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Notebook 5 — Which strategies matter?

SWC ENC 2026 · behaviour module

The AUC in Notebook 4 told us how well the model predicts, but not which of the five strategies is doing the work. To find out, we ablate: remove a strategy, refit, and see how much the model's predictions suffer. The strategies whose removal hurts most are the ones the mouse is really using.

We'll measure "how much it hurts" with the held-out log-likelihood: the likelihood you built in Notebook 3, scored on held-out folds using the cross-validation from Notebook 4 — a fair, overfitting-resistant currency. (The paper uses a more elaborate quantity, model evidence; that's the optional advanced section at the end, for anyone who wants the full picture.)

In this notebook you will:

1. Score a model by its cross-validated log-likelihood.
2. Ablate each strategy and read off how much it contributes.
3. Turn those contributions into the paper's strategy index (Fig 2D–F).
4. (Optional, advanced) Redo the ablation with model evidence.

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The paper panels this notebook reproduces

Fig. 2B — the fitted strategy weights, averaged across sessions.

Fig. 2D — how much the model's evidence drops when each strategy is removed (ablation). This notebook builds that ablation and the strategy indices it feeds.

import numpy as np
import matplotlib.pyplot as plt
import swcbehav as sb

def get_Xy(sess):
    # Observable design matrix, per-image outcome, and column names.
    X, cols = sb.build_design_matrix(sess.table)
    y = sess.table["bout_start"].to_numpy().astype(float)
    return X, y, cols

1. Scoring a model by the probability it assigns

We need a way to rank models by how well they predict licking. A natural score comes straight from the likelihood we built in Notebook 3. On a single image the model assigns probability $p_t$ to a bout and $1-p_t$ to no bout, so the probability it placed on what actually happened is

$$\text{(prob of the observed outcome)} = \begin{cases} p_t & \text{if a bout started} \\ 1 - p_t & \text{if not.} \end{cases}$$

A good model puts high probability on what really happened. Take the log of that number on each image and average over images, and you get the log-likelihood per image — our score. It is always negative (the log of a probability below 1), and higher (closer to zero) means the model was, on average, less surprised by the data. Crucially we compute it on held-out images — using the cross-validation from Notebook 4 — so a model can't score well just by memorizing the images it was fit on.

A note on units. Because we took the natural log, the score comes out in units called nats; with $\log_2$ it would be in bits. The unit is immaterial here — we only ever compare these numbers to one another. A gap of, say, $0.1$ nats/image means one model assigned about $e^{0.1}\approx 1.1\times$ more probability, per image, to the held-out licking than the other.

Why not just reuse the AUC? The AUC from Notebook 4 only asks whether bouts are ranked above quiet images; it ignores the actual probability values. The log-likelihood cares about those values, so it rewards a model whose probabilities are calibrated — when it says "30% chance of a bout," a bout really does happen about 30% of the time. A model that orders images well but is over-confident (saying 99% when the truth is 60%) earns a fine AUC but a poor log-likelihood. That sensitivity to the actual probabilities is exactly what we want when asking whether adding a strategy genuinely buys the model anything.

Exercise 1 (~8 min). Build the score yourself, from pieces you already have.

1. mean_log_likelihood(w, X, y): the average over images of the log-probability the model put on what happened. With $z_t = x_t \cdot w$ and $p_t = \sigma(z_t)$, that is $\log p_t$ on bout images and $\log(1-p_t)$ on quiet ones. The compact form you met in Notebook 3, $y_t z_t - \log(1 + e^{z_t})$, covers both cases at once; np.logaddexp(0, z) computes $\log(1 + e^{z})$ without overflow.
2. cross_val_loglik(X, y, n_folds): the same contiguous-fold loop as Notebook 4's cross_val_auc, but fit on the training folds with sb.fit_static and score the held-out fold with your mean_log_likelihood instead of the AUC. Return the mean over folds.

Check / unstuck. At $w = 0$ every image gets $p = 0.5$, so the mean log-likelihood is $\log 0.5 \approx -0.693$. For the visual mouse the cross-validated score should be about −0.217 nats/image, a shade below the in-sample −0.215. Stuck? sb.mean_log_likelihood and sb.cross_val_loglik.

