These scripts allow to apply multivariate Bayesian inversion (MBI) for classification (MBC) and regression (MBR) to measured multivariate signals ("features") in order to predict distinct categories ("classes") or continuous variables ("targets"), possibly accounting for confounding variables ("covariates"). A manuscript describing this novel machine learning approach to supervised learning is currently in preparation.

These tools rely on functions from the cvLME package (MGLM_Bayes.m) and from the ML4ML toolbox (ML_CV.m) which for simplicity are currently included in this repository.

Let Y1 and Y2 be feature matrices from training and test data and let x1 and x2 be natural numbers representing classes in training and test data. Then, the simplest version of MBC can be used to calculate posterior class probabilities in the test data as follows:

% multivariate Bayesian classification
MBA1 = mbitrain(Y1, x1, [], [], 'MBC');
PP2  = mbitest(Y2, x2, [], [], MBA1, []);

Let Y1 and Y2 be feature matrices from training and test data and let x1 and x2 be real numbers representing targets in training and test data. Then, the simplest version of MBR can be used to calculate posterior target densities in the test data as follows:

% multivariate Bayesian regression
MBA1 = mbitrain(Y1, x1, [], [], 'MBR');
PP2  = mbitest(Y2, x2, [], [], MBA1, []);

The output PP2 is a matrix of posterior probabilities in the test data, with one row corresponding to one data point and each row either representing discrete probability masses or a continuous probability density.

Let Y be an entire feature matrix and let x be an entire vector of class labels or target values across all data points. Then, cross-validated MBC or MBR (number of CV folds k) can be easily implemented as follows:

% cross-validated MBC
CV  = ML_CV(x, k, 'kfc');                  % k-folds on points per class
MBC = ML_MBI(Y, x, [], [], CV, 'MBC', []);
DA  = MBC.perf.DA;                         % decoding accuracy

% cross-validated MBR
CV  = ML_CV(numel(x), k, 'kf');            % k-folds cross-validation
MBR = ML_MBI(Y, x, [], [], CV, 'MBR', []);
r   = MBR.perf.r;                          % predictive correlation

For extra input parameters of ML_MBI.m and fields of the output structure MBC/MBR, type help ML_MBI into the command window. For manipulating the prior probabilities for prediction in the test data, type help mbitest.

These tools rely on the cvLME package (cvBMS.py) which is currently included in the repository.

For using the Python code, you have to import the MBI module somewhere in your script:

import MBI

Let Y1 and Y2 be feature matrices from training and test data and let x1 and x2 be natural numbers representing classes in training and test data. Then, the simplest version of MBC can be used to calculate posterior class probabilities in the test data as follows:

# multivariate Bayesian classification
m1   = MBI.model(Y1, x1, mb_type='MBC')
MBA1 = m1.train()
m2   = MBI.model(Y2, x2, mb_type='MBC')
PP2  = m2.test(MBA1)

Let Y1 and Y2 be feature matrices from training and test data and let x1 and x2 be real numbers representing targets in training and test data. Then, the simplest version of MBR can be used to calculate posterior target densities in the test data as follows:

# multivariate Bayesian regression
m1   = MBI.model(Y1, x1, mb_type='MBR')
MBA1 = m1.train()
m2   = MBI.model(Y2, x2, mb_type='MBR')
PP2  = m2.test(MBA1)

The output PP2 is a matrix of posterior probabilities in the test data, with one row corresponding to one data point and each row either representing discrete probability masses or a continuous probability density.

Let Y be an entire feature matrix and let x be an entire vector of class labels or target values across all data points. Then, cross-validated MBC or MBR (number of CV folds k) can be easily implemented as follows:

# cross-validated MBC
MBC = MBI.cvMBI(Y, x, mb_type='MBC')
MBC.crossval(k=10, cv_mode='kfc')    # k-folds on points per class
MBC.predict()
DA  = MBC.evaluate('DA')             # decoding accuracy

# cross-validated MBR
MBR = MBI.cvMBI(Y, x, mb_type='MBR')
MBR.crossval(k=10, cv_mode='kf')     # k-folds cross-validation
MBR.predict()
r   = MBR.evaluate('r')              # predictive correlation

For extra input parameters of MBI.cvMBI and attributes of the objects MBC/MBR, see these lines of code. For manipulating the prior probabilities for prediction in the test data, see these lines of code.