Probabilistic cosmic shear inference
without calibration
Benjamin Remy
Princeton UniversityISSC collaboration meeting, Cambridge, MA
Ph.D. under the supervision of Jean-Luc Starck and François Lanusse
slides at b-remy.github.io/talks/ISSC2024
ESA/Euclid/Euclid Consortium/NASA, image processing by J.-C. Cuillandre (CEA Paris-Saclay), G. Anselmi, CC BY-SA 3.0 IGO
Weak gravitational lensing
Galaxy shapes as estimators for gravitational shear
$$ e = \gamma + e_i \qquad \mbox{ with } \qquad
e_i \sim \mathcal{N}(0, I)$$
- Standard methods are trying the measure the ellipticity $e$ of galaxies as an estimator for the gravitational shear $\gamma$
Measuring galaxies ellipticity
Moments-based method
$\begin{align} F &= \sum G(x, y)I(x, y)\\ \sigma^2 &= \sum G(x, y)^2 \sigma^2(I(x, y)) \\ T &= \sum G(x, y)I(x, y)(x^2 + y^2) \\ M_1 &= \sum G(x, y)I(x, y)(x^2 - y^2) \\ M_2 &= \sum G(x, y)I(x, y)2xy \\ \color{#B1CB11}{e_1} &\color{#B1CB11}{=} \color{#B1CB11}{M_1 / T}\\ \color{#B1CB11}{e_2} &\color{#B1CB11}{=} \color{#B1CB11}{M_2 / T} \\ S/N &= F / \sigma(F) \\ \end{align}$
$\begin{align} F &= \sum G(x, y)I(x, y)\\ \sigma^2 &= \sum G(x, y)^2 \sigma^2(I(x, y)) \\ T &= \sum G(x, y)I(x, y)(x^2 + y^2) \\ M_1 &= \sum G(x, y)I(x, y)(x^2 - y^2) \\ M_2 &= \sum G(x, y)I(x, y)2xy \\ \color{#B1CB11}{e_1} &\color{#B1CB11}{=} \color{#B1CB11}{M_1 / T}\\ \color{#B1CB11}{e_2} &\color{#B1CB11}{=} \color{#B1CB11}{M_2 / T} \\ S/N &= F / \sigma(F) \\ \end{align}$
Measuring galaxies ellipticity
Model fitting
$\begin{align} I(x, y) &= F \cdot \mathcal{N}\left( 0, \color{#B1CB11}{\Sigma} \right)\\ \color{#B1CB11}{(e_1, e_2)} &\color{#B1CB11}{=} \color{#B1CB11}{f(\Sigma)} \end{align}$
$\begin{align} I(x, y) &= F \cdot \mathcal{N}\left( 0, \color{#B1CB11}{\Sigma} \right)\\ \color{#B1CB11}{(e_1, e_2)} &\color{#B1CB11}{=} \color{#B1CB11}{f(\Sigma)} \end{align}$
What about real galaxies?
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$\hat \gamma = c + (1+\color{orange}{m})\gamma_\text{true}, \quad |m|<10^{-3}$
$\hat \gamma = c + (1+\color{orange}{m})\gamma_\text{true}, \quad |m|<10^{-3}$
The ellipticity is not necessarily a well defined quantity for arbitrary galaxies,
leading to bias in cosmic shear estimation... Requires post measurement calibration,
degrading data
leading to bias in cosmic shear estimation... Requires post measurement calibration,
degrading data
Calibrating the multiplicative bias
$\hat \gamma = c +
(1+\color{orange}{m})\gamma_\text{true}, \quad
|m|<10^{-3}$
Metacalibration: Taylor expansion, assuming $\gamma \ll 1$
$e = e|_{\gamma=0} + \underset{R = 1 + m}{\underbrace{\dfrac{\partial e}{\partial \gamma}\Big|_{\gamma = 0}}} \gamma + \mathcal{O}(\gamma^2)$
Response matrix by finite differentiation:
$R_{ij} = \dfrac{e^+_i - e^+_i}{\Delta \gamma_{j}}$
Requires to add anticorrelated noise to the data, due to artificial shear.
