINGS:
% Output fit.logw is an array with ns elements, in which fit.logw(i) is
% the variational lower bound on the marginal log-likelihood for the ith
% setting of the hyperparameters. In many settings, it is good practice
% to account for uncertainty in the hyperparameters when reporting final
% posterior quantities. Provided that (1) the hyperparameter settings
% options.sigma, options.sa and options.logodds adequately represent the
% space of possible hyperparameter settings with high posterior mass, (2)
% the hyperparameter settings are drawn from the same distribution as the
% prior, and (3) the fully-factorized variational approximation closely
% approximates the true posterior distribution, then final posterior
% quantities can be calculated by using fit.logw as (unnormalized)
% log-marginal probabilities. Even when conditions (1), (2) and/or (3)
% are not satisfied, this can approach can still often yield reasonable
% estimates of averaged posterior quantities. For example, do the
% following to compute the posterior mean estimate of sa:
%
% mean_sa = dot(fit.sa(:),fit.w(:));
%
% This is precisely how final posterior quantities are reported by
% varbvsprint (type 'help varbvsprint' for more details). To account for
% discrepancies between the prior on (sigma,sa,logodds) and the sampling
% density used to draw candidate settings of the hyperparameters, adjust
% the log-marginal probabilities weights by setting fit.logw = fit.logw +
% logp/logq, where logp is the log-density of the prior distribution, and
% logq is the log-density of the sampling distribution. (This is
% importance sampling; see, for example, R. M. Neal, "Annealed importance
% sampling", Statistics and Computing, 2001.)
%
% PRIOR ON PROPORTION OF VARIANCE EXPLAINED:
% Specifying the prior variance of the regression coefficients (sa) can
% be difficult, which is why we have included the option of fitting this
% hyperparameter to the data (see input argument update_sa
% above). However, in many settings, especially when a small number of
% variables are included in the regression model, it is preferrable to
% average over candidate settings of sa instead of fitting sa to the
% data. To choose a set of candidate settings for sa, we have advocated
% for setting sa indirectly through a prior estimate of the proportion of
% variance in the outcome explained by the variables (abbreviated as
% PVE), since it is often more natural to specify the PVE rather than the
% prior variance (see references below). This is technically only
% suitable or the linear regression model (family = 'gaussian'), but
% could potentially be used for the linear regression model in an
% approximate way.
%
% For example, one could approximate a uniform prior on the PVE by
% drawing the PVE uniformly between 0 and 1, additionally specifying
% candidate settings for the prior log-odds, then computing the prior
% variance (sa) as follows:
%
% X = X - repmat(mean(X),size(X,1),1);
% sx = sum(var1(X));
% sa = PVE./(1-PVE)./(sigmoid(log(10)*logodds)*sx)}
%
% Note that this calculation will yield sa = 0 when PVE = 0, and sa = Inf
% when PVE = 1. The first line sets the mean of each column of X to 0,
% which is needed to ensure that var1 correctly computes the sample
% variance of each variable.
%
% Also, bear in mind that if there are additional covariates (Z) included
% in the linear regression model that explain variance in Y, then it will
% usually make more sense to first remove the linear effects of these
% covariates before performing this calculation. The PVE would then
% represent the prior proportion of variance in the residuals of Y that
% are explained by the candidate variables. For an example of how to do
% this, see varbvs.m, under "preprocessing steps". Alternatively, one
% could include the matrix Z in the calculation above, taking care to
% ensure that the covariates are included in the model with
% probability 1.
%
% MEMORY REQUIREMENTS:
% Finally, we point out that the optimization procedures were carefully
% designed so that they can be applied to very large data sets; to date,
% this code has been tested on data sets with >500,000 variables and
% >10,000 samples. An important limiting factor is the ability to store
% the data matrix X in memory. To reduce memory requirements, in the
% MATLAB interface we require that X be single precision (type 'help
% single'). Additionally, we mostly avoid generating intermediate
% products that are of the same size as X. Only one such intermediate
% product is generated when family = 'gaussian', and none for family =
% 'binomial'.
