11 The TAP equations: the term mean field forgot

11.1 Where Week 5 left off

Week 5 ended with a diagnosis: mean field discards every correlation and over-predicts order, and where it fails is where variational inference needs corrections. Naive mean field will turn out to be the first order of a controlled expansion, its leading correction is a single term with a crisp physical meaning, and the size of that term will tell us — quantitatively, model by model — when Week 5’s theory was trustworthy and when it was quietly wrong.

The history behind that term frames the whole of Module 2, and it runs through Week 2’s own lineage. In 1975, Sherrington and Kirkpatrick proposed a mean-field spin glass — Gaussian random couplings, every spin coupled to every other — and, with disarming confidence, titled the paper Solvable Model of a Spin-Glass (Sherrington and Kirkpatrick 1975). They then solved it, by the replica method, and their solution carried a defect they themselves flagged: at low temperature the entropy goes negative — for discrete spins, an impossibility, the thermodynamic equivalent of a proof ending in \(1 = 2\). Two years later, Thouless, Anderson, and Palmer published the corrected mean-field theory under the title Solution of “Solvable Model of a Spin Glass” (Thouless, Anderson, and Palmer 1977) — the quotation marks serving as a referee report. Their correction to the mean-field equations is one term, which we derive twice: once by a short physical argument, and once by an expansion that identifies exactly what order of what series the argument computed. (The full resolution of the SK problem, Parisi’s replica symmetry breaking, lies outside this course; the outlook states how it relates to the cavity route taken here.)

The first correction to mean field removes from each spin’s effective field the part that the spin itself put there, reflected back by its neighbors. Its size is governed by \(\sum_j J_{ij}^2\): of order \(1/N\) for the uniform ferromagnet of Week 5, which is why naive mean field was exact there, but of order one for disordered couplings: the SK model, Week 2’s Hebb-built networks, and generally models whose couplings were set by data rather than by symmetry.

Take-home 1

Naive mean field is the first order of a systematic expansion, and the second order (the Onsager reaction term, Thouless, Anderson, and Palmer (1977)) gives the TAP equations; higher orders follow mechanically. The correction is \(\mathcal{O}(1/N)\) for uniform dense couplings and \(\mathcal{O}(1)\) for disordered ones.

11.2 The echo: the field you feel versus the field you cause

We begin with the physical argument. The naive effective field of Equation 9.5, \(\sum_j J_{ij} m_j + h_i\), treats each neighbor’s magnetization as independent evidence about what spin \(i\) should do. But look at where \(m_j\) comes from: \(j\) sits, in part, in the field of \(i\) itself. Spin \(i\) polarizes its neighbor through the bond; the polarized neighbor reports back through the same bond; and the naive field counts that report as one more vote for what \(i\) already is. A consistent theory must not let a spin respond to its own echo.

One line of Week 1 makes the echo quantitative. A single spin in a field \(h\) has \(m = \tanh(\beta h)\), so its susceptibility — its readiness to respond — is

\[ \chi_j \;=\; \frac{\partial m_j}{\partial h_j} \;=\; \beta\,\big(1 - m_j^2\big): \]

single-site linear response

an undecided spin (\(m_j \approx 0\)) responds maximally; a saturated one (\(m_j \to \pm 1\)) barely responds at all. That is the only ingredient the correction needs, and Week 1 already supplied it. Assemble the echo (Figure 11.1): spin \(i\) shifts its neighbor by \(\delta m_j = \chi_j J_{ij} m_i\); the shifted neighbor feeds back the field \(\delta h_i = J_{ij}\, \delta m_j = \beta (1 - m_j^2)\, J_{ij}^2\, m_i\). Note the square of the coupling — echoes travel out along the bond and back along the same bond, so they always carry \(J^2\). Summing over neighbors and subtracting gives the corrected field:

\[ h_i^\text{cav} \;=\; h_i + \sum_j J_{ij}\, m_j \;-\; \beta\, m_i \sum_j J_{ij}^2 \big(1 - m_j^2\big) \tag{11.1}\]

