Appendix C — Reparameterization Techniques

A reparameterization replaces one mathematical form of a parameter with another that is equivalent for inference but better behaved for HMC, easier to place priors on, or more directly interpretable in cognitive terms. This appendix collects every reparameterization used across the book, organized by the type of constraint the original parameter imposes.

The key intuition throughout: HMC works in unconstrained space. Stan handles the Jacobian of any declared constraint (<lower=0>, simplex, etc.) automatically, but the efficiency of the sampler depends on the shape of the posterior in that unconstrained space. A good reparameterization flattens and regularizes that shape.

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C.1 Part I — Mapping Bounded Parameters to Unbounded Space

C.1.1 1. Probabilities [0, 1] — the Logit Transform

First introduced: Chapter 5 (Chapter 4, logit bias model)

A probability \(\theta \in [0,1]\) cannot use a Normal prior directly — Normal has infinite support, and the boundaries create hard walls that distort HMC trajectories. The logit transform maps \([0,1]\) to \((-\infty, +\infty)\):

\[\text{logit}(\theta) = \log\!\left(\frac{\theta}{1-\theta}\right), \qquad \text{logit}^{-1}(x) = \frac{1}{1+e^{-x}} \equiv \texttt{inv\_logit}(x)\]

Stan pattern:

parameters {
  real theta_logit;           // unconstrained — Normal prior works cleanly
}
transformed parameters {
  real<lower=0,upper=1> theta = inv_logit(theta_logit);
}
model {
  theta_logit ~ normal(0, 1.5);   // ≈ uniform on probability scale
}

Choosing the prior width on the logit scale:

normal(0, σ) on logit scale	Implied prior on probability scale
σ = 0.5	Concentrated near 0.5; rarely below 0.2 or above 0.8
σ = 1.0	Moderately diffuse
σ = 1.5	Approximately uniform on [0,1] — the default for uninformative use
σ = 3.0	Heavy mass near 0 and 1; implies extreme determinism

Cognitive uses: choice bias, probability-matching weight, mixing proportion, learning rate, any rate parameter bounded by [0,1].

Stan shortcut: bernoulli_logit_lpmf(y | theta_logit) accepts the log-odds directly, so the inv_logit transform in transformed parameters is only needed when you want to report \(\theta\) on the probability scale in generated quantities.

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C.1.2 2. Positive-Only Parameters — the Log Transform

First introduced: Chapter 12 (Chapter 11, GCM sensitivity \(c\) and decay \(\lambda\))

A parameter that must be strictly positive (\(\theta > 0\)) — sensitivity, scale, precision, rate — can be declared as real<lower=0>, but this creates a boundary at zero that HMC must negotiate. An unconstrained real log_theta with a Normal prior is geometrically smoother.

\[\log\theta \in (-\infty, +\infty), \qquad \theta = e^{\log\theta} > 0\]

Stan pattern:

parameters {
  real log_c;          // log sensitivity — unconstrained
  real log_lambda;     // log decay rate — unconstrained
}
transformed parameters {
  real<lower=0> c      = exp(log_c);
  real<lower=0> lambda = exp(log_lambda);
}
model {
  log_c      ~ normal(0, 1);    // prior on log scale
  log_lambda ~ normal(-1, 1);   // informative: moderate decay expected
}

Interpreting the log-scale prior:

Prior on log θ	Implied range for θ	Cognitive interpretation
normal(0, 1)	≈ [0.05, 20] (2 SD)	Broad uninformative prior for a scale parameter
normal(log(2), 0.5)	Centred at 2, roughly [0.7, 5.5]	Informative: moderate sensitivity expected
normal(-1, 0.5)	Centred at 0.37, roughly [0.15, 0.9]	Slow decay / weak forgetting

Cognitive uses: GCM sensitivity \(c\), exponential decay rate \(\lambda\), RL learning rate \(\alpha\) (if modeled as positive but not bounded above 1), noise precision, inverse temperature \(\beta\).

TipLog transform vs. <lower=0> declaration

Declaring real<lower=0> sigma in parameters applies an exponential-map transform internally, so it is mathematically equivalent to declaring real log_sigma and computing sigma = exp(log_sigma). The difference is:

• real<lower=0> sigma → Stan chooses the unconstrained transform; you place the prior directly on \(\sigma\).
• real log_sigma → you control the transform and place the prior on \(\log\sigma\).

The second form is preferred when you have informative prior beliefs expressible on the log scale or when the gradient geometry near zero is poor.