%
% LICENSE: GPL v3
%
% DATE: February 21, 2016
%
% AUTHORS:
% Algorithm was designed by Peter Carbonetto and Matthew Stephens.
% R, MATLAB and C code was written by Peter Carbonetto.
% Depts. of Statistics and Human Genetics, University of Chicago,
% Chicago, IL, USA, and AncestryDNA, San Francisco, CA, USA
%
% REFERENCES:
% P. Carbonetto, M. Stephens (2012). Scalable variational inference
% for Bayesian variable selection in regression, and its accuracy in
% genetic association studies. Bayesian Analysis 7: 73-108.
%
% Y. Guan, M. Stephens (2011). Bayesian variable selection regression for
% genome-wide association studies and other large-scale problems. Annals
% of Applied Statistics 5, 1780–-1815.
%
% X. Zhou, P. Carbonetto, M. Stephens (2013). Polygenic modeling with
% Bayesian sparse linear mixed models. PLoS Genetics 9, e1003264.
%
% SEE ALSO:
% varbvsprint, varbvscoefcred
%
% EXAMPLES:
% See demo_qtl.m and demo_cc.m for examples.
%
function fit = varbvs (X, Z, y, labels, family, options)
% Part of the varbvs package, https://github.com/pcarbo/varbvs
%
% Copyright (C) 2012-2017, Peter Carbonetto
%
% This program is free software: you can redistribute it under the
% terms of the GNU General Public License; either version 3 of the
% License, or (at your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANY; without even the implied warranty of
% MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. See the GNU
% General Public License for more details.
%
% Get the number of samples (n) and variables (p).
[n p] = size(X);
% (1) CHECK INPUTS
% ----------------
% Input X must be single precision, and cannot be sparse.
if issparse(X)
error('Input X cannot be sparse');
end
if ~isa(X,'single')
X = single(X);
end
% If input Z is not empty, it must be double precision, and must have as
% many rows as X.
if ~isempty(Z)
if size(Z,1) ~= n
error('Inputs X and Z do not match.');
end
Z = double(full(Z));
end
% Add intercept.
Z = [ones(n,1) Z];
ncov = size(Z,2) - 1;
% Input y must be a double-precision column vector with n elements.
y = double(y(:));
if length(y) ~= n
error('Inputs X and y do not match.');
end
% The labels must be a cell array with p elements, or an empty array.
if nargin < 4
labels = [];
end
if isempty(labels)
labels = cellfun(@num2str,num2cell(1:p)','UniformOutput',false);
else
labels = labels(:);
if (~iscell(labels) | length(labels) ~= p)
error('labels must be a cell array with numel(labels) = size(X,2)');
end
end
% By default, Y is a quantitative trait (normally distributed).
if nargin < 5
family = 'gaussian';
end
if isempty(family)
family = 'gaussian';
end
if ~(strcmp(family,'gaussian') | strcmp(family,'binomial'))
error('family must be gaussian or binomial');
end
% (2) PROCESS OPTIONS
% -------------------
% If the 'options' input argument is not specified, all the options are
% set to the defaults.
if nargin < 6
options = [];
end
% OPTIONS.TOL
% Set the convergence tolerance of the co-ordinate ascent updates.
if isfield(options,'tol')
tol = options.tol;
else
tol = 1e-4;
end
% OPTIONS.MAXITER
% Set the maximum number of inner-loop iterations.
if isfield(options,'maxiter')
maxiter = options.maxiter;
else
maxiter = 1e4;
end
if ~isfinite(maxiter)
error('options.maxiter must be a finite number');
end
% OPTIONS.VERBOSE
% Determine whether to output progress to the console.
if isfield(options,'verbose')
verbose = options.verbose;
else
verbose = true;
end
% OPTIONS.SIGMA
% Get candidate settings for the variance of the residual, if provided.
