Abstract
The probability density function for the visible sector of a Riemann-Theta Boltzmann machine can be taken conditional on a subset of the visible units. We derive that the corresponding conditional density function is given by a reparameterization of the Riemann-Theta Boltzmann machine modelling the original probability density function. Therefore the conditional densities can be directly inferred from the Riemann-Theta Boltzmann machine.
Highlights
Modelling the underlying probability density function of a dataset is a non-trivial problem, already in the low dimensional setting
Several techniques to model probability densities of unknown functional form can be found in the literature
Each technique comes with its own advantages and drawbacks, and it is fair to say that so far no general use technique is at hand
Summary
Modelling the underlying probability density function of a dataset is a non-trivial problem, already in the low dimensional setting. Like the visible sector probability density function, can be calculated in closed form involving Riemann-Theta functions [3]. The appealing property of this new kind of Boltzmann machine is that the normalization (summation over all states) is given in closed-form in terms of the Riemann-Theta function. The closed form solution allows to keep full analytic control, and in particular to derive related quantities, like for example the corresponding cumulative distribution function or conditional densities. The latter will be discussed in this note. RTBM The visible sector probability density function of the Riemann-Theta Boltzmann machine is given by [2]. Once we learned an approximation P (v) of a multi-variate density via a RTBM, we obtain all the conditional densities automatically, as we will show
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