def mean_log_likelihood(w, X, y):
    z = X @ w
    return np.mean(y * z - np.logaddexp(0.0, z))

def cross_val_loglik(X, y, n_folds=5):
    n = len(y)
    folds = np.array_split(np.arange(n), n_folds)
    lls = []
    for i in range(n_folds):
        test = folds[i]
        train = np.setdiff1d(np.arange(n), test)
        w = sb.fit_static(X[train], y[train])
        lls.append(mean_log_likelihood(w, X[test], y[test]))
    return np.mean(lls)

sess = sb.make_mouse("visual", seed=0)
X, y, cols = get_Xy(sess)

w0 = np.zeros(X.shape[1])
print(f"sanity check at w = 0: {mean_log_likelihood(w0, X, y):.4f}   (log 0.5 = {np.log(0.5):.4f})")
w_full = sb.fit_static(X, y)
print(f"in-sample log-likelihood       : {mean_log_likelihood(w_full, X, y):.4f} nats/image")
full_ll = cross_val_loglik(X, y)
print(f"cross-validated log-likelihood : {full_ll:.4f} nats/image   (backend: {sb.cross_val_loglik(X, y):.4f})")

sanity check at w = 0: -0.6931   (log 0.5 = -0.6931)
in-sample log-likelihood       : -0.2153 nats/image
cross-validated log-likelihood : -0.2171 nats/image   (backend: -0.2171)

2. Ablation: does a strategy earn its place?

To ask what a strategy contributes, we compare the full model against a smaller one that is missing that strategy — its column deleted from the design matrix. That smaller model is a special case of the full one: it's exactly what you'd get by forcing the strategy's weight to zero. When one model is a restricted version of another like this, we say it is nested inside it. Comparing nested models is the clean way to isolate one ingredient, because the two are identical except for the piece we're testing.

Since held-out log-likelihood rewards real predictive power (not mere flexibility), the logic is simple: if a strategy is genuinely used, the full model predicts held-out licking better than the nested one without it; if the strategy is idle, the two score about the same. So we measure the drop in held-out log-likelihood when a strategy is removed,

$$\Delta_{\text{strategy}} = \text{LL}_{\text{full}} - \text{LL}_{\text{without it}},$$

reading a large positive $\Delta$ as "the mouse relies on this strategy" and $\Delta \approx 0$ as "it isn't using it."

Exercise 2 (~6 min). Complete ablation_deltas: for each non-bias strategy, build the design matrix without that column and compute $\Delta$. Hint: np.delete(X, k, axis=1) removes column k.

Check / unstuck. For the visual mouse, expect visual ≈ +0.24 and the other three ≈ 0 (it reuses the cross_val_loglik you just wrote). Stuck? Use sb.ablation_loglik_deltas(X, y, cols).

def ablation_deltas(X, y, cols, n_folds=5):
    full_ll = cross_val_loglik(X, y, n_folds)
    deltas = {}
    for k, name in enumerate(cols):
        if name == "bias":
            continue
        X_without = np.delete(X, k, axis=1)
        deltas[name] = full_ll - cross_val_loglik(X_without, y, n_folds)
    return deltas

deltas = ablation_deltas(X, y, cols)
names = list(deltas)
plt.figure(figsize=(5, 3.2))
plt.bar(names, [deltas[n] for n in names], color=[sb.WEIGHT_COLORS[n] for n in names])
plt.ylabel("drop in held-out LL"); plt.title("visual mouse: strategy contributions")
plt.xticks(rotation=45, ha="right"); plt.axhline(0, color="k", lw=0.8)
plt.tight_layout(); plt.show()
for n in names:
    print(f"  {n:14s} {deltas[n]:+.4f}")

  visual         +0.2407
  omission       -0.0002
  post_omission  -0.0004
  timing         -0.0004

Exactly what we'd hope: removing visual collapses the model's predictions, while removing omission, post-omission, or timing barely registers. On a single-strategy mouse the ablation points, unambiguously, at the one strategy we built in — the clean result that makes this the right place to trust the method.

Across the population (Figure 2D)

One mouse gives one clean answer; the paper's Figure 2D shows the whole distribution across the dataset. Let's reproduce its shape over a population spanning the strategy spectrum (a coarser 3-fold CV keeps it quick, but the fits still take ~10 s — that's normal). To match the paper we plot the change in the model's held-out score when a strategy is removed (each dot a mouse, the black bar the population mean).