% Note that this option is not valid for a binary trait.
if isfield(options,'sigma')
sigma = double(options.sigma(:)');
update_sigma = false;
if strcmp(family,'binomial')
error('options.sigma is not valid with family = binomial');
end
else
sigma = var(y);
update_sigma = true;
end
% OPTIONS.SA
% Get candidate settings for the prior variance of the coefficients, if
% provided.
if isfield(options,'sa')
sa = double(options.sa(:)');
update_sa = false;
else
sa = 1;
update_sa = true;
end
% OPTIONS.LOGODDS
% Get candidate settings for the prior log-odds of inclusion. This may
% either be specified as a vector, in which case this is the prior applied
% uniformly to all variables, or it is a p x ns matrix, where p is the
% number of variables and ns is the number of candidate hyperparameter
% settings, in which case the prior log-odds is specified separately for
% each variable. A default setting is only available if the number of
% other hyperparameter settings is 1, in which case we select 20 candidate
% settings for the prior log-odds, evenly spaced between log10(1/p) and
% -1.
if isfield(options,'logodds')
logodds = double(options.logodds);
elseif isscalar(sigma) & isscalar(sa)
logodds = linspace(-log10(p),-1,20);
else
error('options.logodds must be specified');
end
if size(logodds,1) == p
prior_same = false;
else
prior_same = true;
logodds = logodds(:)';
end
% Here is where I ensure that the numbers of candidate hyperparameter
% settings are consistent.
ns = max([numel(sigma) numel(sa) size(logodds,2)]);
if isscalar(sigma)
sigma = repmat(sigma,1,ns);
end
if isscalar(sa)
sa = repmat(sa,1,ns);
end
if size(logodds,2) == 1
logodds = repmat(logodds,1,ns);
end
if numel(sigma) ~= ns | numel(sa) ~= ns | size(logodds,2) ~= ns
error('options.sigma, options.sa and options.logodds are inconsistent');
end
% OPTIONS.UPDATE_SIGMA
% Determine whether to update the residual variance parameter. Note
% that this option is not valid for a binary trait.
if isfield(options,'update_sigma')
update_sigma = options.update_sigma;
if strcmp(family,'binomial')
error('options.update_sigma is not valid with family = binomial');
end
end
% OPTIONS.UPDATE_SA
% Determine whether to update the prior variance of the regression
% coefficients.
if isfield(options,'update_sa')
update_sa = options.update_sa;
end
% OPTIONS.SA0
% Get the scale parameter for the scaled inverse chi-square prior.
if isfield(options,'sa0')
sa0 = options.sa0;
else
sa0 = 1;
end
% OPTIONS.N0
% Get the number of degrees of freedom for the scaled inverse chi-square
% prior.
if isfield(options,'n0')
n0 = options.n0;
else
n0 = 10;
end
% OPTIONS.ALPHA
% Set initial estimates of variational parameter alpha.
initialize_params = true;
if isfield(options,'alpha')
alpha = double(options.alpha);
initialize_params = false;
if size(alpha,1) ~= p
error('options.alpha must have as many rows as X has columns');
end
if size(alpha,2) == 1
alpha = repmat(alpha,1,ns);
end
else
alpha = rand(p,ns);
alpha = alpha ./ repmat(sum(alpha),p,1);
end
% OPTIONS.MU
% Set initial estimates of variational parameter mu.
if isfield(options,'mu')
mu = double(options.mu);
initialize_params = false;
if size(mu,1) ~= p
error('options.mu must have as many rows as X has columns');
end
if size(mu,2) == 1
mu = repmat(mu,1,ns);
end
else
mu = randn(p,ns);
end
% OPTIONS.ETA
% Set initial estimates of variational parameter eta. Note this is only
% relevant for logistic regression.
