# A Bayesian Modeling Approach to Situated Design of Personalized Soundscaping Algorithms

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Problem Statement and Proposed Solution Framework

#### 2.1.1. Source Modeling

#### 2.1.2. Source Separation

#### 2.1.3. Soundscaping

#### 2.2. Model Specification

#### 2.2.1. Source Mixing Model 1: Algonquin Model

#### 2.2.2. Source Mixing Model 2: Gaussian Scale Sum Model

#### 2.2.3. Source Model: Gaussian Mixture Model

#### 2.3. Factor Graphs and Message Passing-Based Inference

#### 2.3.1. Forney-Style Factor Graphs

#### 2.3.2. Sum-Product Message Passing

#### 2.3.3. Variational Message Passing

#### 2.3.4. Automating Inference and Variational Free Energy Evaluation

#### 2.3.5. Message Passing-Based Inference in the Algonquin Model

#### 2.3.6. Message Passing-Based Inference in the Gaussian Scale Sum Model

#### 2.3.7. Implementation Details

## 3. Experimental Validation

#### 3.1. Experimental Overview

#### 3.2. Data Selection

- The LibriSpeech data set (the LibriSpeech data set is available at https://www.openslr.org/12, accessed on 31 March 2021) [51], which is a corpus of approximately 1000 h of 16 kHz read English speech.
- The FSDnoisy18k data set (the FSDnoisy18k data set is available at https://www.eduardofonseca.net/FSDnoisy18k, accessed on 13 April 2021) [52], is an audio data set, which has been collected with the aim of fostering the investigation of label noise in sound event classification. It contains 42.5 h of audio samples across 20 sound classes, including a small amount of manually-labeled data and a larger quantity of real-world noise data.

#### 3.3. Preprocessing

#### 3.4. Performance Evaluation

#### 3.5. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Inference for Learning and Signal Processing

#### Appendix A.1. Source Modeling

#### Appendix A.2. Source Separation

#### Appendix A.3. Soundscaping

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**Figure 1.**An overview of the source modeling stage. The user records short fragments of the observed signals and infers the corresponding model parameters.

**Figure 2.**An overview of the source separation stage. Based on the observed signal and the trained source models the latent states corresponding to the individual signal are tracked and extracted.

**Figure 3.**An overview of the soundscaping stage. The user controls the weights for the individual signals and performs source-specific amplification and suppression.

**Figure 4.**Forney-style factor graph representation of (17).

**Figure 5.**The Forney-style factor graph representation of the Algonquin-based generative model fully specified by (8), (12)–(16). The $\mathrm{GMM}$ composite node represents the Gaussian mixture node as specified in ([37], A.2). The $\mathrm{Algonquin}$ composite node has been summarized in Table 1. The dashed rectangular bounding boxes here denote plates, which represent repeating parts of the graph. Edges that are intersected by the plate boundaries are implicitly connected between plates using equality nodes.

**Figure 6.**The Forney-style factor graph representation of the Gaussian scale sum-based generative model fully specified by (11)–(16). The $\mathrm{GMM}$ composite node represents the Gaussian mixture node as specified in ([37], A.2). The Gaussian scale sum composite node has been summarized in Table 2. The dashed rectangular bounding boxes here denote plates, which represent repeating parts of the graph. Edges that are intersected by the plate boundaries are implicitly connected between plates using equality nodes.

**Figure 8.**Example of a situational sketch of (22) where a variational message $\overrightarrow{\nu}\left({x}_{j}\right)$ flows out of an arbitrary node $f({x}_{1},\phantom{\rule{4pt}{0ex}}{x}_{2},\dots ,{x}_{n})$ with marginals $q\left({x}_{\setminus j}\right)$.

**Figure 9.**Overview of the performance metrics for the first experiment as described in Section 3.1. For a varying input SNR, the output SNR (

**left**), the PESQ (

**middle**) and the STOI (

**right**) are evaluated for both models from Section 2.2 for 1 noise mixture component. For comparison the baseline performance of the noisy signal has been included, in addition to the results obtained by the Wiener filter. It should be noted that the input SNR refers to the signal before entering the frequency-warped filter, causing the offset in the left plot.