⚠️ Sign flip. These are the same numbers as the bar chart above, negated. The bar showed the drop (positive = important); here we show the change when removed (negative = important), to match the paper. A big drop up there is a big dip down here.

population = [sb.make_mouse(name, seed=s)
              for s in range(8) for name in ["visual", "timing", "mixed"]]
order = ["omission", "post_omission", "visual", "timing"]
dist = {s: [] for s in order}
for sess in population:
    X, y, cols = get_Xy(sess)
    d = sb.ablation_loglik_deltas(X, y, cols, n_folds=3)
    for s in order:
        dist[s].append(d[s])

plt.figure(figsize=(5.6, 4))
jit = np.random.default_rng(0)
for i, s in enumerate(order):
    change = -np.array(dist[s])        # change when removed = -(drop): negative if the strategy helps
    plt.scatter(i + jit.uniform(-0.15, 0.15, len(change)), change, s=25, alpha=0.6,
                color=sb.WEIGHT_COLORS[s])
    plt.hlines(change.mean(), i - 0.25, i + 0.25, color="k", lw=2)
plt.axhline(0, color="0.7", lw=0.8)
plt.xticks(range(len(order)), order, rotation=45, ha="right")
plt.ylabel("change in held-out LL when removed")
plt.title("strategy contributions across the population (cf. Fig 2D)")
plt.tight_layout(); plt.show()

Visual and timing plunge well below zero, with a wide spread — deep because mice rely on them, spread because which of the two dominates depends on the individual mouse (a timing mouse's visual change is near zero, and vice versa). Omission and post-omission stay at zero. Removing a strategy the mouse actually uses makes the model worse — exactly the negative excursions of Figure 2D.

3. The strategy index

Piet et al. summarize each session with two numbers — how much the model leans on the visual vs the timing strategy — and their difference, the strategy index:

$$\text{strategy index} = \Delta_{\text{visual}} - \Delta_{\text{timing}}.$$

Positive = visual-dominant, negative = timing-dominant. Let's compute it across a population of single-strategy mice and reproduce the layout of Figure 2F.

def strategy_index(sess, n_folds=5):
    X, y, cols = get_Xy(sess)
    d = ablation_deltas(X, y, cols, n_folds)
    return d["visual"], d["timing"], d["visual"] - d["timing"]

plt.figure(figsize=(5.2, 4.4))
colors = {"visual": sb.WEIGHT_COLORS["visual"], "timing": sb.WEIGHT_COLORS["timing"], "mixed": "purple"}
for name in ["visual", "timing", "mixed"]:
    for seed in range(4):
        vi, ti, si = strategy_index(sb.make_mouse(name, seed=seed))
        plt.scatter(vi, ti, color=colors[name], s=40,
                    label=name if seed == 0 else None)
plt.xlabel("visual index (drop in LL)"); plt.ylabel("timing index (drop in LL)")
plt.title("strategy space (cf. Fig 2F)"); plt.legend()
plt.axline((0, 0), slope=1, color="0.7", ls="--")
plt.tight_layout(); plt.show()

Visual mice sit in the bottom-right (high visual index, low timing), timing mice in the top-left, and mixed mice near the diagonal — each strategy lands where it should. In Notebook 6 we'll push this on mice that genuinely blend strategies.

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4. (Advanced, optional) Model evidence

Everything below is optional. If you're short on time, skip straight to §4.3 for the result — the ablation verdict comes out the same as above.

Ablation by held-out log-likelihood is intuitive and reuses the cross-validation you already know. The paper instead compares models by their evidence, also called the marginal likelihood: the probability the model assigns to the data once the weights have been averaged over, rather than fixed at a single best value,

$$p(y \mid \text{model}) = \int p(y \mid w)\, p(w)\, dw.$$

Here $p(w)$ is the prior over the weights, and the integral sweeps over every possible weight vector, weighting each by how well it explains the data. Averaging instead of fitting one best $w$ builds in an automatic penalty for spare parameters: a model with an extra strategy must spread its prior over a larger weight-space, so unless that strategy genuinely helps, the average probability of the data goes down. That is a built-in Occam's razor, and it needs no separate cross-validation.

4.1 What the Laplace approximation is (optional deep-dive)

That integral has no closed form for logistic regression, so we approximate it. The Laplace approximation is a general trick for integrals of the form $\int e^{g(w)}\,dw$ whose integrand has a single sharp peak: nearly all the area sits under the peak, so we replace $g$ by its second-order Taylor expansion there, which turns $e^{g}$ into a plain Gaussian bump we can integrate exactly.

In one dimension: let $w^{*}$ be the location of the peak, where $g'(w^{*}) = 0$. Expanding to second order (the first-order term vanishes at the peak),

$$g(w) \approx g(w^{*}) - \tfrac{1}{2}\,|g''(w^{*})|\,(w - w^{*})^2,$$

so $e^{g(w)}$ is approximately $e^{g(w^{*})}$ times a Gaussian in $w$. Using the standard Gaussian integral $\int e^{-\frac12 a (w-\mu)^2}\,dw = \sqrt{2\pi/a}$,

$$\int e^{g(w)}\,dw \;\approx\; e^{g(w^{*})}\,\sqrt{\frac{2\pi}{|g''(w^{*})|}} \;=\; \underbrace{e^{g(w^{*})}}_{\text{peak height}} \times \underbrace{\sqrt{2\pi/|g''(w^{*})|}}_{\text{peak width}}.$$

The width comes from the curvature $|g''(w^{*})|$: a sharper peak (large curvature) is narrower and encloses less area. In many dimensions the same formula holds with $|g''|$ replaced by the determinant of the Hessian (the matrix of second derivatives) — which is exactly what sb.log_evidence_laplace computes.

