Previous generation survey: SDSS
Current generation survey: DES
LSST precursor survey: HSC
Observed $y$
$f_\theta$
Unknown $x$
“"amount of information" obtained about one random variable through observing the other random variable”
We want to solve for the Maximum A Posterior solution:
$$\arg \max - \frac{1}{2} \parallel {\color{Orchid} y} - {\color{SkyBlue} x} \parallel_2^2 + \log p_\theta({\color{SkyBlue} x})$$ This can be done by gradient descent as long as one has access to the score function $\frac{\color{orange} d \color{orange}\log \color{orange}p\color{orange}(\color{orange}x\color{orange})}{\color{orange} d \color{orange}x}$.$\Longrightarrow$ We can see which parts of the image are well constrained by data, and which regions are uncertain.
import jax_cosmo as jc
# Defining a Cosmology
cosmo = jc.Planck15()
# Define a redshift distribution with smail_nz(a, b, z0)
nz = jc.redshift.smail_nz(1., 2., 1.)
# Build a lensing tracer with a single redshift bin
probe = probes.WeakLensing([nz])
# Compute angular Cls for some ell
ell = np.logspace(0.1,3)
cls = angular_cl(cosmo_jax, ell, [probe])
# Fisher matrix in just one line:
F = - jax.hessian(gaussian_likelihood)(theta)
No derivatives were harmed by finite differences in the computation of this Fisher!
“Given (g)riz photometry, find a tomographic bin assignment method that optimizes a 3x2pt analysis.”
import tensorflow as tf
import flowpm
# Defines integration steps
stages = np.linspace(0.1, 1.0, 10, endpoint=True)
initial_conds = flowpm.linear_field(32, # size of the cube
100, # Physical size
ipklin, # Initial powerspectrum
batch_size=16)
# Sample particles and displace them by LPT
state = flowpm.lpt_init(initial_conds, a0=0.1)
# Evolve particles down to z=0
final_state = flowpm.nbody(state, stages, 32)
# Retrieve final density field
final_field = flowpm.cic_paint(tf.zeros_like(initial_conditions),
final_state[0])
Thank you !