In higher dimensions

**Theorem A.A.2** (Estimator MSE and differential entropy in multivariate case). For any random vector $\theta \in \mathbb{R}^D$ and any estimator $\hat{\theta}$,

$$ \mathbb{E} [\|\theta - \hat{\theta}\|^2] \geq \frac{D}{2\pi e} \exp \left( \frac{2}{D} h(\theta) \right), \qquad (A.30) $$

with equality if and only if $\theta$ is Gaussian, $\hat{\theta} = \mathbb{E}[\theta]$ and the covariance matrix of $\theta$ is proportional to the identity.

*Proof.* Let $\hat{\theta}$ be any estimator of $\theta$.

$$ \mathbb{E} [\|\theta - \hat{\theta}\|^2] \geq \min_{\hat{\theta}} \mathbb{E} [\|\theta - \hat{\theta}\|^2] = \mathrm{Tr} \, \mathbf{C}, \qquad (A.31) $$

with $\mathbf{C} \in \mathcal{M}_D(\mathbb{R})$ the covariance matrix of $\theta$. Then, to get a relation between the trace of the covariance matrix of $\theta$ and its entropy, one uses the fact that for a random vector with given covariance matrix, the multivariate Gaussian distribution has maximum entropy. Let $Z \sim \mathcal{N}(0, \mathbf{C})$. Then,

$$ h(\theta) \leq h(Z) = \frac{1}{2} \ln(\det(2\pi e \mathbf{C})), \qquad (A.32) $$

which yields

$$ \det(\mathbf{C}) \geq \frac{1}{(2\pi e)^D} \exp(2h(\theta)). \qquad (A.33) $$

Using the arithmetic mean geometric mean inequality, i.e.,

$$ [\det \mathbf{C}]^{\frac{1}{D}} \leq \frac{\mathrm{Tr} \, \mathbf{C}}{D}, \qquad (A.34) $$

we get

$$ \mathrm{Tr} \, \mathbf{C} \geq D [\det \mathbf{C}]^{\frac{1}{D}} \geq \frac{D}{2\pi e} \exp \left( \frac{2}{D} h(\theta) \right), \qquad (A.35) $$

with equality if $\theta$ is Gaussian and $C$ is proportional to the identity.

□

**Corollary A.A.2.1.** Given side information $y$ and estimator $\hat{\theta}(y)$, it follows that

$$ \mathbb{E}_{(\theta, y)} [\|\theta - \hat{\theta}(y)\|^2] \geq \frac{D}{2\pi e} \exp \left( \frac{2}{D} h(\theta|y) \right) \qquad (A.36) $$

*Proof.* The proof of this corollary is similar to the previous one: we rewrite the MSE to get an average covariance matrix for $\theta|y$ (averaged over $y$), and apply the theorem to get the inequality with $h(\theta|y)$.

