`log_modified_bessel_first_kind` has zero derivatives in the order at v = 0 and v = 1

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调研方向

Start with the implementation of log_modified_bessel_first_kind and the source comment about autodiff at v = 0, then inspect the reverse-mode test area under test/unit/math/rev/fun/. The issue proposes checking whether the special-case fits can be limited to non-autodiff orders so autodiff uses the general power series. Add the supplied order-gradient checks at v = 0 and v = 1 and confirm they match the expected values.

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描述

AI use disclosure: This was written with the help of Claude Code (Opus 5.5).

Description

For v == 0 and v == 1, log_modified_bessel_first_kind(v, z) evaluates Boost's fits for $I_0$ and $I_1$, which do not depend on v. When v is an autodiff variable, every derivative with respect to v is therefore 0, in reverse and forward mode. The exact first derivative is $-K_0(z)/I_0(z)$ at v = 0 (decaying like $e^{-2z}$) and about $-1/z$ at v = 1 for large z. The source comments acknowledge this ("WARNING: will not autodiff for v = 0 correctly").

A sampler, optimizer or diagnose can start exactly there: with init=0, a real<lower=0> order starts at exp(0) = 1 and a real<lower=-1> order at 0.

do_lkj_constant, the normalizing constant of lkj_corr_lpdf, lkj_corr_cholesky_lpdf and lkj_cov_lpdf, has the same defect at eta == 1.0: develop drops the η-derivatives of the constant there, so the η-gradient keeps only the data term and the second η-derivative is 0. Open PR #3434 fixes this as part of its rewrite: on its head (35435f53dc), the first and second η-derivatives at η = 1 are within 3e-16 of the exact values (K = 2). Its tests have no η = 1 point.

Possible fix: take the two shortcuts only when v is not autodiff (!is_autodiff_v<T1> &&). The test below then passes, and an autodiff order goes through the general power series.

Example

New file test/unit/math/rev/fun/log_modified_bessel_first_kind_test.cpp:

#include <stan/math/rev.hpp>
#include <test/unit/math/rev/util.hpp>
#include <gtest/gtest.h>

TEST_F(AgradRev, log_modified_bessel_first_kind_order_gradient) {
  using stan::math::var;
  auto d_dv = [](double v_val) {
    var v = v_val;
    stan::math::log_modified_bessel_first_kind(v, 0.5).grad();
    double g = v.adj();
    stan::math::recover_memory();
    return g;
  };
  // d/dv log I_v(0.5) from mpmath (50 digits); at v = 0 it is -K_0 / I_0
  EXPECT_NEAR(-0.86923697789771742858, d_dv(0), 1e-12);
  EXPECT_NEAR(-1.8244901560201558681, d_dv(1), 1e-12);
}

It fails on develop:

[ RUN      ] AgradRev.log_modified_bessel_first_kind_order_gradient
d_dv(0) evaluates to 0
d_dv(1) evaluates to 0
[  FAILED  ] AgradRev.log_modified_bessel_first_kind_order_gradient

Expected Output

The order gradient at z = 0.5 is −0.86923697789771743 at v = 0 and −1.8244901560201559 at v = 1 (mpmath).

Current Version:

v5.4.0 (develop at 1f83dc5dc4)

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