A Tutorial on Fisher Information

1. Introduction

• Mathematical psychologists use quantitative models to describe behavior and latent processes, while different statistical paradigms apply Fisher information to testing, priors, and model complexity.The tutorial addresses frequentist confidence intervals and hypothesis tests, Bayesian default priors, and MDL model-complexity measurement for graduate students and researchers in cognitive modeling and mathematical statistics.Source: §1 · p. 2

1.1. Notation and key concepts

• The tutorial defines statistical models through f(x_i ∣θ), relating parameters to potential outcomes, and primarily treats one-dimensional θ while referring vector-valued extensions to Appendix A.It distinguishes probability density and mass functions according to whether outcomes are continuous or discrete, while using pmf as a general shorthand.Source: §1.1 · p. 3
• For n trials, the iid assumption means observations share one θ and have a joint distribution given by the product of their individual probability functions.This setup models sequences of roughly equally difficult psychological items, such as repeated correct or incorrect responses.Source: §1.1 · pp. 3–4

1.2. Definition of Fisher information

• A statistic T is sufficient for θ when the conditional distribution of the raw data given T does not depend on θ.In the coin-flip example, after observing the number of heads Y, the conditional probability of the raw sequence is independent of θ, so no information about θ remains in the raw data beyond Y (Fisher, 1920; Stigler, 1973).Source: §1.2 · pp. 4–5
• Fisher information measures the expected sensitivity of a model’s functional relationship to changes in θ, using the score function’s sensitivity weighted by outcome probabilities.For n iid trials, information is additive: I_Xn(θ) = nI_X(θ), making the unit information I_X(θ) proportional to the number of trials.Source: §1.2 · p. 5
• In the Bernoulli model, the sufficient statistic Y retains the same Fisher information about θ as the raw data Xn for every θ, making large datasets replaceable by one summary number.The Bernoulli expected information diverges as θ approaches zero or one, while equality of I_Xn(θ) and I_Y(θ) shows that using Y loses no statistical information.Source: §1.2 · p. 6

2. The Role of Fisher Information in Frequentist Statistics

• The data-generative view defines the MLE by selecting the parameter value that makes the observed data most likely; for 7 heads in 10 trials, it is ˆθobs = 0.7.The observation has probability 26.7% under θ = 0.7 versus 11.7% under θ = 0.5.Source: §2 · p. 7, Figure 2 · p. 7

2.1. Using Fisher information to design an experiment

• For large n, Fisher information approximates the MLE’s sampling distribution, with standard error 1/√(nI_X(θ∗)); increasing n or unit information improves precision.The approximation assumes iid data and general conditions under which the Fisher information exists; finite samples incur approximation error.Source: §2.1 · pp. 7–8
• Choosing n through Fisher information can ensure that MLEs are sufficiently accurate: n = 25 yields estimates within 0.1 of θ∗ with 68% expected frequency in the Bernoulli worst case.The worst case is θ = 1/2, and among 100 simulated data sets, at least 68 estimates are expected to be within 0.1 of θ∗.Source: §2.1 · p. 8

2.2. Using Fisher information to construct a null hypothesis test

• Under a postulated θ∗ = θ0, the normal approximation produces an approximately 95% prediction interval for MLE outcomes, enabling a point-null hypothesis test.For H0: θ∗ = θ0 = 0.5 with n = 10, the interval is (0.19, 0.81), and outcomes outside it are rejected at significance level .05.Source: §2.2 · p. 9

2.3. Using Fisher information to compute confidence intervals

• Replacing the unknown true value with the observed MLE yields approximate 95% confidence intervals whose centers and lengths vary with the estimate and Fisher information.For 7 heads in 10 trials, the interval is (0.42,0.98) with length 0.57; for 6 heads, it is (0.29,0.90) with length 0.61.Source: §2.3 · p. 10

3.1. Bayesian updating

• Bayesian updating combines prior information with the likelihood from observed data to form posterior knowledge about the parameter.The prior is specified over the parameter space before observations are incorporated.Source: §3.1 · p. 11

3. The Role of Fisher Information in Bayesian Statistics

• When little is known about θ, a uniform prior appears neutral but is unsuitable as a default because it depends on how the inference problem is parameterized.Jeffreys proposed an information-based alternative for this reason.Source: §3 · p. 10, §3.2 · p. 11, §3.3 · p. 14

3.2. Failure of the uniform distribution on the parameter as a noninformative prior

• Uniform priors can produce different posterior conclusions for equivalent parameterizations: assigning uniformity to angle φ rather than propensity θ yields notably different posterior inference for θ.Before observing data, the prior probability for θ* in Jθ = (0.6,0.8) is 0.20; after observing yobs = 7 out of n = 10, the posterior probability is 0.54.Source: §3.2 · pp. 12–13

3.3. A default prior by Jeffreys’s rule

• Jeffreys’s prior, constructed from Fisher information, produces the same posterior regardless of whether the model is parameterized by θ or φ.This parameterization invariance avoids the differing conclusions produced by uniform priors on equivalent representations.Source: §3.3 · pp. 14–15

