LISA Data analysis

The data

The “Global fit”

Analyze all the data, simultaneously, block-by-block

$<10^5$ parameters in the full problem

SGWB estimation methods

High precision reconstruction required to extract an SGWB signal

SBI

Traditional problem

$$ p(\theta|d) = \frac{\mathcal{L}(d|\theta)\pi(\theta)}{\color{red}{Z(d)}}= \frac{\mathcal{L}(d|\theta)\pi(\theta)}{\color{red}{\int_{\theta}\mathcal{L}(d|\theta)\pi(\theta) d\theta}} $$

• Monte Carlo: e.g. Rejection sampling
• Markov-chain MC: e.g. Metropolis-Hastings, NUTS
• Variational Inference: surrogate $p(\theta|d)$

What if we dont have $\mathcal{L}(d|\theta)$ ?

Simulation based inference:

New term for:

• Approximate Bayes Computation,
• Likelihood free inference,
• Indirect inference,
• Synthetic likelihood

Algorithm

Compare the ‘simulated’ data to the ’true’ data

Different SBI methods:

• Classical: Rejection ABC (‘97), MCMC-ABC (‘03)
• Neural density:
  • Neural posterior estimator
  • Neural likelihood estimator
  • Neural ratio estimator (Lnl/evid)
• Types of NN:
  • Mixture density networks
  • Normalising flows

Goals for NN + SBI:

• Speed: Training faster than MCMC
• Scalability: Doesn’t fall apart with high D
• Pre-existing research: Leverage modern ML tools (flows, NNs …)

MCMC, VI, SBI

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SBI Math

Skipping this, can come back if folks interested

$D_{KL}$ “Loss” function for training

$$D_{\rm KL}(\tilde{p}, p) = \int \tilde{p}(x) \log \frac{\tilde{p}(x)}{p(x)}\ dx$$

$D_{KL}$ is not symmetric

• $D_{\rm KL}(\tilde{p}, p)$: Variational inference (LnL based)
• $D_{\rm KL}(p, \tilde{p})$: NPE (Simulation based)

PROBLEM: how do we avoid evaluating the $p(\theta|d)$?

KL-Divergence and VI

$$D_{\rm KL} [\tilde{p}, p] (\theta) \sim \mathbb{E}_{\theta\sim\tilde{p}(\theta|d)} \log \left[ \frac{\tilde{p}(\theta|d)}{\mathcal{L}(d|\theta)\pi(\theta)} \right] + C$$

• PROBLEM: $p(\theta|d)$ is $$$
• SOLUTION:
  • $p(\theta|d) \sim \mathcal{L}(d|\theta)\pi(\theta)$
  • $0\leq D_{\rm KL} [\tilde{p}, p]\leq Z(d)$
  • Train $\tilde{p}(\theta|d)$

KL-Divergence and SBI

$$D_{\rm KL}[p, \tilde{p}] (\theta, d) \sim -\mathbb{E}_{(\theta,d)\sim p(\theta,d)} \log \tilde{p}(\theta| d) + C $$

• PROBLEM: $p(\theta|d)$ is $$$
• SOLUTION:
  • sample from $p_{\rm joint}(\theta, d) = \mathcal{L}(d|\theta)\pi(\theta)$
  • Train $\tilde{p}(\theta|d)$

Marginal SBI vs VI

Variatinal inference

• variational posterior $\tilde{p}(\vec{\theta}|d)$ must conver all params likelihoodd model condditioned on

SBI Marginal inference

• Can replace $\tilde{p}(\vec{\theta}|d)$ for $\tilde{p}(\theta_1|d)$ without need of doing integrals

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Truncated Marginal Neural Ratio Estimation (TMNRE)

Active learning loop

Network architecture

Truncation example

Alvey+ Signal and noise model

• Noise model (only amplitudes parameterised – shape fixed):

  • $\small S^{\rm N}(A, P, f) \sim A^2 s^{TM}(f) + P^2 s^{OMS}(f)$

• Two signal models (one chosen):

  • $\tiny {\rm Power Law}: \Omega(\alpha, \gamma, f) \sim 10^\alpha\ f^\gamma$
  • $\tiny {\rm N-Power Laws}:\Omega(\vec{\alpha}, \vec{\gamma}, \vec{f}_{\rm range}, f) \sim \sum^N 10^\alpha_i\ f^\gamma_i\ \Theta[f_i^{\rm min}, f_i^{\rm max}]$

BASE Model consists of

• data(t) = noise(t) + $\sum^{\rm signals}$ s_i(t)
• Single TDI channel
• 12 days of data (split into 100 segments, 1 segment ~ 2.9 hours)
• $\Delta f\sim0.1\ {\rm mHz}$

Model with transients:

• Same as BASE mode
• In each segement Inject 1 massive BH merger (priors below) if U[0,1] < p

    Mc = U(8e5, 9e5)
    eta = U(0.16, 0.25)
    chi1 = U(-1.0, 1.0)
    chi2 = U(-1.0, 1.0)
    dist_mpc = U(5e4, 1e5)
    tc = 0.0
    phic = 0.0

MLA training:

“Several numerical settings should be chosen for the general structure of the algorithm as well as the network architechture”

• 500K simulations (9:1 train:val split)
• 50 epochs (512 batch size)
• save model weights with the lowest validation loss

Results + Discussion

MCMC vs SBI fit

Some thoughts

• The good:
  • ‘Implicit marginalisation’ may enable focused study (without global fit)!
  • Fewer evaluations of the model needed!
• The $\tiny{\rm bad}$ not so good:
  • Doest use LnL even when known (no gradients)
  • Requires robust models for noise (slow!)
  • Need to model all signals in data generation?
  • MLA architecture…
• The ugly:
  • unfair MCMC comparison for data with transients

Future work

• More complex noise model
• Longer data duration
• Additional data channels
• other “SBI” blocks for the global fit