Hugo SIMON-ONFROY,
PhD student supervised by Arnaud DE MATTIA and François LANUSSE
CSI, 2024/10/18
Hugo SIMON-ONFROY,
PhD student supervised by Arnaud DE MATTIA and François LANUSSE
DESI Marseille meeting, 2024/07/12
Hugo SIMON-ONFROY,
PhD student supervised by Arnaud DE MATTIA and François LANUSSE
Rodolphe Clédassou Summer School, 2024/08
$$\frac{H}{H_0} = \sqrt{\Omega_r + \Omega_b + \Omega_c+ \Omega_\kappa + \Omega_\Lambda}$$
instantaneous expansion rate
energy content
Cosmological principle + Einstein equation
+ Inflation
\(\delta_L \sim \mathcal G(0, \mathcal P)\)
\(\sigma_8:= \sigma[\delta_L * \boldsymbol 1_{r \leq 8}]\)
initial field
primordial power spectrum
std. of fluctuations smoothed at \(8 \text{ Mpc/h}\)
\(\Omega := \{ \Omega_c, \Omega_b, \Omega_\Lambda, H_0, \sigma_8, n_s,...\}\)
Linear matter spectrum
Structure growth
$$\begin{align*}\operatorname{\boldsymbol{H}}(\Omega\mid \delta_g) &= \boldsymbol{H}(\delta_g \mid \Omega) + \boldsymbol{H}(\Omega) - \boldsymbol{H}(\delta_g)\\&= \boldsymbol{H}(\Omega) - \boldsymbol{I}(\Omega;\delta_g) \leq \boldsymbol{H}(\Omega)\end{align*}$$
$$\boldsymbol{I}(\Omega; \delta_g)$$
\(\boldsymbol{H}(X)\) = missing info on \(X\)
$$\boldsymbol{H}(\Omega)$$
$$\boldsymbol{H}(\delta_g)$$
$$\boldsymbol{H}(\Omega\mid\delta_g)$$
\(\Omega := \{ \Omega_c, \Omega_b, \Omega_\Lambda, H_0, \sigma_8, n_s,...\}\)
Linear matter spectrum
Structure growth
How to perform this marginalization?
\(\Omega := \{ \Omega_c, \Omega_b, \Omega_\Lambda, H_0, \sigma_8, n_s,...\}\)
Linear matter spectrum
Structure growth
How to perform this marginalization?
$$\boldsymbol{H}(X\mid Y_1)$$
$$\boldsymbol{H}(X)$$
$$\boldsymbol{H}( Y_1)$$
$$\boldsymbol{I}(X; Y_1)$$
$$\boldsymbol{H}( Y_2)$$
$$\boldsymbol{I}(X\mid Y_1; Y_2)$$
$$\boldsymbol{H}(X\mid Y_1,Y_2)$$
\(\boldsymbol{H}(X)\) = missing information on \(X\) = amount of bits to communicate \(X\)
$$\boldsymbol{H}(X\mid Y_1)$$
$$\begin{align*}\operatorname{\boldsymbol{H}}(X\mid Y) &= \boldsymbol{H}(Y \mid X) + \boldsymbol{H}(X) - \boldsymbol{H}(Y)\\&= \boldsymbol{H}(X) - \boldsymbol{I}(X;Y) \leq \boldsymbol{H}(X)\end{align*}$$
Marginalize then sample
summary stat inference
A tractable candidate: the power spectrum
$$0-$$
$$\boldsymbol H(\delta_g)-$$
$$0-$$
$$\boldsymbol H(\delta_g)-$$
How much information can we still gain (not lose)?
Model based field-level inference
summary stat inference
Model Based Inference at the field level
(explicitly solve)
How much information can we still gain (not lose)?
field-level inference
summary stat inference
For full field marginalize then sample hardly tractable,
so sample then marginalize
Which sampling methods can scale to high dimensions?
gradient descent
posterior mode
Brownian
exploding Gaussian
Langevin
posterior
+
=
Variations around HMC
Variations around HMC
3) Hamiltonian dynamic
1) mass \(M\) particle at \(q\)
2) random kick \(p\)
2)random kick \(p\)
1) mass \(M\) particle at \(q\)
3) Hamiltonian dynamic
Reconstruct the initial field simultaneously, yielding posterior on full universe history
number of evaluations to yield one effective sample: the higher the worse
1st author paper (< December 2024)
Leverage modern computational tools to build fast and differentiable cosmological model
Field-level inference can scale to Stage-IV galaxy surveys, and is relevant to fully capture cosmological information from data
Field-level inference may be relevant to fully capture cosmological information in data.
Leverage modern computational tools to build fast and differentiable cosmological model.
Standardized benchmark for comparing MCMC samplers on field-level inference tasks, selecting proposed methods for Stage-IV galaxy surveys.
2 fields, 1 power spectrum: Gaussian or N-body?
$$\boxed{\min_s \operatorname{\boldsymbol{H}}(\Omega\mid s(\delta_g))} = \boldsymbol{H}(\Omega) - \max_s \boldsymbol{I}(\Omega ; s(\delta_g))$$
$$\boldsymbol{H}(\Omega)$$
$$\boldsymbol{H}(\delta_g)$$
$$\boldsymbol{H}(\mathcal s_1)$$
$$\boldsymbol{H}(\mathcal s_2)$$
$$\boldsymbol{H}(\mathcal P)$$
non-Gaussianities
relevant stat
(low info but high mutual info)
irrelevant stat
(high info but low mutual info)
also a relevant stat
(high info and mutual info)
Which stats are relevant for cosmo inference?
Gaussianities
import jax.numpy as np
# then enjoy
function = jax.jit(function)
# function is so fast now!
gradient = jax.grad(function)
# too bad if you love chain ruling by hand
vfunction = jax.vmap(function)
pfunction = jax.pmap(function)
# for-loops are for-loosers
def model():
z = sample('Z', dist.Normal(0, 1))
x = sample('X', dist.Normal(z**2, 1))
return x
render_model(model, render_distributions=True)
x_sample = dict(x=seed(model, 42)())
obs_model = condition(model, x_sample)
logp_fn = lambda z: log_density(obs_model,(),{},{'Z':z})[0]
from jax import jit, vmap, grad
score_vfn = jit(vmap(grad(logp_fn)))
kernel = infer.NUTS(obs_model)
mcmc = infer.MCMC(kernel, num_warmup, num_samples)
mcmc.run(PRGNKey(43))
samples = mcmc.get_samples()