the cavity field: \(i\)’s field with \(i\)’s own influence removed

— the cavity field, so called because it is the field at site \(i\) computed as if \(i\) sat in a cavity: the neighbors’ magnetizations taken as they would be in \(i\)’s absence. (Conceptually, the correct inputs are the cavity magnetizations \(m_j^{(i)}\) — \(j\)’s value with \(i\) removed — and the reaction term is exactly the bookkeeping that converts the full \(m_j\) into them, to the order computed: \(m_j = m_j^{(i)} + \chi_j J_{ij} m_i + \mathcal{O}(J_{ij}^2)\), whose remainder enters the field only at \(\mathcal{O}(J^3)\). That distinction matters loosely today and exactly in the next lecture, because keeping “who was removed” straight is precisely what message passing does.) The idea is much older than spin glasses: Onsager (1936) introduced the reaction field for dielectrics — a molecular dipole polarizes the liquid around it, and the returning polarization field must not be counted when computing the dipole’s own alignment (Figure 11.1, right). TAP imported a forty-year-old piece of electrostatics into disordered magnetism; this week we carry it into inference.

Figure 11.1: The Onsager reaction term. Left: mean field’s double-count — spin \(i\) polarizes its neighbor \(j\) by \(\delta m_j = \beta(1-m_j^2)J_{ij}m_i\) (orange), and the polarization reports back as a field \(\propto J_{ij}^2 m_i\) (red, struck out): a spin’s own influence, returning as spurious evidence. Subtracting the echo yields the *cavity field* Equation 11.1 — the field at \(i\) computed with \(i\) absent. Right: the physical ancestor (Onsager 1936) — a dipole polarizes the surrounding medium, and the medium’s *reaction field* must not act back on the dipole’s own alignment.

Writing \(m_i = \tanh(\beta h_i^\text{cav})\) already gives the corrected self-consistency — the TAP equations, in fact. At this point, however, the approximation is uncontrolled: we do not know what the argument neglected, what “order” it computed, or how to go further. The Plefka expansion answers all three.

11.3 The Plefka expansion

Setup. We want an expansion whose zeroth order is free spins, whose knob is the interaction strength, and whose variables are the magnetizations — so that each order corrects the theory of \(m\), not just the free energy. The right object is the constrained (Gibbs) free energy \(G(\lambda, m)\): the free energy of the Hamiltonian with its exchange part scaled by an interpolation parameter \(\lambda \in [0,1]\) (\(J \to \lambda J\); the external fields ride along unscaled), evaluated with the magnetizations pinned to prescribed values \(m_i\). The pinning is done by auxiliary Lagrange fields \(b_i\) adjusted at every \(\lambda\) so that \(\langle s_i \rangle = m_i\) exactly ; the Legendre-transform bookkeeping is spelled out in Plefka (1982) and Georges and Yedidia (1991). What matters here is that at \(\lambda = 0\) the constrained measure factorizes, so every average is computable, and derivatives of \(G\) in \(\lambda\) are averages of the interaction in that computable ensemble. Expand around \(\lambda = 0\), then set \(\lambda = 1\). (One notational flag: the literature writes this parameter \(\alpha\); we rename it \(\lambda\) because \(\alpha\) is this course’s storage load, and that collision is not worth carrying.)

Order zero — independent spins. At \(\lambda = 0\), pinned free spins are Week 5’s factorized ansatz exactly, and \(G\) is pure entropy: \(\beta G_0 = -\sum_i s(m_i)\), the binary entropy per site, minus the field term \(\beta \sum_i h_i m_i\).

Order one — Week 5, recovered. The first derivative is the average of the exchange energy in the factorized reference, where \(\langle s_i s_j \rangle = m_i m_j\):

\[ G_1 \;=\; \Big\langle -\!\sum_{(ij)} J_{ij}\, s_i s_j \Big\rangle_0 \;=\; -\sum_{(ij)} J_{ij}\, m_i m_j . \]

At \(\lambda = 1\), \(G_0 + \lambda G_1\) is, symbol for symbol, Week 5’s mean-field free energy. This reframes the whole of last week: naive mean field is the first-order truncation of this expansion. It is the first rung of a ladder (Figure 11.2) that continues.