3.4. Geometrical properties of Fisher information

• The geometric treatment shows that Fisher information defines Jeffreys’s prior as uniform in model space, with normalizing constant V = π across parameterizations.It also interprets the shaded prior area geometrically.Source: §3.4 · p. 16

3.4.1. The model space M

• Representing probability mass functions by twice their square roots places the complete binary model space on the positive part of a circle, making distances geometrically meaningful.This representation reveals why a parameter-space uniform prior may be informative in model space.Source: Figure 6 · p. 16, §3.4.1 · p. 17

3.4.2. Uniform on the parameter space versus uniform on the model space

• The Bernoulli parameterizations θ and φ generate the same saturated candidate model space, M_Θ = M = M_Φ.Thus, equivalent parameterizations trace the same set of probability models even though uniform parameter priors differ.Source: §3.4.2 · p. 17
• Uniform priors on θ or φ do not produce uniform assessments of pmfs because equal parameter intervals can correspond to unequal distances in model space.A uniform prior on φ favors pmfs with φ close to zero, while the Jeffreys prior compensates for this geometric distortion by assigning more mass toward the boundaries, especially for φ.Source: §3.4.2 · p. 18
• The Jeffreys prior uses the functional relationship between parameters and data, whereas a prior specified without that relationship imposes parameter-space geometry on the model and neglects problem context.For φ, this neglect forces the geometry of Φ = (−π,π) onto the model M_Φ.Source: §3.4.2 · p. 19

3.5. Uniform prior on the model

• The Jeffreys prior is exactly the uniform prior on model space, yielding parameterization-invariant conclusions by focusing on the data-generating pmf rather than a parameter value.Fisher information converts parameter-space displacements into model-space distances, allowing a uniform model prior to be expressed in parameterizations.Source: §3.5 · pp. 19–20
• 53%: after observing 7 successes out of 10, the posterior probability that the true data-generating pmf lies in the interval J_m rises from 14% under the Jeffreys prior.The interval is J_m = (m_0.6(X), m_0.8(X)).Source: §3.5 · p. 19

4.0.1. The description length of a model

• FIA approximates MDL description length by combining a model’s goodness of fit with dimensionality and geometrical complexity, whose integral corresponds to the volume induced by Jeffreys’s prior.The model with the shortest description length is selected; its fit capacity is proportional to its volume in model space.Source: §4.0.1 · p. 22

4.1. A new running example and the geometry of a random variable with w = 3 outcomes

• The three-outcome example represents probability mass functions on a simplex and transforms them onto a spherical model space, making MDL selection interpretable through geometric distance.The outcomes are L, M, and R, and the transformed representation gives all probability vectors the same distance from the origin.Source: §4.1 · p. 23

4.2. The individual-word and the only-mixed strategy

• M1 and M2 are distinct one-dimensional, nonnested models that overlap only at one probability distribution, so AIC and BIC compare them by goodness of fit while FIA additionally exposes Fisher-information complexity.Both models define proper subsets of the full model space and have identical dimensionality terms.Source: §4.2 · p. 25
• For n = 30 trials, FIA prefers the only-mixed strategy M2 when the empirical probability distribution falls between the models’ non-decision curves, including their shared cross.The complexity penalty is most visible at the shared distribution, where observations are automatically assigned to the simpler model M2.Source: Figure 10 · p. 26

4.3.1. The maximum likelihood estimators

• Goodness of fit measures distance from observations, whereas FIA additionally penalizes model volume to prevent overfitting that can arise from labeling sampling error as structure.A more voluminous nested model is closer to every data set, while AIC and BIC penalize dimensionality and FIA also accounts for model volume.Source: §4.3.1 · p. 28

4.3.2. Geometrical complexity

• The individual-word strategy is more complex than the only-mixed strategy because its larger model volume incurs an additional penalty of 1/2 log(2).Both models have the same dimensionality term, so the difference arises from geometrical complexity.Source: §4.3.2 · p. 28

4.4. Model selection based on the minimum description length principle

• FIA selects the model closest to the data except in overlap regions, where its complexity penalty can favor the simpler M2 even when M1 generates the data.The simpler-model preference decreases as n increases; for n = 120, the middle observation is better described by M1 as decision bounds move.Source: §4.4 · p. 29
• For large n, FIA selects the model closest to the empirical pmf and therefore asymptotically closest to the true pmf, yielding the best projected predictions.This agrees with the goodness-of-fit interpretation and produces the same selection as BIC in the large-sample quadrant division.Source: §4.4 · pp. 29–30

4.5. Fisher information and generalizability

• Fisher information expands parameter-estimation error into prediction error, so larger Fisher information pushes predictions farther from the true pmf and worsens generalization.Smaller models generally have smaller Fisher information at θ* and consequently produce more stable predictions closer to the data-generating pmf.Source: §4.5 · p. 30