$e = e|_{\gamma=0} + \underset{R = 1 + m}{\underbrace{\dfrac{\partial e}{\partial \gamma}\Big|_{\gamma = 0}}} \gamma + \mathcal{O}(\gamma^2)$
Response matrix by finite differentiation:
$R_{ij} = \dfrac{e^+_i - e^+_i}{\Delta \gamma_{j}}$
Requires to add anticorrelated noise to the data, due to artificial shear.
$\gamma \ll 1$ no longer true close to a cluster
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Requires to compute $\dfrac{\partial^2 e}{\partial
\gamma^2}\Big|_{\gamma = 0}$, $\dfrac{\partial^3 e}{\partial
\gamma^3}\Big|_{\gamma = 0}$, ...
Requires to compute $\dfrac{\partial^2 e}{\partial
\gamma^2}\Big|_{\gamma = 0}$, $\dfrac{\partial^3 e}{\partial
\gamma^3}\Big|_{\gamma = 0}$, ...
Let's do forward modeling instead
Let us build a probabilistic model of galaxy images
$\longrightarrow$
$g_\theta$
$g_\theta$
$\longrightarrow$
shear $\gamma$
shear $\gamma$
$\longrightarrow$
PSF
PSF
$\longrightarrow$
Noise
Noise
Probabilistic model
$$ x \sim ? $$
$$ x \sim \mathcal{N}(z, \Sigma) \quad z \sim ? $$
latent $z$ is a denoised galaxy image
latent $z$ is a denoised galaxy image
$$ x \sim \mathcal{N}(\Pi \ast z, \Sigma)
\quad z \sim ? $$
latent $z$ is a deconvolved, and denoised galaxy image
latent $z$ is a deconvolved, and denoised galaxy image
$\begin{align}
x \sim \mathcal{N}(\Pi \ast (z \otimes \gamma), \Sigma) \quad & z \sim ? \\
& \gamma \sim \mathcal{N}(0, .05)
\end{align}$
latent $z$ is a unsheared deconvolved, and denoised galaxy image
latent $z$ is a unsheared deconvolved, and denoised galaxy image
$\begin{align}
x \sim \mathcal{N}(\Pi \ast
(g_\theta(z) \otimes \gamma), \Sigma) \quad & z \sim \mathcal{N}(0,
\mathbf{I}) \\
& \gamma \sim \mathcal{N}(0, .05)
\end{align}$
latent $z$ are morphological parameters
$\theta$ are global parameters of the model
latent $z$ are morphological parameters
$\theta$ are global parameters of the model
$\Longrightarrow$
Decouples the morphology model from the observing conditions.
Bayesian modeling of cosmic shear
We aim to model the posterior distribution $p(\gamma|\mathcal{D})$
$\begin{align}
p(\gamma|\mathcal{D}) &= \int p(\gamma, z, \Pi|\mathcal{D}) ~dz~d\Pi \\
\end{align}$
$\begin{align}
~~~~~~~~~~~&= \int \color{#B1CB11}{\underbrace{p(\mathcal{D}|\gamma, z, \Pi)}_{\text{likelihood}}} \underbrace{p(\gamma)p(z)p(\Pi)}_{\text{priors}} ~dz~d\Pi
\end{align}$
The likelihood $\color{#B1CB11}{p(\mathcal{D}|\gamma, z, \Pi)}$ is naturally built from the forward model
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Joint inference using a parametric model for the morphology
Let's assume that $g(z)$ is a sersic model, i.e. $z = \{n, r_\text{hlr}, F, e_1, e_2, s_x, s_y\}$ and
$$g(z) = F \times I_0 \exp \left( -b_n \left[\left( \frac{r}{r_\text{hlr}}\right)^{\frac{1}{n}} -1\right] \right)$$
...due to model misspecification $\longrightarrow$ Let's learn a more flexible $g_\theta$
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$\mathbf{P} (\Pi \ast g_\theta(z))$
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The joint inference of $p(z, \gamma | \mathcal{D})$ leads to a biased posterior...