Order two — the Onsager term, derived. The second derivative of \(G\) in \(\lambda\) is (minus \(\beta\) times) a variance: the fluctuation of the exchange energy about its mean, in the constrained reference. A few lines of moment algebra do the evaluation. The fluctuating object is \(\sum_{(ij)} J_{ij}(s_i s_j - m_i m_j)\), and with \(\delta s_i = s_i - m_i\) each bond splits as

\[ s_i s_j - m_i m_j \;=\; m_j\, \delta s_i + m_i\, \delta s_j + \delta s_i\, \delta s_j . \]

The linear pieces correlate bonds that share a spin: \(\langle (s_i s_j - m_i m_j)(s_i s_k - m_i m_k) \rangle_0 = m_j m_k (1 - m_i^2) \neq 0\), so the square of the sum is not yet a sum of independent bond variances. The constraint removes them. The Lagrange fields \(b_i\) are readjusted at each order so that \(\langle \delta s_i \rangle\) stays zero, and this projection absorbs every term linear in a single-spin fluctuation, cross-bond and same-bond alike. What remains per bond is \(\delta s_i\, \delta s_j\), uncorrelated across distinct bonds, with variance equal to the full single-bond variance \(1 - m_i^2 m_j^2\) minus the two linear pieces \(m_j^2(1 - m_i^2)\) and \(m_i^2(1 - m_j^2)\):

\[ \big(1 - m_i^2 m_j^2\big) - m_j^2\big(1 - m_i^2\big) - m_i^2\big(1 - m_j^2\big) \;=\; \big(1 - m_i^2\big)\big(1 - m_j^2\big) \]

per bond: the product of the two ends’ single-site variances — that is, of their susceptibilities, the very objects that carried the echo. The second-order term is therefore \(G_2 = -\tfrac{\beta}{2} \sum_{(ij)} J_{ij}^2 (1-m_i^2)(1-m_j^2)\), where the \(\tfrac12\) is the Taylor coefficient of \(\lambda^2\), and truncating the ladder here gives the TAP free energy:

\[ \boxed{\; F_\text{TAP}(m) \;=\; \underbrace{-\sum_{(ij)} J_{ij}\, m_i m_j - \sum_i h_i m_i - T \sum_i s(m_i)}_{F_\text{naive}(m)\ \text{(Week 5)}} \;-\; \frac{\beta}{2} \sum_{(ij)} J_{ij}^2 \big(1 - m_i^2\big)\big(1 - m_j^2\big) \;} \tag{11.2}\]

TAP free energy

Stationarity gives the equations. Differentiate in \(m_i\): the naive part gives Week 5’s terms (\(-h_i^\text{eff}\) from the energy, \(T\operatorname{artanh}(m_i)\) from the entropy), and the new term contributes \(+\beta\, m_i \sum_j J_{ij}^2 (1 - m_j^2)\). Setting the sum to zero and inverting:

\[ \boxed{\; m_i \;=\; \tanh\!\Big( \beta \Big[ \sum_j J_{ij}\, m_j + h_i \Big] \;-\; \beta^2\, m_i \sum_j J_{ij}^2 \big(1 - m_j^2\big) \Big) \;} \tag{11.3}\]

TAP equations

— the TAP equations (Thouless, Anderson, and Palmer 1977): Week 5’s self-consistency with one new term inside the \(\tanh\), and that term is precisely the echo of Equation 11.1. The two derivations serve different purposes: the physical route says why, the expansion says what (second order in the coupling) and what next (the third and higher rungs are generated mechanically by the algorithm of Georges and Yedidia (1991)).

Figure 11.2: Mean field as a truncation. The constrained free energy expands in the interaction strength as \(\beta G = \beta G_0 + \lambda\,\beta G_1 + \lambda^2 \beta G_2 + \cdots\): order zero is the entropy of pinned free spins, order one is Week 5’s mean-field theory, and order two is the Onsager reaction term, whose per-bond weight \((1-m_i^2)(1-m_j^2)\) is the product of the two ends’ susceptibilities. Truncating after \(G_1\) gives W5·L1, after \(G_2\) TAP; the ladder continues mechanically (Georges and Yedidia 1991).