if isfield(options,'eta')
eta = double(options.eta);
initialize_params = false;
optimize_eta = false;
if ~strcmp(family,'binomial')
error('options.eta is only valid for family = binomial');
end
if size(eta,1) ~= n
error('options.eta must have as many rows as X');
end
if size(eta,2) == 1
eta = repmat(eta,1,ns);
end
else
eta = ones(n,ns);
optimize_eta = true;
end
% OPTIONS.OPTIMIZE_ETA
% Determine whether to update the variational parameter eta. Note this
% is only relevant for logistic regression.
if isfield(options,'optimize_eta')
optimize_eta = options.optimize_eta;
if ~strcmp(family,'binomial')
error('options.optimize_eta is only valid for family = binomial');
end
end
% OPTIONS.INITIALIZE_PARAMS
% Determine whether to find a good initialization for the variational
% parameters.
if isfield(options,'initialize_params')
initialize_params = options.initialize_params;
if initialize_params & ns == 1
error(['options.initialize_params = true has no effect when ' ...
'there is only one hyperparameter setting.']);
end
end
% OPTIONS.NR
% This is the number of samples to draw of the proportion of variance
% in Y explained by the fitted model.
if isfield(options,'nr')
nr = options.nr;
else
nr = 100;
end
% TO DO: Allow specification of summary statistics ('Xb') from "fixed"
% variational estimates for an external set of variables.
clear options
% (3) PREPROCESSING STEPS
% -----------------------
% Adjust the genotypes and phenotypes so that the linear effects of
% the covariates are removed. This is equivalent to integrating out
% the regression coefficients corresponding to the covariates with
% respect to an improper, uniform prior; see Chipman, George and
% McCulloch, "The Practical Implementation of Bayesian Model
% Selection," 2001.
if strcmp(family,'gaussian')
% Here I compute two quantities that are used here to remove linear
% effects of the covariates (Z) on X and y, and later on (in function
% "outerloop"), to efficiently compute estimates of the regression
% coefficients for the covariates.
SZy = (Z'*Z)\(Z'*y);
SZX = double((Z'*Z)\(Z'*X));
if ncov == 0
X = X - repmat(mean(X),length(y),1);
y = y - mean(y);
else
% This should give the same result as centering the columns of X and
% subtracting the mean from y when we have only one covariate, the
% intercept.
y = y - Z*SZy;
X = X - Z*SZX;
end
else
SZy = [];
SZX = [];
end
% Provide a brief summary of the analysis.
if verbose
fprintf('Welcome to ');
fprintf('-- * * \n');
fprintf('VARBVS version 2.4-0 ');
fprintf('-- | | | \n');
fprintf('large-scale Bayesian ');
fprintf('-- || | | | || | | | \n');
fprintf('variable selection ');
fprintf('-- | || | | | | || || |||| || | || \n');
fprintf('*********************');
fprintf('********************************************************\n');
fprintf('Copyright (C) 2012-2017 Peter Carbonetto.\n')
fprintf('See http://www.gnu.org/licenses/gpl.html for the full ');
fprintf('license.\n');
fprintf('Fitting variational approximation for Bayesian variable ');
fprintf('selection model.\n');
fprintf('family: %-8s',family);
fprintf(' num. hyperparameter settings: %d\n',numel(sa));
fprintf('samples: %-6d',n);
fprintf(' convergence tolerance %0.1e\n',tol);
fprintf('variables: %-6d',p);
fprintf(' iid variable selection prior: %s\n',tf2yn(prior_same));
fprintf('covariates: %-6d',ncov);
fprintf(' fit prior var. of coefs (sa): %s\n',tf2yn(update_sa));
fprintf('intercept: yes ');
if strcmp(family,'gaussian')
fprintf('fit residual var. (sigma): %s\n',tf2yn(update_sigma));
elseif strcmp(family,'binomial')
fprintf('fit approx. factors (eta): %s\n',tf2yn(optimize_eta));
end
end
% (4) INITIALIZE STORAGE FOR THE OUTPUTS
% --------------------------------------
% Initialize storage for the variational estimate of the marginal log-
% likelihood for each hyperparameter setting (logw), the variances of the
% regression coefficients (s), and the posterior mean estimates of the
% coefficients for the covariates, including the intercept (muz).