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C.1.3 3. Parameters Bounded in [0, 1] with a Structural Interpretation — the Beta Reparameterization

Used in: Chapter 5 (Chapter 4, forgetting rate), Chapter 11 (Chapter 10, evidence weight \(\rho\), allocation \(p\))

Sometimes a [0,1] parameter is best understood as a mean with an associated uncertainty rather than as a log-odds. The Beta distribution has two equivalent parameterizations:

Parameterization	Parameters	Interpretation
Standard	\(\alpha, \beta > 0\)	Shape parameters — not directly interpretable
Mean + concentration	\(\mu \in (0,1)\), \(\kappa > 0\)	Mean \(= \mu\); concentration \(= \kappa\); variance \(= \mu(1-\mu)/(\kappa+1)\)

\[\alpha = \mu\kappa, \qquad \beta = (1-\mu)\kappa\]

// Mean+concentration parameterization — more interpretable priors
parameters {
  real<lower=0,upper=1> mu;     // expected value
  real<lower=0> kappa;          // concentration (larger = tighter around mu)
}
transformed parameters {
  real<lower=0> alpha = mu * kappa;
  real<lower=0> beta  = (1 - mu) * kappa;
}
model {
  mu    ~ beta(2, 2);          // soft peak at 0.5; any value plausible
  kappa ~ exponential(0.1);    // most agents moderately uncertain
  y     ~ beta(alpha, beta);
}

When to prefer mean+concentration over logit: When the cognitive claim is about the mean allocation (e.g., “this agent allocates \(\rho\) of its weight to direct evidence”), and you want to reason separately about average tendency (\(\mu\)) and consistency (\(\kappa\)).

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C.1.4 4. Weight Vectors Summing to One — Simplex Reparameterizations

Used in: Chapter 12 (Chapter 11, GCM attention weights \(\mathbf{w}\))

A \(K\)-dimensional simplex (\(w_k \geq 0\), \(\sum_k w_k = 1\)) can be parameterized in several ways. For \(K = 2\) (two features), logit-normal NCP is always preferred (see Section C.3.1). For \(K > 2\), the choice is less clear-cut.

C.1.4.1 4a. Dirichlet directly (simple, but problematic for hierarchical models)

parameters { simplex[K] w; }
model      { w ~ dirichlet(alpha); }  // alpha: K-vector of concentration hyperparameters

Works well for single-subject models where concentration alpha is fixed. Avoid in hierarchical models: the Dirichlet + hyperprior-on-concentration geometry creates funnels identical to the NCP problem (Ch. 11 case study).

C.1.4.2 4b. Stick-breaking (unconstrained representation, K−1 free parameters)

Decompose a \(K\)-simplex into \(K-1\) unconstrained values using the sequential stick-breaking transform. Stan implements this automatically for simplex declarations — you never need to code it manually. But understanding it helps with custom implementations:

\[v_k = \text{logit}(w_k^* + \epsilon_k), \quad w_k = w_k^* \cdot \prod_{j<k}(1 - w_j^*), \quad w_K = 1 - \sum_{k<K} w_k\]

For \(K = 2\): reduces exactly to the logit transform on \(w_1\), with \(w_2 = 1 - w_1\).

C.1.4.3 4c. Logit-normal NCP for \(K = 2\) (recommended for hierarchical models)

parameters {
  real          pop_logit_w_mean;
  real<lower=0> pop_logit_w_sd;
  vector[J]     z_w;                        // standard-normal offsets
}
transformed parameters {
  for (j in 1:J) {
    real lw  = pop_logit_w_mean + pop_logit_w_sd * z_w[j];
    w[j][1]  = inv_logit(lw);
    w[j][2]  = 1.0 - w[j][1];
  }
}
model { z_w ~ std_normal(); }

This was the fix for the 4 000 max_treedepth warnings in the multilevel decay GCM (Chapter 11). The centered Dirichlet + kappa ~ exponential(0.1) produced a funnel; the logit-normal NCP eliminated it entirely.

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C.1.5 5. K-Way Choice Probabilities — the Softmax Transform

Mentioned in: Chapter 10 (Chapter 9), Chapter 12 (Chapter 11)

When an agent chooses among \(K > 2\) options and the evidence for each option is a real-valued score \(v_k\), the softmax maps scores to probabilities:

\[p_k = \frac{e^{v_k / \tau}}{\sum_{j=1}^K e^{v_j / \tau}}\]

where \(\tau > 0\) is an inverse temperature (higher \(\tau\) = more random; lower = more deterministic). For \(K = 2\), softmax reduces to the logistic function.