11.4 When the reaction term matters, and its limitations

The scaling. The reaction term is controlled by \(\sum_j J_{ij}^2\), whose value across coupling ensembles is summarized in Figure 11.3:

Uniform ferromagnet, \(J_{ij} = J/N\): \(\sum_j J_{ij}^2 = J^2 (N-1)/N^2 \to 0\). The echo vanishes in the thermodynamic limit, which accounts for Week 5’s finding that naive mean field is exact on this model: the couplings are individually so weak that no spin meaningfully polarizes any neighbor.
SK disorder, \(J_{ij} \sim \mathcal{N}(0, J^2/N)\): \(\sum_j J_{ij}^2 \to J^2\) by the law of large numbers — order one, at every temperature. Individual couplings are strong enough (\(\sim 1/\sqrt{N}\)) to polarize neighbors, and the echoes, though random in sign as fields, add their squares. For disordered couplings the reaction term is not a refinement; it is the difference between a wrong theory and a usable one — which is why SK’s “solvable” model needed TAP. On the SK model the term takes a famously compact form: \(\sum_j J_{ij}^2 (1 - m_j^2) \to J^2 (1 - q)\) with \(q = \frac{1}{N}\sum_j m_j^2\) — the Edwards–Anderson overlap, Week 2’s \(q\).
Sparse graph, degree \(z\), bonds of fixed strength \(J\): \(\sum_j J_{ij}^2 = zJ^2\) — the correction matters whenever individual bonds stay strong, however few.

The same physics appeared in Week 2, without its name. The AGS retrieval theory’s noise-amplification factor \(r = (1-\chi)^{-2}\) in Equation 4.5 — the term that moved the capacity from \(2/\pi\) to \(0.138\) — is the network responding to its own errors: flipped bits polarize the fields that flipped them. That was a reaction effect, quoted in Week 2 and derived here. Hebb couplings at extensive load are of the disordered, \(\mathcal{O}(1)\) kind, and so are learned couplings generally: learning writes disordered couplings.

Show code

rng = np.random.default_rng(6)
Ns = np.unique(np.logspace(0.9, 3.3, 12).astype(int))
fig, ax = plt.subplots(figsize=(8.2, 4.2))
for ens, col, lab in [("ferro", NAVY, r"ferromagnet $J_{ij}=J/N$:  $\to J^2/N$"),
                      ("sk", ORANGE, r"SK $J_{ij}\sim\mathcal{N}(0,J^2/N)$:  $\to J^2$")]:
    means = []
    for N in Ns:
        vals = []
        for _ in range(20):
            if ens == "ferro":
                J = (np.ones((N, N)) - np.eye(N)) / N
            else:
                J = rng.normal(0, 1 / np.sqrt(N), (N, N))
                J = (J + J.T) / np.sqrt(2)
                np.fill_diagonal(J, 0)
            vals.append((J**2).sum(axis=1).mean())
        means.append((np.mean(vals), np.min(vals), np.max(vals)))
    m, lo, hi = map(np.array, zip(*means))
    ax.loglog(Ns, m, "o-", color=col, lw=2, ms=5, label=lab)
    ax.fill_between(Ns, lo, hi, color=col, alpha=0.2)
ax.axhline(1.0, color=GRAY, ls=":", lw=1)
ax.text(11, 1.45, r"$J^2$ — the echo is $\mathcal{O}(1)$: TAP required", color=ORANGE, fontsize=10.5)
ax.text(150, 0.05, r"echo $\to 0$: naive mean field exact",
        color=NAVY, fontsize=10.5, ha="left")
ax.set_xlabel(r"system size $N$")
ax.set_ylabel(r"echo strength  $\overline{\sum_j J_{ij}^2}$")
ax.legend(frameon=False, fontsize=10, loc="lower left")
fig.tight_layout(); plt.show()

Figure 11.3: The scaling, sampled: the echo’s driver \(\frac{1}{N}\sum_i \sum_j J_{ij}^2\) against system size for the two coupling ensembles (20 instances per size; bands span the instances). For the uniform ferromagnet \(J_{ij} = J/N\) (navy) the sum decays as \(J^2/N\) — the reaction term vanishes in the thermodynamic limit, which is why naive mean field is exact on this model. For SK disorder \(J_{ij} \sim \mathcal{N}(0, J^2/N)\) (orange) it converges to \(J^2\): order one at every size and every temperature. Learned, data-built couplings are of the disordered kind.