logw = zeros(1,ns);
s = zeros(p,ns);
mu_cov = zeros(ncov+1,ns);
% (5) FIT BAYESIAN VARIABLE SELECTION MODEL TO DATA
% -------------------------------------------------
if ns == 1
% Find a set of parameters that locally minimize the K-L
% divergence between the approximating distribution and the exact
% posterior.
if verbose
fprintf(' variational max. incl variance params\n');
fprintf(' iter lower bound change vars sigma sa\n');
end
[logw sigma sa alpha mu s eta mu_cov] = ...
outerloop(X,Z,y,family,SZy,SZX,sigma,sa,logodds,alpha,mu,eta,tol,...
maxiter,verbose,[],update_sigma,update_sa,optimize_eta,...
n0,sa0);
if verbose
fprintf('\n');
end
else
% If a good initialization isn't already provided, find a good
% initialization for the variational parameters. Repeat for each
% candidate setting of the hyperparameters.
if initialize_params
if verbose
fprintf('Finding best initialization for %d combinations of ',ns);
fprintf('hyperparameters.\n');
fprintf('-iteration- variational max. incl variance params\n');
fprintf('outer inner lower bound change vars sigma sa\n');
end
% Repeat for each setting of the hyperparameters.
for i = 1:ns
[logw(i) sigma(i) sa(i) alpha(:,i) mu(:,i) s(:,i) eta(:,i) ...
mu_cov(:,i)] = ...
outerloop(X,Z,y,family,SZy,SZX,sigma(i),sa(i),logodds(:,i),...
alpha(:,i),mu(:,i),eta(:,i),tol,maxiter,verbose,i,...
update_sigma,update_sa,optimize_eta,n0,sa0);
end
if verbose
fprintf('\n');
end
% Choose an initialization common to all the runs of the coordinate
% ascent algorithm. This is chosen from the hyperparameters with
% the highest variational estimate of the marginal likelihood.
[ans i] = max(logw);
alpha = repmat(alpha(:,i),1,ns);
mu = repmat(mu(:,i),1,ns);
if optimize_eta
eta = repmat(eta(:,i),1,ns);
end
if update_sigma
sigma = repmat(sigma(i),1,ns);
end
if update_sa
sa = repmat(sa(i),1,ns);
end
end
% Compute a variational approximation to the posterior distribution
% for each candidate setting of the hyperparameters.
if verbose
fprintf('Computing marginal likelihood for %d combinations of ',ns);
fprintf('hyperparameters.\n');
fprintf('-iteration- variational max. incl variance params\n');
fprintf('outer inner lower bound change vars sigma sa\n');
end
% For each setting of the hyperparameters, find a set of parameters that
% locally minimize the K-L divergence between the approximating
% distribution and the exact posterior.
for i = 1:ns
[logw(i) sigma(i) sa(i) alpha(:,i) mu(:,i) s(:,i) eta(:,i) ...
mu_cov(:,i)]=...
outerloop(X,Z,y,family,SZy,SZX,sigma(i),sa(i),logodds(:,i),...
alpha(:,i),mu(:,i),eta(:,i),tol,maxiter,verbose,i,...
update_sigma,update_sa,optimize_eta,n0,sa0);
end
if verbose
fprintf('\n');
end
end
% (6) CREATE FINAL OUTPUT
% -----------------------
% Compute the normalized importance weights and the posterior inclusion
% probabilities (PIPs) and mean regression coefficients averaged over the
% hyperparameter settings.
if ns == 1
w = 1;
pip = fit.alpha;
beta = fit.mu;
else
w = normalizelogweights(logw);
pip = alpha * w(:);
beta = mu * w(:);
end
if strcmp(family,'gaussian')
fit = struct('family',family,'n',n,'labels',{labels},'n0',n0,'sa0',sa0,...