**Figure 10.**Overview of the performance metrics for the second experiment as described in Section 3.1. For a varying input SNR, the output SNR (

**left**), the PESQ (

**middle**) and the STOI (

**right**) are evaluated for both models from Section 2.2 for 2 noise mixture components. For comparison the baseline performance of the noisy signal has been included, in addition to the results obtained by the Wiener filter. It should be noted that the input SNR refers to the signal before entering the frequency-warped filter, causing the offset in the left plot.

**Table 1.**Table containing (a) the Forney-style factor graph representation of the generalized Algonquin node. (b) The likelihood function corresponding to the generalized Algonquin node. (c) An overview of the chosen approximate posterior distributions. Here the $\widehat{\xb7}$ accent refers to the parameters of these distributions. (d) The derived variational messages for the generalized Algonquin node. Here the $\sigma {(\xb7)}_{k}$ represents the $k\mathrm{th}$ output of the softmax function. (e) The derived local variational free energy, in which $\psi (\xb7)$ denotes the digamma function. All derivations are available at Supplementary Materials https://github.com/biaslab/SituatedSoundscaping, accessed on 1 July 2021.

Factor graph for the Algonquin node | |
---|---|

Node function | |

$p(x\mid {s}_{1},\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}{s}_{K},\phantom{\rule{4pt}{0ex}}\gamma )=\mathcal{N}\left(\right)open="("\; close=")">x\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}ln\left(\right)open="("\; close=")">\sum _{k=1}^{K}exp\left({s}_{k}\right),\phantom{\rule{4pt}{0ex}}{\gamma}^{-1}$ | |

Marginals | Functional form |

$q\left(x\right)$ | $\mathcal{N}\left(\right)open="("\; close=")">x\mid {\widehat{m}}_{x},\phantom{\rule{4pt}{0ex}}{\widehat{\gamma}}_{x}^{-1}$ |

$q\left({s}_{k}\right)$ | $\mathcal{N}\left(\right)open="("\; close=")">{s}_{k}\mid {\widehat{m}}_{{s}_{k}},\phantom{\rule{4pt}{0ex}}{\widehat{\gamma}}_{{s}_{k}}^{-1}$ |

$q\left(\gamma \right)$ | $\mathsf{\Gamma}\left(\right)open="("\; close=")">\gamma \mid \widehat{\alpha},\phantom{\rule{4pt}{0ex}}\widehat{\beta}$ |

Messages | Functional form |

$\overrightarrow{\nu}\left(x\right)$ | $\mathcal{N}\left(\right)open="("\; close=")">x\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}ln\left(\right)open="("\; close=")">\sum _{k=1}^{K}exp\left({\widehat{m}}_{{s}_{k}}\right),\phantom{\rule{4pt}{0ex}}\frac{\widehat{\beta}}{\widehat{\alpha}}$ |

$\overleftarrow{\nu}\left({s}_{k}\right)$ | $\mathcal{N}\left(\right)open="("\; close=")">{s}_{k}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{\widehat{m}}_{x}-\frac{ln\left(\right)open="("\; close=")">{\sum}_{i=1}^{K}exp\left({m}_{{s}_{i}}\right)}{-}\sigma {\left({\widehat{\mathit{m}}}_{s}\right)}_{k},\phantom{\rule{4pt}{0ex}}\frac{\widehat{\beta}}{\widehat{\alpha}\sigma {\left({\widehat{\mathit{m}}}_{s}\right)}_{k}^{2}}$ |

$\overleftarrow{\nu}\left(\gamma \right)$ | $\mathsf{\Gamma}\left(\right)open="("\; close=")">\gamma \phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\frac{3}{2},\phantom{\rule{4pt}{0ex}}\frac{1}{2}\left(\right)open="("\; close=")">\frac{1}{{\widehat{\gamma}}_{x}}+\sum _{k=1}^{K}\frac{\sigma {\left({\widehat{\mathit{m}}}_{s}\right)}_{k}^{2}}{{\widehat{\gamma}}_{{s}_{k}}}+{\left(\right)}^{{\widehat{m}}_{x}}2$ |