The bound is lost. Week 5’s one-sided guarantee does not survive the correction. \(F_q \geq F\) held because \(F_q\) was the exact variational functional evaluated on a restricted family. \(F_\text{TAP}\) is a truncated expansion — the exact functional of no family at all — and the guarantee does not survive truncation: \(F_\text{TAP}\) is generally much closer to the truth and can sit on either side of it (in the example below it sits below the exact \(F\) at every temperature shown). This is a real loss, and it recurs across the course’s methods: contrastive divergence traded unbiasedness for tractability (Week 3); TAP trades the bound for accuracy. (The loss is not inevitable: any exact variational functional on a richer tractable family — structured mean field, for instance, which keeps chains or trees of spins intact — retains the bound. The next lecture’s Bethe free energy restores the variational character at the pairwise level, its stationary points being the fixed points of belief propagation, but off trees it is not a bound either.)

Many solutions, and trouble iterating. A further limitation is physical. At high temperature the TAP equations have one solution; lower the temperature into the glass region of a disordered model and they develop many — exponentially many for SK — each a metastable valley of the corrected free-energy landscape, the many states of Mézard, Parisi, and Virasoro (1987). This is Week 2’s ruggedness surfacing inside a deterministic approximation: the TAP solutions are the stationary points of the truncated \(F_\text{TAP}(m)\), and their multiplicity is the two ferromagnetic wells of last week, multiplied by disorder into a landscape. No inference reading is implied: SK observes nothing, and \(F_\text{TAP}\), being the functional of no family, is neither a variational free energy nor a negative ELBO. Where TAP replaces naive mean field on a clamped system, with visibles observed and hidden units inferred, the same multiplicity means several candidate sets of posterior magnetizations. Practical corollary: naive fixed-point iteration of Equation 11.3 can oscillate or diverge exactly where the physics is interesting (our example uses damping), and the iteration whose error can be tracked exactly, step by step, arrives in the next lecture under an algorithm’s name.

The ML echo. The same machinery runs inference in learning systems directly: Kappen and Rodríguez (1998) showed that a Boltzmann machine’s intractable statistics — Week 3’s negative phase \(\langle s_i s_j \rangle_\text{model}\) — can be estimated deterministically by solving the TAP equations and reading correlations from linear response, replacing the sampling chains of contrastive divergence with a corrected mean-field computation. The correction term remains a working component of inverse-Ising and Boltzmann-machine inference.

Trap

“Second order” does not mean small — and \(F_\text{TAP}\) is not a bound. ① For disordered couplings the second-order term is \(\mathcal{O}(1)\) — the same size as the first — at every temperature; order in the expansion is position on a ladder, not magnitude, and the magnitude is set by \(\sum_j J_{ij}^2 (1 - m_j^2)\), not by the order. ② Truncating the ladder destroys Week 5’s one-sidedness: \(F_\text{TAP}\) can dip below the true \(F\) and promises no error sign. A bound and an estimate are different objects, and inside the glass phase, where there are exponentially many TAP solutions, no single one is the solution.

Take-home 2

The size of the reaction term is set by \(\sum_j J_{ij}^2\): it vanishes for the uniform ferromagnet (\(J^2/N\)) and is of order one for SK-type disorder, where it becomes \(J^2(1-q)\) with the Edwards–Anderson overlap \(q\).

11.5 Example: one term on a disordered instance

We take one disordered instance small enough to enumerate, so that every claim can be checked directly: \(N = 10\) spins with SK couplings \(J_{ij} \sim \mathcal{N}(0, J^2/N)\), \(J = 1\), and small random fields (\(h_i \sim \mathcal{N}(0, 0.2^2)\), to break the global flip symmetry); \(2^{10} = 1024\) states gives the exact \(F\) and \(\langle s_i \rangle\) by brute force. Solve Week 5’s naive equations and today’s TAP equations by damped iteration from the same start, and compare, with the same protocol as Week 5’s example:

\(T\)	RMS error, naive MF	RMS error, TAP	\(F_\text{exact}\)	\(F_\text{naive}\)	\(F_\text{TAP}\)
\(2.0\)	\(0.051\)	\(0.009\)	\(-15.06\)	\(-14.01\)	\(-15.15\)
\(1.5\)	\(0.227\)	\(0.027\)	\(-11.96\)	\(-10.69\)	\(-12.10\)
\(1.2\)	\(0.419\)	\(0.064\)	\(-10.21\)	\(-9.01\)	\(-10.43\)
\(0.8\)	\(0.460\)	\(0.225\)	\(-8.19\)	\(-7.49\)	\(-8.65\)