Stan:

// v is a vector of K utility/evidence values; tau is inverse temperature
vector[K] p = softmax(v / tau);
target += categorical_lpmf(choice | p);

// Or equivalently, in log space (more numerically stable):
target += categorical_logit_lpmf(choice | v / tau);

TipSampling tau vs. sampling beta = 1/tau

Convention varies across the literature. Stan’s categorical_logit_lpmf takes the log-odds vector directly, so it is natural to sample \(\beta = 1/\tau\) (inverse temperature) on the log scale:

real log_beta;                        // log inverse-temperature
// ...
target += categorical_logit_lpmf(choice | exp(log_beta) * v);

This places a Normal prior on \(\log\beta\), keeping sampling unconstrained.

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C.2 Part II — Priors on Scale Parameters

Scale parameters (\(\sigma\), \(\tau\), \(\kappa\)) must be positive and appear in hierarchical models as between-subject standard deviations. The prior on these parameters strongly affects sampling geometry near zero.

C.2.1 6. Exponential, Half-Normal, and Half-Cauchy

Used in: Chapter 7 (Chapter 6, sigma_theta ~ exponential(lambda))

Prior	Stan syntax	Shape	Recommendation
Exponential(\(\lambda\))	sigma ~ exponential(lambda)	Monotone decreasing from 0; most mass near 0	Default in this book; works well when small \(\sigma\) is plausible
Half-Normal(\(\sigma_0\))	sigma ~ normal(0, sigma_0) T[0,]	Bell-shaped; puts more mass away from 0	Better when you expect moderate between-subject variance
Half-Cauchy(\(s\))	sigma ~ cauchy(0, s) T[0,]	Heavy tails	Use when outlier subjects are plausible

The book’s choice (exponential throughout) reflects a regularizing stance: most cognitive-parameter hierarchies are relatively homogeneous, so placing more mass near \(\sigma = 0\) acts as a mild shrinkage prior that prevents the posterior from drifting to implausibly large between-subject variances.

Parameterizing the exponential: exponential(lambda) has mean \(1/\lambda\). Setting lambda = 1 means the prior expects \(\sigma \approx 1\) on the log-odds scale. For parameters whose population mean is on a different scale, adjust accordingly.

// Exponential prior for between-subject SD on the logit scale
sigma ~ exponential(1);    // mean = 1 log-odds unit ≈ ±25 percentage points

// Half-Normal alternative — more permissive
sigma ~ normal(0, 0.5) T[0,];  // 95% prior mass below ~1 log-odds unit

NoteWhy not <lower=0> real sigma + normal(0, σ₀)?

Stan accepts sigma ~ normal(0, 0.5) even when sigma is declared <lower=0>. This is a truncated normal — Stan evaluates the density only on \([0, \infty)\). For HMC, declaring the constraint <lower=0> is important; the prior statement handles the shape. The explicit T[0,] truncation notation is only needed inside stan_code blocks where Stan cannot infer the constraint from the declaration.

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C.3 Part III — Structural Reparameterizations

C.3.1 7. Non-Centered Parameterization (NCP)

Covered in detail in Section B.8 (Appendix B)

The NCP separates the location (\(\mu\)) and scale (\(\sigma\)) of a hierarchical parameter from the individual-level deviations (\(z_j \sim \mathcal{N}(0,1)\)).

// Centered: theta_j ~ normal(mu, sigma)  ← funnel when sigma → 0
// Non-Centered: theta_j = mu + sigma * z_j, z_j ~ std_normal()  ← always Gaussian

The NCP is the single most impactful reparameterization for hierarchical cognitive models. See Section B.8 for full details, the K-simplex extension, and the Chapter 11 case study.

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C.3.2 8. Probit Link — Signal Detection Theory

The probit link uses the standard Normal CDF \(\Phi\) in place of the logistic function:

\[p = \Phi(d'), \qquad d' = \Phi^{-1}(p) \equiv \text{probit}(p)\]

Stan function: Phi(x) (Normal CDF), Phi_approx(x) (fast rational approximation).

