Simplify the inverse of SimplexToOrderedTransform

[haansn08]

I tested this change with $ python -m pytest test/test_transforms.py -k test_bijective_transforms.

[github-actions]

Benchmark report

this PR SimplexToOrderedTransform-inv at 8d83a11d vs baseline master at 6e0fe1fc

- run time:     1 slower, 0 faster
  compile time: unchanged across 32 benchmarks

Significant changes (1)

                       ─────── run time ───────     ───── compile time ─────
  benchmark            baseline  this PR      Δ     baseline  this PR      Δ
────────────────────────────────────────────────────────────────────────────
- nuts_eight_schools     1.16 s   1.24 s  +6.6%       2.69 s   2.92 s  +8.6%

_{Red is slower, green is faster; a row is coloured by the worse of its two columns. A delta in parentheses cleared the threshold on a measurement below the resolution floor, so it is shown without being called a change. † marks a benchmark that could not be compared — see below.}

Full results

distributions

                                 ───────── run time ────────     ────── compile time ─────
  benchmark                      baseline  this PR         Δ     baseline   this PR      Δ
──────────────────────────────────────────────────────────────────────────────────────────
  biject_to_constraints            3.9 ms   3.9 ms     -0.4%     393.3 ms  379.4 ms  -3.5%
  categorical_log_prob             2.5 ms   2.5 ms     -3.2%      73.1 ms   74.0 ms  +1.2%
  dirichlet_log_prob               626 µs   634 µs     +1.3%     468.9 ms  485.2 ms  +3.5%
  dirichlet_sample                57.7 ms  57.6 ms     -0.2%     878.4 ms  861.7 ms  -1.9%
  gamma_log_prob                   2.2 ms   2.2 ms     -2.9%       2.32 s    2.33 s  +0.7%
  gamma_sample                    26.9 ms  27.1 ms     +0.7%     812.0 ms  828.1 ms  +2.0%
  lkj_cholesky_sample              5.9 ms   6.0 ms     +0.9%       1.16 s    1.26 s  +8.9%
  mixture_same_family_log_prob     982 µs   1.0 ms     +4.0%      87.8 ms   87.5 ms  -0.3%
  multivariate_normal_log_prob     342 µs   298 µs  (-12.7%)     160.0 ms  173.8 ms  +8.6%
  normal_log_prob                  593 µs   571 µs     -3.8%      62.1 ms   64.2 ms  +3.3%
  normal_sample                   23.5 ms  23.9 ms     +1.6%     216.3 ms  219.8 ms  +1.6%
  stick_breaking_transform         6.5 ms   6.7 ms     +2.3%     222.5 ms  226.4 ms  +1.8%
  student_t_log_prob               3.2 ms   3.2 ms     -0.2%      82.5 ms   86.2 ms  +4.4%
  truncated_normal_log_prob        708 µs   725 µs     +2.5%      55.7 ms   55.0 ms  -1.3%

handlers

                                  ──────── run time ───────     ────── compile time ──────
  benchmark                       baseline   this PR      Δ     baseline   this PR       Δ
──────────────────────────────────────────────────────────────────────────────────────────
  initialize_model_hierarchical    38.9 ms   40.7 ms  +4.4%       4.09 s    4.22 s   +3.1%
  log_density_hierarchical          3.0 ms    3.0 ms  +0.5%       1.33 s    1.34 s   +1.0%
  nested_handler_stack              1.2 ms    1.2 ms  +0.2%       897 µs    782 µs  -12.7%
  potential_energy_and_grad          26 µs     25 µs  -2.6%      99.2 ms  103.2 ms   +4.1%
  predictive_forward_sampling     725.3 ms  725.1 ms  -0.0%     211.6 ms  196.0 ms   -7.4%
  trace_seeded_model                672 µs    683 µs  +1.7%     585.8 ms  602.7 ms   +2.9%

mcmc

                             ──────── run time ───────     ───── compile time ─────
  benchmark                  baseline   this PR      Δ     baseline  this PR      Δ
───────────────────────────────────────────────────────────────────────────────────
  hmc_logistic_regression    762.5 ms  770.7 ms  +1.1%       3.60 s   3.66 s  +1.7%
  nuts_dense_mass_funnel       1.19 s    1.22 s  +2.3%       2.91 s   3.04 s  +4.2%
- nuts_eight_schools           1.16 s    1.24 s  +6.6%       2.69 s   2.92 s  +8.6%
  nuts_hierarchical_glm        4.95 s    4.96 s  +0.3%       5.07 s   5.43 s  +7.2%
  nuts_logistic_regression     1.14 s    1.13 s  -0.9%       3.56 s   3.66 s  +2.8%
  nuts_vectorized_chains       2.64 s    2.56 s  -3.0%       2.89 s   3.03 s  +4.9%

svi

                                             ──────── run time ───────     ───── compile time ─────
  benchmark                                  baseline   this PR      Δ     baseline  this PR      Δ
───────────────────────────────────────────────────────────────────────────────────────────────────
  svi_autodelta_map_logistic                 321.2 ms  325.0 ms  +1.2%       3.45 s   3.55 s  +2.9%
  svi_autodiagonalnormal_hierarchical          1.07 s    1.02 s  -4.5%       5.41 s   5.45 s  +0.8%
  svi_automultivariatenormal_eight_schools   764.8 ms  789.0 ms  +3.2%       4.35 s   4.47 s  +2.6%
  svi_autonormal_logistic                    804.3 ms  790.5 ms  -1.7%       3.92 s   3.75 s  -4.3%
  svi_multi_particle_elbo                      1.53 s    1.54 s  +0.3%       3.78 s   3.98 s  +5.4%
  svi_trace_mean_field_elbo                    1.40 s    1.35 s  -3.5%       5.73 s   5.97 s  +4.2%

Methodology and environment

Each benchmark is set up untimed, then called once with the JAX caches cleared and several more times warm. Run is the fastest warm call; compile is the first call minus that, i.e. the tracing, lowering and XLA compilation the warm calls did not have to pay for.

Both refs were measured on the same runner over 2 interleaved round(s), taking the best observation per benchmark. A result is called neutral when it moves less than ±5% (run) or ±25% (compile), or when the measurement itself is under 1 ms (run) / 50 ms (compile) — a shared CI runner cannot resolve changes below that. Compile time gets the looser band because it is measured once per round rather than best-of-N, and swings by roughly 20% even between two runs of identical code. A delta shown in parentheses did clear its threshold, but on a measurement below the resolution floor, so it is reported without being called a change.

	baseline	this PR
ref	master	SimplexToOrderedTransform-inv
commit	6e0fe1fc	8d83a11d
numpyro	0.21.0	0.21.0
jax	0.11.1	0.11.1
backend	cpu	cpu
python	3.14.7	3.14.7

Runner: Linux-6.17.0-1022-azure-x86_64-with-glibc2.39, 4 CPUs.

_{Produced by this benchmark run.}

[Qazalbash]

Thanks @haansn08