In the error columns, at \(T = 1.5\) the single reaction term cuts the magnetization error eightfold (\(0.227 \to 0.027\)); at \(T = 1.2\), sixfold. A single term makes the theory quantitatively useful on a model whose \(\sum_j J_{ij}^2\) is of order one. The free-energy columns show the loss of the bound directly: \(F_\text{naive}\) sits above the truth at every temperature (Week 5’s bound, holding faithfully — off by \(1.3\) at \(T = 1.5\)), while \(F_\text{TAP}\) sits far closer (\(0.15\) at \(T = 1.5\)) but below the truth. The last row shows the remaining limitation: at \(T = 0.8\), below the glass temperature \(T = J = 1\) of the SK model in the thermodynamic limit, TAP’s own error grows to \(0.23\) — the territory of many solutions and higher-order physics, where no truncation of the ladder is safe and the next lecture’s algorithmic ideas become necessary. Figure 11.4 draws the \(T = 1.5\) magnetizations and the free-energy curves. For contrast, on the uniform ferromagnet (\(J_{ij} = J/N\), same \(N\)) at \(T = 1.5\), the naive and TAP magnetizations agree to \(0.01\), a gap of order \(1/N\) that shrinks with size.

Show code

def make_sk(N, J, seed):
    rng = np.random.default_rng(seed)
    Jm = rng.normal(0, J / np.sqrt(N), (N, N))
    Jm = (Jm + Jm.T) / np.sqrt(2)
    np.fill_diagonal(Jm, 0.0)
    h = rng.normal(0, 0.2, N)
    return Jm, h

def exact(Jm, h, T):
    N = len(h)
    S = np.array([[(c >> i & 1) * 2 - 1 for i in range(N)] for c in range(2**N)], dtype=float)
    E = -0.5 * np.einsum("ci,ij,cj->c", S, Jm, S) - S @ h
    w = np.exp(-(E - E.min()) / T)
    p = w / w.sum()
    return -T * np.log(w.sum()) + E.min(), (p[:, None] * S).sum(axis=0)

def solve(Jm, h, T, tap, damp, iters=20000, tol=1e-12):
    m = np.full(len(h), 0.01)
    J2 = Jm**2
    for _ in range(iters):
        field = Jm @ m + h
        if tap:
            field = field - (1 / T) * m * (J2 @ (1 - m**2))   # the reaction term
        m_new = np.tanh(field / T)
        if np.abs(m_new - m).max() < tol:
            return m_new
        m = (1 - damp) * m + damp * m_new
    return m

def free_energies(Jm, h, T, m, tap):
    S_ent = np.sum(-(1 + m) / 2 * np.log((1 + m) / 2) - (1 - m) / 2 * np.log((1 - m) / 2))
    F = -0.5 * m @ Jm @ m - h @ m - T * S_ent
    if tap:
        F -= (1 / (2 * T)) * np.sum(np.triu(Jm**2, 1) * np.outer(1 - m**2, 1 - m**2))
    return F

N = 10
Jm, h = make_sk(N, 1.0, seed=8)
fig, (a1, a2) = plt.subplots(1, 2, figsize=(10.4, 4.1))

T0 = 1.5
_, mex = exact(Jm, h, T0)
mmf = solve(Jm, h, T0, tap=False, damp=0.5)
mtap = solve(Jm, h, T0, tap=True, damp=0.3)
a1.plot([-1, 1], [-1, 1], color=GRAY, ls="--", lw=1)
a1.plot(mex, mmf, "o", color=NAVY, ms=8, label=rf"naive MF (RMS {np.sqrt(np.mean((mmf-mex)**2)):.3f})")
a1.plot(mex, mtap, "s", color=ORANGE, ms=7, label=rf"TAP (RMS {np.sqrt(np.mean((mtap-mex)**2)):.3f})")
a1.set_xlabel(r"exact $\langle s_i \rangle$"); a1.set_ylabel("estimate")
a1.set_title(rf"magnetizations at $T = {T0}$", fontsize=12)
a1.legend(frameon=False, fontsize=10, loc="upper left")
a1.set_xlim(-1.05, 1.05); a1.set_ylim(-1.05, 1.05)