'mu_cov',{mu_cov},'update_sigma',update_sigma,'update_sa',...
update_sa,'prior_same',prior_same,'logw',{logw},'w',{w},...
'sigma',{sigma},'sa',{sa},'logodds',{logodds},'alpha',...
{alpha},'mu',{mu},'s',{s},'eta',[],'pip',{pip},'beta',...
{beta},'optimize_eta',false);
% Compute the proportion of variance in Y, after removing linear
% effects of covariates, explained by the regression model.
if verbose
fprintf('Estimating proportion of variance in Y explained by model.\n');
end
fit.model_pve = varbvspve(X,fit,nr);
% Compute the proportion of variance in Y, after removing linear
% effects of covariates, explained by each variable.
fit.pve = zeros(p,ns);
sx = var1(X);
for i = 1:ns
fit.pve(:,i) = sx.*(mu(:,i).^2 + s(:,i))/var(y,1);
end
elseif strcmp(family,'binomial')
fit = struct('family',family,'n',n,'labels',{labels},'n0',n0,'sa0',sa0,...
'mu_cov',{mu_cov},'update_sa',update_sa,'optimize_eta',...
optimize_eta,'prior_same',prior_same,'logw',{logw},...
'w',{w},'sa',{sa},'logodds',{logodds},'alpha',{alpha},...
'mu',{mu},'s',{s},'eta',{eta},'pip',{pip},'beta',{beta},...
'update_sigma',false,'pve',[]);
end
% ------------------------------------------------------------------
% This function implements one iteration of the "outer loop".
function [logw, sigma, sa, alpha, mu, s, eta, mu_cov] = ...
outerloop (X, Z, y, family, SZy, SZX, sigma, sa, logodds, alpha, ...
mu, eta, tol, maxiter, verbose, outer_iter, ...
update_sigma, update_sa, optimize_eta, n0, sa0)
p = length(alpha);
if isscalar(logodds)
logodds = repmat(logodds,p,1);
end
% Note that we need to multiply the prior log-odds by log(10), because
% varbvsnorm, varbvsbin and varbvsbinz define the prior log-odds using
% the natural logarithm (base e).
if strcmp(family,'gaussian')
% Optimize the variational lower bound for the Bayesian variable
% selection model.
[logw err sigma sa alpha mu s] = ...
varbvsnorm(X,y,sigma,sa,log(10)*logodds,alpha,mu,tol,maxiter,...
verbose,outer_iter,update_sigma,update_sa,n0,sa0);
% Adjust the variational lower bound to account for integral over the
% regression coefficients corresponding to the covariates.
logw = logw - logdet(Z'*Z)/2;
% Compute the posterior mean estimate of the regression
% coefficients for the covariates under the current variational
% approximation.
mu_cov = SZy - SZX*(alpha.*mu);
elseif strcmp(family,'binomial')
% Optimize the variational lower bound for the Bayesian variable
% selection model.
if size(Z,2) == 1
[logw err sa alpha mu s eta] = ...
varbvsbin(X,y,sa,log(10)*logodds,alpha,mu,eta,tol,maxiter,verbose,...
outer_iter,update_sa,optimize_eta,n0,sa0);
else
[logw err sa alpha mu s eta] = ...
varbvsbinz(X,Z,y,sa,log(10)*logodds,alpha,mu,eta,tol,maxiter,...
verbose,outer_iter,update_sa,optimize_eta,n0,sa0);
end
% Compute the posterior mean estimate of the regression
% coefficients for the covariates under the current variational
% approximation.
Xr = X*(alpha.*mu);
d = slope(eta);
mu_cov = (Z'*diag(sparse(d))*Z)\(Z'*(y - 0.5 - d.*Xr));
end
logw = logw(end);
% ------------------------------------------------------------------
function y = tf2yn (x)
if x
y = 'yes';
else
y = 'no';
end