Optimal experimental design — designing the observation before you take it - progenax — differentiable initial conditions for N-body simulations

Optimal experimental design¶

Every other demo on this site answers the same kind of question: given data, can you recover a parameter? This section asks the question that comes before any data exist. You have been awarded a fixed amount of telescope time — call it a budget of $N$ stars you can afford to measure. Where on the sky should you point, how faint should you go, and which measurement (a radial velocity? a proper motion?) should you make on each star, so that you learn the most about the one number you actually care about?

That is optimal experimental design (OED): you optimise the observation itself, not just the analysis you do afterwards. It is rare in astronomy — almost no simulator can compute $\partial(\text{information})/\partial(\text{observing strategy})$ — and it is exactly what a differentiable forward model makes possible. The precision you will achieve is a known function of the observing strategy, computable before you collect a single photon, through the Fisher information. So you can treat the observing strategy as a free variable and maximise the information you expect to extract. The output is not just “a good plan”; it is a quantified plan: “this allocation reaches the same precision with several-fold fewer stars,” or “this is the exact depth at which more telescope time stops helping.”

The worked examples¶

Example	Design lever	Target	Headline
Anisotropy — where the proper motions belong	radial × channel (RV ↔ PM) allocation	anisotropy radius $r_a$	PMs to the outskirts; $3.66\times$ fewer stars at equal precision on $r_a$
Dynamical mass — how deep to survey	survey depth (limiting magnitude $m_{\rm lim}$ )	dynamical mass $M$	an interior optimal depth ( $m_{\rm lim}\approx13.3$ ); deeper or shallower is worse
Concentration — where the proper motions belong, redux	radial × channel (RV ↔ PM) allocation	concentration $W_0$	PMs to the core (the mirror image of anisotropy); $\approx3.4\times$ fewer stars at equal precision on $W_0$
Robustness — when binaries lie to your mass	binary-aware vs binary-blind model (radial RV allocation)	dynamical mass $M$ under misspecification	a binary-blind design biases $M$ by $+184\%$ with $41\times$ false confidence; the binary-aware design removes it

What OED can do with progenax¶

The worked examples above are instances of one general capability: every differentiable progenax forward model, paired with a likelihood, is a Fisher matrix Equation — and therefore an OED problem. Because the Fisher is additive, the design Fisher for any cluster-science parameter is assembled by computing per-star (or per-bin) blocks once and optimising the weights. The same machinery extends to other design levers (survey depth, observing epochs and cadence, multi-instrument fusion), other targets (the IMF slope, binary fraction, tidal radius, rotation, multi-population separation), and other optimality criteria (D/A for all parameters, model-discrimination designs across the multiple forward models progenax carries for the same observable). Since the forward stack is differentiable end to end, $\partial(\text{information})/\partial(\text{observing strategy})$ is computable for essentially any of these — which turns “how should we observe?” from a heritage decision into a gradient-ascent problem.

The four designs above were prototyped to demonstrate this; the OED tooling itself is planned for a separate package and is not part of progenax v0.1.0.