Ts = np.linspace(0.8, 2.5, 60)
Fex, Fmf, Ftap = [], [], []
for T in Ts:
    F, _ = exact(Jm, h, T)
    m1 = solve(Jm, h, T, tap=False, damp=0.5)
    m2 = solve(Jm, h, T, tap=True, damp=0.3)
    Fex.append(F)
    Fmf.append(free_energies(Jm, h, T, m1, tap=False))
    Ftap.append(free_energies(Jm, h, T, m2, tap=True))
a2.plot(Ts, Fex, color=GRAY, lw=2.6, label=r"exact $F$")
a2.plot(Ts, Fmf, color=NAVY, ls="--", lw=2.0, label=r"$F_\mathrm{naive}$ (bound: always above)")
a2.plot(Ts, Ftap, color=ORANGE, lw=2.0, label=r"$F_\mathrm{TAP}$ (closer — no side promised)")
a2.set_xlabel(r"temperature $T$"); a2.set_ylabel("free energy")
a2.set_title("free energies against temperature", fontsize=12)
a2.legend(frameon=False, fontsize=9.5, loc="upper right")
fig.tight_layout(); plt.show()

Figure 11.4: The correction at work, on an enumerable SK instance (\(N = 10\), \(J_{ij} \sim \mathcal{N}(0, 1/N)\), small random fields; exact values from all \(1024\) states). Left: per-spin magnetizations at \(T = 1.5\), estimate against truth — naive mean field (navy) scatters off the diagonal (RMS error \(0.227\)); adding the one reaction term (TAP, orange) collapses the points onto it (RMS \(0.027\)). Right: free energies across temperature. \(F_\mathrm{naive}\) (navy dashed) sits above the exact \(F\) everywhere — Week 5’s variational bound, intact. \(F_\mathrm{TAP}\) (orange) hugs the truth an order of magnitude closer — and from *below*: the truncated expansion has no definite side of the truth. Inside the glass phase (left edge) all truncations degrade — the many-states territory.

A spin must not respond to its own echo: subtracting it is the first correction to mean field, and on disordered couplings it is of order one.

The cavity field derived above is the object that the belief-propagation chapter of Mézard and Montanari (2009) passes along the edges of a graph, and the next lecture takes it onto a sparse graph.

11.6 Outlook: from cavity to messages

The variational road now has two stations: the bound (Week 5 — exact functional, restricted family, guaranteed sign) and the correction (today — truncated expansion, better numbers, no sign), with the ladder above them mechanical (Georges and Yedidia 1991). The next lecture adds a third: put the cavity field on a sparse graph, where “compute \(j\)’s magnetization with \(i\) removed” becomes a quantity passed along an edge, and the self-consistency becomes belief propagation — exact on trees, equipped with its own free energy (Bethe’s, in the role Gibbs–Bogoliubov played), and the ancestor of algorithms that decode modern error-correcting codes. The convergence trouble flagged today has an algorithmic resolution there.

One connection to Week 2 remains open. When W2·L2 quoted the AGS phase diagram, it deferred the machinery behind quenched averages of \(\log Z\) to Module 2 and named the replica method as the technology. The replica route and the cavity route are the two great roads through disordered systems; they give the same answers where both apply. This course’s acquisition is the cavity route — Onsager’s echo, TAP, and from the next lecture message passing — because it is the road that became algorithms: belief propagation, approximate message passing, the inference engines of modern high-dimensional statistics. The replica method remains, in this course, a named landmark: you now know what it computes (the quenched average W2 could not take), what went wrong in its naive form (SK’s negative entropy, \(S(0) = -1/(2\pi) \approx -0.16\) per spin in the replica-symmetric solution — the impossibility that provoked TAP), and what fixed it (Parisi’s replica symmetry breaking (Parisi 1979), the many-states structure reflected in TAP’s multiple solutions, recognized by the 2021 Nobel Prize; Nishimori (2001) carries the full story, and Mézard, Parisi, and Virasoro (1987) is its book). The AGS diagram stays a quoted result — but the physics inside its equations, feedback and reaction and self-consistency, has now been derived within Module 2’s framework.