4.2 Try it on a coin (optional exercise)

Let's watch the approximation work on a case we can solve exactly. Flip a coin $n$ times, observe $k$ heads, and put a flat prior on the head-probability $p$. The evidence is

$$\int_0^1 p^{\,k}(1-p)^{\,n-k}\,dp \;=\; \frac{k!\,(n-k)!}{(n+1)!}$$

(a standard Beta integral — the exact value is on the right). Now approximate the same integral with Laplace, taking $g(p) = k\log p + (n-k)\log(1-p)$, whose peak sits at $p^{*} = k/n$.

Exercise (optional) (~5 min). Fill in the Laplace estimate: the curvature is $|g''(p^{*})| = k/p^{*2} + (n-k)/(1-p^{*})^2$, and the estimate is $e^{g(p^{*})}\sqrt{2\pi/|g''(p^{*})|}$.

Check / unstuck. The Laplace/exact ratio should approach 1 as $n$ grows (≈ 1.07, 1.01, 1.00). Fully optional — skip to §4.3 if short on time.

from math import factorial, log, sqrt, pi, exp

def coin_evidence_exact(k, n):
    return factorial(k) * factorial(n - k) / factorial(n + 1)

def coin_evidence_laplace(k, n):
    p = k / n                                       # peak of g
    g_peak = k * log(p) + (n - k) * log(1 - p)      # log integrand at the peak
    curv = k / p**2 + (n - k) / (1 - p)**2          # |g''(p*)|
    return exp(g_peak) * sqrt(2 * pi / curv)

for k, n in [(3, 10), (30, 100), (300, 1000)]:
    exact, approx = coin_evidence_exact(k, n), coin_evidence_laplace(k, n)
    print(f"k={k:4d} n={n:5d}:  exact={exact:.3e}  laplace={approx:.3e}  ratio={approx/exact:.3f}")

k=   3 n=   10:  exact=7.576e-04  laplace=8.077e-04  ratio=1.066
k=  30 n=  100:  exact=3.371e-28  laplace=3.394e-28  ratio=1.007
k= 300 n= 1000:  exact=1.840e-267  laplace=1.842e-267  ratio=1.001

The ratio approaches $1$ as $n$ grows: more data makes the peak sharper and more Gaussian, so Laplace gets better. That's the same regime our 4800-image sessions live in.

4.3 Ablation by evidence

sb.log_evidence_laplace applies this exact idea to the logistic model — many dimensions, so a Hessian determinant stands in for $|g''|$ — combining the fit at the peak, the prior, and the peak width. Redo the ablation with it:

def evidence_deltas(X, y, cols, prior_var=100.0):
    full_ev = sb.log_evidence_laplace(X, y, prior_var)
    out = {}
    for k, name in enumerate(cols):
        if name == "bias":
            continue
        out[name] = full_ev - sb.log_evidence_laplace(np.delete(X, k, axis=1), y, prior_var)
    return out

ev = evidence_deltas(X, y, cols)
print("change in log-evidence when each strategy is removed:")
for n in names:
    print(f"  {n:14s} evidence {ev[n]:+9.1f}   held-out-LL {deltas[n]:+.4f}")

change in log-evidence when each strategy is removed:
  visual         evidence    +556.8   held-out-LL +0.2407
  omission       evidence      -3.4   held-out-LL -0.0002
  post_omission  evidence      -3.5   held-out-LL -0.0004
  timing         evidence    +242.0   held-out-LL -0.0004

Both methods crown visual by a wide margin and rank the rest near zero. The evidence differences are large in absolute terms (they sum log-probabilities over all 4800 images rather than averaging), which is why the paper reports them as a percent change in evidence (Fig 2D). The takeaway is the same: on these mice, one strategy carries the model.

Wrap-up

You measured what each strategy contributes by ablation, and distilled it into the strategy index that organizes the paper's Figure 2 — verified on mice whose answer we already knew.

Next (Notebook 6): stress-test the whole fit → evaluate → ablate cycle on mice that truly mix strategies, where the index has to earn its keep.

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