LEARNING LATENT FIELDS · COMPUTED TEACHING LABS
Change the observations. Inspect the likelihood. Let posterior beliefs teach a power model.
Off-diagonal entries carry relative phase. Diagonal sensor powers alone give a flat bearing likelihood.
Press Play or Step. The initial power model has not seen data.
Synthetic Gaussian array example. Known first bearing and powers. These are computed curves, not paper benchmark results.
Each snapshot is z = ρ₁a(θ₁) + ρ₂a(θ₂) + w. Independent amplitudes satisfy ρₙ ∼ CN(0,γₙ), and w ∼ CN(0,ηI). The six-element, half-wavelength array uses unit-norm steering vectors. Here γ₁ = γ₂ = 1 and the slider is 10 log₁₀(γ/η). Random draws are reproducible from the seed.
After marginalizing amplitudes, Σ = ηI + γ₁a₁a₁ᴴ + γ₂a₂a₂ᴴ. For S independent snapshots, log L = −S log det Σ − Σₛ zₛᴴΣ⁻¹zₛ, up to a constant. The green curve varies the second bearing, holding the first bearing and powers known. A complex Cholesky factorization evaluates the likelihood. A uniform grid prior gives the plotted relative posterior.
The orange curve is conventional beam power, the mean of |a(θ)ᴴz|². It uses cross-sensor products and retains angular information. Local peaks above the adjustable threshold are detection candidates. The gray curve instead retains only sensor-wise powers, or diagonal covariance. Every sensor has the same modeled power, so this particular compression is direction-independent.
Both scans use the same samples. The full-likelihood scan has stronger information about the scene: a known first source and known powers. Its accuracy is therefore not a fair end-to-end tracker comparison. Change the separation, noise or number of snapshots to inspect ambiguity. The covariance heat map shows the actual sample estimate, not the expected matrix.
One always-present object traverses x ∈ [−0.85, 0.85] with a known motion model, using x = sin(bearing). Four array sensors produce three independent complex snapshots per frame. Two finite-grid Bayesian filters infer position and log power from exactly the same data. Both know the noise model. The SNR slider refers to the coefficient-to-noise ratio at reference power γ = 2; instantaneous SNR varies with the field. One uses a fixed broad lognormal power prior and one uses the learned conditional prior.
The field plot shows conditional median power exp μ(c), which isolates the learned location dependence. The learned family uses t = log γ = μθ(c) + σε with ε ∼ N(0,1). This is a conditional affine flow in log power, followed by exponentiation. Both filters use the same fixed log-power standard deviation, σ = 1.1. The synthetic field has σ = 0.45, so this restricted family learns the context-dependent mean of log power and deliberately cannot match the full density. The context c is the preceding posterior position estimate, frozen during the current frame. Shared coefficients θ make information from one traversal useful at other positions and later times.
After inference, the learner stores the frozen context and posterior E[log γ]. A regularized regression update fits μθ(c) to these soft targets. This is a small, regularized version of the paper's expected log-density objective. There are no true positions or powers in the training targets. Numerical integration takes place on a log-power grid with the correct change of variables.
This lab fixes the object count and sets existence probability to one. It omits births, deaths, multiobject interference, belief propagation and the paper's flexible conditional normalizing flow. Hidden truth appears only for visual comparison and evaluation. RMSE is calculated from the displayed synthetic run and need not improve for every setting. It is not a reproduction of any reported experiment.