The probit link is the natural choice whenever the cognitive model is derived from signal detection theory (SDT): the sensitivity parameter \(d'\) is directly the standardized distance between signal and noise distributions, and the criterion \(c\) is the decision threshold in the same units.

parameters {
  real d_prime;     // sensitivity (positive = above chance)
  real criterion;   // decision criterion
}
model {
  d_prime   ~ normal(0, 2);
  criterion ~ normal(0, 1);
  for (i in 1:N) {
    if (signal[i] == 1)
      target += log(Phi( d_prime/2 - criterion));   // hit
    else
      target += log(Phi(-d_prime/2 - criterion));   // correct rejection
  }
}

Logit vs. probit in practice: The two links are very similar numerically (probit ≈ logit × 0.607), and prior sensitivity analyses rarely distinguish them. Choose probit when the theoretical framework is SDT and interpretability in \(d'\) units matters; choose logit for all other binary-outcome models because bernoulli_logit_lpmf is faster and numerically more stable.

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C.3.3 9. Correlation Matrices — LKJ and Cholesky

Covered in Section B.9 (Appendix B); introduced in Chapter 7 (Chapter 6)

The LKJ distribution is the natural prior for correlation matrices. Its single parameter \(\eta\) controls how much mass is placed near the identity matrix:

\(\eta\)	Implied prior on correlations
\(\eta = 1\)	Uniform over all valid correlation matrices
\(\eta = 2\)	Mild regularization toward identity (off-diagonal correlations shrunk toward 0)
\(\eta \to \infty\)	All mass on the identity (zero correlations)

Always sample the Cholesky factor cholesky_factor_corr[K] L_Omega and apply the lkj_corr_cholesky prior. Recover \(\Omega = L_\Omega L_\Omega^\top\) in generated quantities. See Section B.9 for full code and a cost comparison.

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C.4 Part IV — Numerical Stability

C.4.1 10. Log-Space Arithmetic

Used throughout: log_mix, log_sum_exp, log1m in Chapters 8–11

Computing probabilities directly (multiplying small numbers together, then taking logs) leads to underflow for long trial sequences. Staying in log space throughout avoids this.

Operation	Unstable form	Stable Stan function
\(\log(p_1 + p_2)\)	log(exp(lp1) + exp(lp2))	log_sum_exp(lp1, lp2)
\(\log(\pi p_1 + (1-\pi) p_2)\)	log(pi*exp(lp1) + (1-pi)*exp(lp2))	log_mix(pi, lp1, lp2)
\(\log(1 + e^x)\)	log(1 + exp(x))	log1p_exp(x)
\(\log(1 - e^x)\) (requires \(x < 0\))	log(1 - exp(x))	log1m_exp(x)
\(\log(1 - p)\)	log(1 - p)	log1m(p)
\(\log \Phi(x)\) (Normal CDF)	log(Phi(x))	log(Phi_approx(x)) or normal_lcdf(x \| 0,1)

// Mixture likelihood in log space — numerically stable
for (i in 1:N) {
  real lp1 = bernoulli_logit_lpmf(y[i] | alpha);
  real lp2 = bernoulli_logit_lpmf(y[i] | nu);
  target  += log_mix(pi, lp1, lp2);   // never calls exp() on small values
}

// Accumulated log-probability for a trajectory
real log_p = 0.0;
for (t in 1:T)
  log_p += log_mix(pi[t], lp_A[t], lp_B[t]);
target += log_p;

WarningWhen log_sum_exp is not available

For vectors: log_sum_exp(v) sums all elements of v in log-space. For accumulation inside a loop: maintain a running log_p and add each term. Never compute sum(exp(log_probs)) and then take the log — this is the numerically unstable version that Stan’s log-space functions exist to replace.

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C.5 Quick Reference

Parameter type	Constraint	Recommended reparameterization	Chapter
Probability / rate	[0, 1]	Logit → real theta_logit, inv_logit in TP	Ch 4
Positive scale / rate	> 0	Log → real log_theta, exp in TP	Ch 11
[0,1] with mean+variance story	[0, 1]	Beta mean+concentration (μ, κ)	Ch 10
Two-feature attention weight	simplex[2]	Logit-normal NCP	Ch 11
K-feature attention weight	simplex[K]	Stan simplex (auto stick-break); NCP for hierarchical	Ch 11
K-way choice probability	probability K-vector	Softmax of unconstrained scores / categorical_logit	Ch 9
Individual params drawn from population	real	Non-centered: \(z \sim \mathcal{N}(0,1)\), \(\theta = \mu + \sigma z\)	Ch 6
Between-subject standard deviation	> 0	exponential(1) or normal(0, 0.5) T[0,]	Ch 6
Correlation matrix	positive-definite, diag = 1	Cholesky factor + LKJ prior	Ch 6
SDT sensitivity	unconstrained	Probit: Phi(d_prime)	—
Sum of log-probabilities	\((-\infty, 0]\)	log_sum_exp, log_mix, log1p_exp	Ch 8–11