diff --git a/ISSC2024/index.html b/ISSC2024/index.html index 5072768..9e1ec22 100644 --- a/ISSC2024/index.html +++ b/ISSC2024/index.html @@ -1,42 +1,1673 @@ - - - - - Template Presentation - - - - - - - - - - - -
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Slide 1
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Slide 2
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+ + + + + + ISSC talk 2024 + + + + + + + + + + + + + +
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+

Probabilistic cosmic shear inference
without calibration + + +

+ +
+ +
+ +

Benjamin Remy

+ Princeton University +
+

ISSC collaboration meeting, + Cambridge, MA

+ +

Ph.D. under the supervision of Jean-Luc Starck and François + Lanusse

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+ ESA/Euclid/Euclid Consortium/NASA, image processing by J.-C. Cuillandre (CEA Paris-Saclay), G. Anselmi, CC BY-SA 3.0 IGO +
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+

Weak gravitational lensing

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+ + +
+
+ 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 $\gamma$ +
    • +
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+ + +
+

Measuring galaxies ellipticity

+ +
+
+ +
+ +
+ +
+ +
+ + + Moments-based method

+ + $\begin{align} + F &= \sum G(x, y)I(x, y)\\ + \sigma^2 &= \sum G(x, y)^2 \sigma^2(I(x, y)) \\ + T &= \sum G(x, y)I(x, y)(x^2 + y^2) \\ + M_1 &= \sum G(x, y)I(x, y)(x^2 - y^2) \\ + M_2 &= \sum G(x, y)I(x, y)2xy \\ + \color{#B1CB11}{e_1} &\color{#B1CB11}{=} \color{#B1CB11}{M_1 / T}\\ + \color{#B1CB11}{e_2} &\color{#B1CB11}{=} \color{#B1CB11}{M_2 / T} \\ + S/N &= F / \sigma(F) \\ + \end{align}$ +
+ +
+
+ + + + +
+ +
+

Measuring galaxies ellipticity

+ +
+
+ +
+ +
+ +
+ Model fitting

+ + $\begin{align} + I(x, y) &= F \cdot \mathcal{N}\left( 0, \color{#B1CB11}{\Sigma} \right)\\ + + \color{#B1CB11}{(e_1, e_2)} &\color{#B1CB11}{=} \color{#B1CB11}{f(\Sigma)} + \end{align}$ +
+ + +
+ What about real galaxies?
+ +
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+ +
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+
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+ +
+ +
+ +
+ +
+ +
+ + $\hat \gamma = c + + (1+\color{orange}{m})\gamma_\text{true}, \quad |m|<10^{-3}$ + +
+ +
+
+ +
+ 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 +
+
+ +
+

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. +
+
+ +
+ $\gamma \ll 1$ no longer true close to a cluster + + 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 + +

+ +
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+ +
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+ +
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+ +
+ +
+ +
+ +
+ +
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+ +
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+ $\longrightarrow$
+ $g_\theta$ +
+
+ $\longrightarrow$
+ shear $\gamma$ +
+
+ $\longrightarrow$
+ PSF +
+
+ $\longrightarrow$
+ Noise +
+
+ +
+
+
+ + + + + +
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+
+
+
Probabilistic model
+
+
+
+ $$ x \sim ? $$ +
+
+ $$ x \sim \mathcal{N}(z, \Sigma) \quad z \sim ? $$
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 +
+
+ $\begin{align} + x \sim \mathcal{N}(\Pi \ast (z \otimes \gamma), \Sigma) \quad & z \sim ? \\ + & \gamma \sim \mathcal{N}(0, .05) + \end{align}$ + +
latent $z$ is a unsheared deconvolved, + and denoised galaxy image +
+
+ $\begin{align} + x \sim \mathcal{N}(\Pi \ast + (g_\theta(z) \otimes \gamma), \Sigma) \quad & z \sim \mathcal{N}(0, + \mathbf{I}) \\ + & \gamma \sim \mathcal{N}(0, .05) + \end{align}$ + + +
latent $z$ are morphological parameters
+ + $\theta$ are global parameters of the model +
+
+
+
+
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+
+
+
+
+ $\Longrightarrow$ + + 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 + $$g(z) = F \times I_0 \exp \left( -b_n \left[\left( \frac{r}{r_\text{hlr}}\right)^{\frac{1}{n}} -1\right] \right)$$ + +
+ 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 +
+
+
+ +

+ ...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 +
+
+ $\log p(x) \geq \mathbb{E}_{z\sim q_\phi(z|x)} \left[ \log p_\theta(x|z, \Pi, \Sigma) + \mathbb{D}_\text{KL}(q_\phi \| p(z)) \right]$ +
+ +
+ + + + + +
+

+ 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
+ $x_n$ +
+
+ Truth
+ $x_0$ +
+
+
+
+ +
+
+
+
+
+
+ Posterior samples
+ $g_\theta(z)$ +
+
+ +
+
+
+
+ +
+ +
+
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+ $\mathbf{P} (\Pi \ast g_\theta(z))$ +
+
+ Median +
+
+
+ +
+
+
+
+ +
+ +
+
+ Data residuals
+ $x_n - \mathbf{P} (\Pi \ast g_\theta(z))$ +
+
+ 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 +
+
+
+ + +
+ +
+

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 + +
+
+
+ +
+ +
+ +
+ +

Opening remarks

+
+
    +
  • + Shear estimators are known to respond to galaxy selection, + how to include galaxy detection and selection + in the bayesian framework? +
    • + +
  • + Galaxy that are observed blended may be affected by a + different amount of shear. How to handle blended objects? +
    • +
+
+ +
+
+ + + + + + + + +
+
+ + + + + + + + + + + - - - - - - - + \ No newline at end of file