{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a Gaussian regression likelihood with a chart-pulled exponential-family head. The network predicts a natural parameter eta, while each target y is mapped to a bounded probability coordinate u=G(y). Because the sufficient statistic T(u) is bounded, extreme targets cannot create arbitrarily large parameter gradients. The chart determines the tail class, while eta controls the distribution in probability space.",
 "formulas_latex": [
  "$$f_{\\eta}(x)=p_{\\eta}(G(x))\\,g(x),\\qquad p_{\\eta}(u)=h(u)\\exp\\bigl(\\eta T(u)-A(\\eta)\\bigr),\\qquad g(x)=G'(x).$$",
  "$$A(\\eta)=\\log\\int_0^1 h(u)\\exp\\bigl(\\eta T(u)\\bigr)\\,du,\\qquad \\frac{\\partial}{\\partial\\eta}\\log f_{\\eta}(y)=T(G(y))-A'(\\eta).$$",
  "$$b_G(f_{\\eta})=G^{-1}\\bigl(A'(\\eta)\\bigr),\\qquad I(\\eta)=A''(\\eta)=\\operatorname{Var}_{p_{\\eta}}[T(U)].$$",
  "$$G_{\\mathrm{Cauchy}}(x)=\\frac12+\\frac{1}{\\pi}\\arctan(x),\\qquad g_{\\mathrm{Cauchy}}(x)=\\frac{1}{\\pi(1+x^2)},\\qquad \\ell(y;\\eta)=-\\eta T(G(y))+A(\\eta)-\\log g(y).$$"
 ],
 "id": 3132,
 "implementation": "Integrate this as the final scalar probabilistic head of a small MLP or transformer regression model. The backbone produces a scalar z, and use eta=eta_max*tanh(z), with eta_max initially set to 8 to keep quadrature and gradients numerically safe. Use the Cauchy chart u=0.5+atan(y)/pi and statistic t=u-0.5. Choose h(u)=1 on (0,1), so A(eta)=log integral of exp(eta*(u-0.5)) du and A'(eta) is the corresponding mean statistic. For an MVP, precompute A and A' on a grid of 4096 eta values over [-8,8] using Gauss-Legendre quadrature with 64 or 128 nodes, then linearly interpolate during training. For each batch, execute: `z=backbone(x)`; `eta=eta_max*tanh(z)`; `u=0.5+atan(y)/pi`; `t=u-0.5`; `A=interp_A(eta)`; `nll=-eta*t+A-log(1/(pi*(1+y*y)))`; `loss=mean(nll)`. The Jacobian term is necessary for calibrated value-space likelihoods but has no gradient with respect to eta. The formulas provide the chart, score, Fisher information, and barycenter; estimate quadrature tables numerically and measure gradient bounds empirically. First compare against Gaussian NLL, Huber regression, and Student-t NLL on y=x+epsilon with Cauchy noise and with 5 percent extreme outliers. Use a two-layer MLP and equal training budgets. Record per-example gradient magnitude versus |y|, optimization divergence rate, clean-test MAE, test NLL, and 90/95 percent interval coverage. Success is a bounded or saturating score for extreme targets, fewer unstable updates, improved contaminated-data accuracy, and better calibrated intervals at comparable compute.",
 "math_summary": "Let G:R -\u003e (0,1) be a differentiable probability chart, let g(x)=G'(x), and let U=G(X). Start with the ordinary exponential-family density p_eta(u)=h(u) exp(eta T(u)-A(eta)) on (0,1), where h is a base density, T is a bounded sufficient statistic, eta is the neural-network output, and A(eta)=log integral from 0 to 1 of h(u) exp(eta T(u)) du. Pulling this family back to value space gives f_eta(x)=p_eta(G(x))g(x). For an observation y, the eta-dependent log-likelihood score is d log f_eta(y)/d eta = T(G(y))-A'(eta). Since T is bounded, this score is bounded up to the bounded range of A'(eta). The Fisher information is I(eta)=A''(eta)=Var_p_eta[T(U)], and the probability barycenter is b_G(f_eta)=G^{-1}(A'(eta)). Use the Cauchy chart G(x)=1/2+arctan(x)/pi and g(x)=1/(pi(1+x^2)) to obtain Cauchy-equivalent tails. A simple bounded statistic is T(u)=u-1/2, with range [-1/2,1/2].",
 "math_tags": [
  "probability",
  "statistics",
  "information-theory",
  "geometry"
 ],
 "ml_areas": [
  "mlp",
  "loss",
  "training"
 ],
 "paper": {
  "arxiv_id": "2609.03208",
  "arxiv_url": "https://arxiv.org/abs/2609.03208",
  "summary_what_math_gives_to_ml": "The paper gives a constructive way to build heavy-tailed likelihoods without changing the exponential function: map the target through a probability chart G, fit an ordinary exponential family in the bounded coordinate, and pull the density back with the chart Jacobian. The useful neural-network property is that bounded sufficient statistics produce bounded likelihood scores, while the chart controls tail behavior independently of the predicted natural parameter. This can become a robust probabilistic regression head whose gradients are less sensitive to extreme targets than Gaussian likelihood training. The most direct test is contaminated regression with Cauchy or Pareto noise.",
  "title": "G-Exponential Families through Probability Coordinates",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Probability-coordinate robust likelihood head",
 "url": "https://synthcore.org/idea/3132/probability-coordinate-robust-likelihood-head",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}