Local variational free energy | |

$\frac{1}{2}ln\left(2\pi \right)+\frac{1}{2}\left(\right)open="("\; close=")">\psi \left(\widehat{\alpha}\right)-ln\left(\widehat{\beta}\right)+\frac{\widehat{\alpha}}{2\widehat{\beta}}\left(\right)open="("\; close=")">\frac{1}{{\widehat{\gamma}}_{y}}+\sum _{k=1}^{K}\frac{\sigma {\left({\widehat{\mathit{m}}}_{s}\right)}_{k}^{2}}{{\widehat{\gamma}}_{{s}_{k}}}+{\left(\right)}^{{\widehat{m}}_{x}}$ |

**Table 2.**Table containing (a) the Forney-style factor graph representation of the Gaussian scale sum node. (b) The likelihood function corresponding to the Gaussian scale sum node. (c) An overview of the chosen approximate posterior distributions. Here the $\widehat{\xb7}$ accent refers to the parameters of these distributions. (d) The derived variational messages for the Gaussian scale sum node. (e) The derived local variational free energy. All derivations are available at https://github.com/biaslab/SituatedSoundscaping, accessed on 1 July 2021.

Factor graph for the Gaussian scale sum node | |
---|---|

Node function | |

$p(x\mid {s}_{1},\phantom{\rule{4pt}{0ex}}\dots ,\phantom{\rule{4pt}{0ex}}{s}_{K})=\mathcal{N}\left(\right)open="("\; close=")">x\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}\sum _{k=1}^{K}exp\left({s}_{k}\right),\phantom{\rule{4pt}{0ex}}0$ | |

Marginals | Functional form |

$q\left(x\right)$ | ${\mathcal{N}}_{\mathcal{C}}\left(\right)open="("\; close=")">x\mid {\widehat{m}}_{x},\phantom{\rule{4pt}{0ex}}{\widehat{\gamma}}_{x}^{-1},\phantom{\rule{4pt}{0ex}}0$ |

$q\left({s}_{k}\right)$ | $\mathcal{N}\left(\right)open="("\; close=")">{s}_{k}\mid {\widehat{m}}_{{s}_{k}},\phantom{\rule{4pt}{0ex}}{\widehat{\gamma}}_{{s}_{k}}^{-1}$ |

Messages | Functional form |

$\overrightarrow{\nu}\left(x\right)$ | ${\mathcal{N}}_{\mathcal{C}}\left(\right)open="("\; close=")">x\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}0,\phantom{\rule{4pt}{0ex}}\sum _{k=1}^{K}exp\left(\right)open="("\; close=")">{\widehat{m}}_{{s}_{k}},\phantom{\rule{4pt}{0ex}}0$ |

$\overleftarrow{\nu}\left({s}_{k}\right)$ | $\frac{1}{exp\left({s}_{k}\right)+{\sum}_{i\ne k}exp\left({\widehat{m}}_{{s}_{i}}\right)}exp\left(\right)open="("\; close=")">-\frac{({\widehat{\gamma}}_{x}^{-1}+|{\widehat{m}}_{x}{|}^{2})}{exp\left({s}_{k}\right)+{\sum}_{i\ne k}exp\left({\widehat{m}}_{{s}_{i}}\right)}$ |

Local variational free energy | |

$ln\left(\pi \right)+ln\left(\right)open="("\; close=")">\sum _{k=1}^{K}exp\left({\widehat{m}}_{{s}_{k}}\right)+\frac{1}{{\sum}_{k=1}^{K}exp\left({\widehat{m}}_{{s}_{k}}\right)}({\widehat{\gamma}}_{x}^{-1}+|{\widehat{m}}_{x}{|}^{2})$ |

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**MDPI and ACS Style**

van Erp, B.; Podusenko, A.; Ignatenko, T.; de Vries, B.
A Bayesian Modeling Approach to Situated Design of Personalized Soundscaping Algorithms. *Appl. Sci.* **2021**, *11*, 9535.
https://doi.org/10.3390/app11209535

**AMA Style**

van Erp B, Podusenko A, Ignatenko T, de Vries B.
A Bayesian Modeling Approach to Situated Design of Personalized Soundscaping Algorithms. *Applied Sciences*. 2021; 11(20):9535.
https://doi.org/10.3390/app11209535

**Chicago/Turabian Style**

van Erp, Bart, Albert Podusenko, Tanya Ignatenko, and Bert de Vries.
2021. "A Bayesian Modeling Approach to Situated Design of Personalized Soundscaping Algorithms" *Applied Sciences* 11, no. 20: 9535.
https://doi.org/10.3390/app11209535