Marginal shear posterior $p(\gamma|\mathcal{D})$
Maximum a posteriori fit and residuals
Marginal shear posterior $p(\gamma|\mathcal{D})$
Maximum a posteriori fit and residuals
...due to model misspecification $\longrightarrow$ Let's learn a more flexible $g_\theta$
Learning from corrupted data
Lanusse et al. 2020 
$\longrightarrow$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Inference network}$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Generator network}$
$q_\phi(z|x)$
$p_\theta(x|z)$
$z \sim q_\phi(z|x)$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$\longrightarrow$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$x' \sim p_\theta(x|z)$
Learning from corrupted data
Lanusse et al. 2020
$\longrightarrow$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Inference network}$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Generator network}$
$q_\phi(z|x)$
$p_\theta(x|z)$
$z \sim q_\phi(z|x)$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$\longrightarrow$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$g_\theta(z)$
$x' \sim p_\theta(x|z)$
Learning from corrupted data
Lanusse et al. 2020
$\longrightarrow$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Inference network}$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Generator network}$
$q_\phi(z|x)$
$p_\theta(x|z)$
$z \sim q_\phi(z|x)$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$\longrightarrow$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$g_\theta(z)$
$\ast$
$\Pi$
$x' \sim p_\theta(x|z, \Pi, \Sigma)$
$\longrightarrow$
Learning from corrupted data
Lanusse et al. 2020
$\longrightarrow$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Inference network}$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Generator network}$
$q_\phi(z|x)$
$p_\theta(x|z)$
$z \sim q_\phi(z|x)$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$\longrightarrow$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$g_\theta(z)$
$\ast$
$\Pi$
$x' \sim p_\theta(x|z, \Pi, \Sigma)$
$\longrightarrow$
$\underbrace{\quad \quad \quad}_{g_\theta(z) \ast \Pi}$
$\longrightarrow$
Learning from corrupted data
Lanusse et al. 2020
$\longrightarrow$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Inference network}$
$\underbrace{\quad \quad \quad \quad \quad \quad}_\textrm{Generator network}$
$q_\phi(z|x)$
$p_\theta(x|z)$
$z \sim q_\phi(z|x)$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$\longrightarrow$
$\rightarrow$
$\rightarrow$
$\rightarrow$
$\longrightarrow$
$g_\theta(z)$
$\ast$
$\Pi$
$x' \sim p_\theta(x|z, \Pi, \Sigma)$
$\longrightarrow$
$\underbrace{\quad \quad \quad}_{g_\theta(z) \ast \Pi}$
$\longrightarrow$
Optimized maximizing the ELBO
$\log p(x) \geq \mathbb{E}_{z\sim q_\phi(z|x)} \left[ \log p_\theta(x|z, \Pi, \Sigma) + \mathbb{D}_\text{KL}(q_\phi \| p(z)) \right]$
$\log p(x) \geq \mathbb{E}_{z\sim q_\phi(z|x)} \left[ \log p_\theta(x|z, \Pi, \Sigma) + \mathbb{D}_\text{KL}(q_\phi \| p(z)) \right]$
A generative model for galaxy morphologies
The Bayesian view of the problem: $$ p(z | x ) \propto
p_\theta(x | z, \Sigma, \mathbf{\Pi}) p(z)$$ where:
- $p( z | x )$ is the posterior
- $p( x | z )$ is the data likelihood, contains the physics
- $p( z )$ is the prior
Data
$x_n$
$x_n$
Truth
$x_0$
$x_0$
Posterior samples
$g_\theta(z)$
$g_\theta(z)$
$\mathbf{P} (\Pi \ast g_\theta(z))$
Median
Data residuals
$x_n - \mathbf{P} (\Pi \ast g_\theta(z))$
$x_n - \mathbf{P} (\Pi \ast g_\theta(z))$
Standard Deviation
$\Longrightarrow$
Uncertainties are fully captured by the posterior.
Joint inference using a generative model for the morphology
Remy, Lanusse, Starck (2022)
Let's use the learned $g_\theta(z)$
The joint inference of $p(z, \gamma | \mathcal{D})$ leads to an unbiased posterior!
Marginal shear posterior $p(\gamma|\mathcal{D})$
Maximum a posteriori fit and residuals
Marginal shear posterior $p(\gamma|\mathcal{D})$
Maximum a posteriori fit and residuals
Takeaway message
- Ellipticity is not a well defined quantity for arbitrary galaxies $\rightarrow$ bias in shear estimation
-
Forward modeling allows to decouple morphology from observing conditions
- Deep generative models can be used to provide flexible light profile model
- Explicit likelihood: uses of all of our physical knowledge
$ + $ Our method can be applied for varying PSF, noise, or even different instruments!
$\Longrightarrow$ Joint inference of morphology and shear leads to unbiased marginal shear posterior