□

Here again, no assumption is made over the distribution of $y$.
Acronyms

ABC approximate Bayes computing

AdaGrad adaptive gradient algorithm

AE absolute error

AIC Akaike information criterion

ALMA Atacama large millimeter/submillimeter array

ANN artificial neural network

ASIS ancillarity-sufficiency interweaving strategy

au astronomical unit

BB black body

BIC Bayesian information criterion

BO Bayesian optimization

cdf cumulative density function

cgs centimetre-gram-second

CI credibility interval

CL Cauchy loss function

CLT central limit theorem

CNM cold neutral medium

DGMC distributed genetic MC

DIC deviance information criterion

DMC darting MC

EES equi-energy sampler

EF error factor

elpd expected log-predictive density

EMC evolutionary MC

ESS effective sample size

EUV extreme ultraviolet

FIR far infrared

FUV far ultraviolet

GA genetic algorithm

GD gradient descent

GELU Gaussian error linear unit

GMC giant molecular cloud

GMM Gaussian mixture model

GP Gaussian process

HIM hot ionized medium

HMC Hamiltonian Monte Carlo

HPR high probability region

HST Hubble space telescope

ICA independent component analysis

i.i.d. independent and identically distributed

I-MH independent Metropolis-Hastings

I-MTM independent multiple-try Metropolis

IR infrared

ISM interstellar medium

ISRF interstellar radiation field

JAMS jumping adaptative multimodal sampler

JWST James Webb spatial telescope

KDE kernel density estimation

KL Kullback-Leibler

KS Kolmogorov-Smirnov

L-BFGS limited memory BFGS

LHS latin hypercube sampling

LM Levenberg-Marquardt

LMC large Magellanic cloud

Ipd log-predictive density

LTE local thermal equilibrium

LVG large velocity gradient

ly light-year

MAE mean absolute error

MALA Metropolis adjusted Langevin algorithm

MAP maximum a posteriori

MBB modified black body

MC Monte Carlo

MCMC Markov chain Monte Carlo

MH Metropolis-Hastings

ML machine learning

MLE maximum likelihood estimator

MMALA manifold Metropolis adjusted Langevin algorithm

MMSE minimum mean square error

MSE mean squared error

MTM multiple-try Metropolis

MW Milky Way
**MYULA** Moreau-Yosida unadjusted Langevin algorithm

**NIR** near infrared

**OLHS** orthogonal LHS

**PAH** polycyclic aromatic hydrocarbons

**pc** parsec

**PCA** principal component analysis

**PCR** principal component regression

**pdf** probability density function

**PDR** photodissociation region

**PGD** preconditioned gradient descent

**PMALA** preconditioned Metropolis adjusted Langevin algorithm

**pmf** probability mass function

**QMC** quasi-Monte Carlo

**RBF** radial basis function

**RDMC** regeneration darting MC

**ReLU** rectified error linear unit

**RF** random forest

**RJ** Rayleigh-Jeans

**RMHMC** Riemannian manifold HMC

**RMSProp** root mean squared propagation

**R-SNR** reconstruction signal-to-noise ratio

**RWMH** random walk Metropolis-Hastings

**SA** simulated annealing

**SDE** stochastic differential equation

**SE** squared error

**SED** spectral energy density

**SFR** star formation rate

**SGD** stochastic gradient descent

**SG-MCMC** stochastic gradient MCMC

**SLED** spectral line energy distribution

**SLHS** symmetric LHS

**SMC** sequential MC

**SNR** signal-to-noise ratio

**UF** uncertainty factor

**ULA** unadjusted Langevin algorithm

**UV** ultraviolet

**VBI** variational Bayes inference

**WAIC** widely applicable information criterion

**WBIC** widely applicable Bayesian information criterion

**WHMC** wormhole Hamiltonian Monte Carlo

**WIM** warm ionized medium

**WNM** warm neutral medium

**YSO** young stellar object
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# Résumé court

## Méthodes d'échantillonnage pour l'inférence statistique de problèmes inverses non linéaires : distribution spatiale des propriétés physico-chimiques du milieu interstellaire

Le milieu interstellaire (MIS) est un milieu très diffus qui remplit l'immense volume entre les objets célestes tels que les étoiles et les trous noirs au sein d'une galaxie. L'étude du MIS soulève des questions fondamentales dont la formation d'étoiles. Les étoiles naissent de l'effondrement gravitationnel de parties de régions froides et denses du MIS appelées nuages moléculaires.

Cette thèse analyse des cartes multispectrales de nuages moléculaires dans les domaines infra-rouge lointain et radio, obtenues par des télescopes spatiaux ou terrestres. L'attention est portée aux nuages illuminés et chauffés par des étoiles massives voisines émettant des photons UV. La couche de surface de tels nuages, où le champ radiatif UV chauffe et dissocie le gaz moléculaire, est appelée région de photodissociation (PDR). Leur cartes multispectrales contiennent typiquement de 1 à 10 000 pixels, où chaque pixel contient l'intensité intégrée de 5 à 30 raies d'émission. Ces intensités peuvent être comparées avec les prédictions d'un modèle numérique du MIS tel que le code PDR de Meudon, qui calcule ces intensités à partir de paramètres physiques. Cette thèse vise à estimer des cartes de paramètres physiques (tels que la pression thermique ou l'intensité du champ UV incident) à partir d'une carte d'observation et du code PDR de Meudon. Ce problème est une instance d'une classe générale de problèmes inverses.

Une nouvelle méthode d'inférence tenant compte d'autant de sources d'incertitudes que possible est introduite. Une procédure générale est proposée pour construire une approximation de modèles numériques. Elle exploite un réseau de neurones spécifique et surpasse les méthodes d'interpolation en termes de précision, de poids mémoire et de durée d'évaluation. Une régularisation spatiale améliore les estimations. Une approche par échantillonnage est considérée pour fournir des quantification d'incertitudes en plus d'estimateurs ponctuels de cartes de paramètres physiques afin de compenser l'absence de vérité terrain, inhérente à l'astrophysique. L'algorithme Monte Carlo Markov chain (MCMC) proposé combine deux échantillonneurs : l'un identifie les minima locaux dans l'espace des paramètres tandis que l'autre les explore efficacement. Finalement, la pertinence du modèle d'observation considéré pour l'inférence est vérifiée. La méthode proposée est appliquée à des données synthétiques pour validation, puis à des observations réelles. Les résultats sont analysés pour fournir des interprétations astrophysiques.

**Mots-clés** – Problèmes inverses, algorithmes MCMC, milieu interstellaire, nuages moléculaires, régions de photodissociation, apprentissage statistique.
# Abstract

Sampling methods for statistical inference of non-linear inverse problems:
spatial distribution of physico-chemical properties of the interstellar medium

The interstellar medium (ISM) is a very diffuse medium that fills the extraordinarily large volume between celestial objects such as stars and black holes in a galaxy. The study of the ISM raises fundamental questions including star formation. Stars are born from the gravitational collapse of a part of cold and dense regions of the ISM called molecular clouds.

This thesis analyzes multispectral maps of molecular clouds in the infrared and radio domains, observed by space or ground telescopes. The focus is put on clouds that are illuminated and heated by nearby massive stars emitting UV photons. The surface layer of such clouds, where the UV irradiation heats and dissociates the molecular gas, is called a photodissociation region (PDR). Their multispectral maps typically contain from 1 to 10 000 pixels, where each pixel contains the integrated intensities of 5 to 30 emission lines. These intensities can be compared with the predictions of an ISM numerical model such as the Meudon PDR code that computes intensities from physical parameters. This thesis aims at estimating maps of physical parameters (such as the thermal pressure or the intensity of the incident UV field) from an observation map and the Meudon PDR code. This problem is an instance of a general class of inverse problems.

A new inference method that accounts for as many uncertainty sources as possible is intro-
duced. A general procedure to derive a surrogate approximation of numerical models is proposed.
It is based on a specific neural network and outperforms interpolation methods in accuracy,
memory weight and evaluation time. A spatial regularization improves estimations. A sampling
approach is considered to provide uncertainty quantification along with the estimated physical
parameter maps to address the absence of ground truth, inherent to astrophysics. The proposed
Monte Carlo Markov Chain (MCMC) algorithm combines two samplers: one identifies local min-
ima in the parameters space while the second efficiently explores them. Finally, the relevance of
the observation model considered for inference is assessed. The proposed method is applied to syn-
thetic data for validation and then to real observations. The results are analyzed for astrophysical
interpretation.

**Keywords** – Inverse problems, MCMC algorithms, interstellar medium, molecular clouds, photodissociation regions, machine learning.