• Fisher information supports frequentist inference, Bayesian default-prior specification, and information-geometric or minimum-description-length measures of model complexity.The tutorial uses these three paradigms to show distinct applications of the same statistical concept.Source: §5 · p. 31
• The paradigms differ in how the likelihood relationship f(x_n∣θ) is treated: frequentist analysis fixes θ, Bayesian analysis fixes observed data, and information geometry varies both arguments.The information-geometric formulation uses f(⋅∣⋅) together with observed data and the maximum likelihood.Source: §5 · p. 31
• The treatment focuses on one-dimensional parameters, with vector-valued generalizations deferred to the appendix and a complete account explicitly beyond a tutorial article’s scope.The authors frame the article as a set of concrete examples rather than a comprehensive treatment of Fisher information.Source: §5 · p. 31
• The tutorial aims to clarify Fisher information for psychologists who propose, develop, and compare mathematical models, while noting further uses in model misspecification, estimation, decision theory, objective priors, and computational statistics.The authors conclude that Fisher information is a key concept in statistical modeling and hope the article provides an accessible account of selected applications.Source: §5 · p. 31

Appendix A: Generalization to Vector-Valued Parameters: The Fisher Information Matrix

• For a d-dimensional parameter vector, Fisher information is a d×d symmetric positive semidefinite matrix.Its entries are defined through expectations involving the score-function derivatives.Source: Appendix A · p. 39
• The score is a d×1 vector of partial derivatives, whose expected-zero components make the Fisher information matrix a covariance matrix.The derivatives are evaluated at the same parameter value used to weight outcomes through the probability mass function.Source: Appendix A · p. 39
• Under mild regularity conditions, each matrix entry can equivalently be computed from the negative expected second-order partial derivatives.The expectation or sum is taken over the outcomes of X, with integration used in the continuous case.Source: Appendix A · p. 39
• For normally distributed variables with parameters μ and σ^2, the unit Fisher information is a 2×2 symmetric positive semidefinite matrix.Its entries can be obtained either from expected score derivatives or equivalently from second-order partial derivatives.Source: Appendix A · pp. 39–40
• Off-diagonal Fisher-information entries are generally nonzero; a zero entry indicates that the corresponding parameters are orthogonal.Orthogonality is therefore a special rather than typical relationship between parameter components.Source: Appendix A · p. 40
• For iid samples of size n, the vector-parameter Fisher information satisfies I_Xn(⃗θ) = nI_X(⃗θ), so information remains additive.The sample-level matrix scales linearly with the number of trials.Source: Appendix A · p. 40

Appendix B: Frequentist Statistics based on Asymptotic Normality

• The frequentist hypothesis tests and confidence intervals rely on the maximum likelihood estimator being asymptotically normal.This asymptotic-normality property underlies their construction in the frequentist section.Source: Appendix B · p. 40

B.1. Asymptotic normality of the MLE for vector-valued parameters

• In regular parametric models, the MLE for vector-valued parameters converges in distribution to a multivariate normal distribution.Source: §B.1 · p. 40
• For large n, the MLE’s sampling error can be approximated by a normal distribution involving the inverse Fisher information matrix at the true parameter θ∗.Source: §B.1 · p. 40
• At fixed n, replacing the true sampling distribution with its asymptotic normal approximation incurs negligible error only when n is sufficiently large.What counts as sufficiently large depends on the true data-generating pmf p∗(x), which is unknown in practice.Source: §B.1 · pp. 40–41
• Approximation error can make nominal 5% hypothesis tests produce a 42% type 1 error rate and nominal 95%-confidence intervals cover the true parameter only 20% of the time.Source: §B.1 · p. 41

General comparison

• The CLT requires only finite population mean and variance, whereas MLE asymptotic normality additionally requires a known functional relationship between parameters and outcomes.The CLT’s distributional form is otherwise irrelevant, while stronger MLE results depend on additional model assumptions.Source: §B.2 · p. 41

Example B.1: Gaussian distribution

• In the Gaussian case, the MLE equals the sample mean and offers no asymptotic gain over the CLT, but normality makes the approximation exact for every finite n.With σ2 = 1 and n = 100, P(∣ˆθ − θ∗∣≤0.196) = 0.95 holds exactly.Source: §B.2 · p. 42

Example B.2: Laplace distribution

• n = 50 samples suffice for the Laplace MLE to achieve approximately 95% probability of estimates within 0.196 of θ∗ when Var(X) = 1.The MLE is the sample median, and exploiting the Laplace model lowers asymptotic variance enough to require twice as few participants as the Gaussian calculation’s n = 100.Source: §B.2 · p. 42

Example B.3: Cauchy distribution

• For the Cauchy distribution, the CLT is unusable because mean and variance are not finite, and the sample mean remains Cauchy-distributed independently of n.Thus, increasing the sample size provides no additional information through the sample mean; the sample median performs better, while the MLE is better still but lacks an explicit data formula.Source: §B.2 · pp. 42–43
• n = 25π2 ≈247 samples for the sample median and n = 200 for the MLE yield approximately 95% probability of estimates within 0.196 of θ∗ in the Cauchy example.These sample-size calculations use normal approximations and therefore hold only approximately when n is large enough.Source: §B.2 · p. 43

Hájek-LeCam convolution theorem

• Among regular estimators, the MLE attains the inverse-Fisher-information lower bound and therefore has the lowest asymptotic variance.The convolution theorem expresses asymptotic variance as Var(Δθ*) + I_X^-1(θ*), while the MLE has Var(Δθ*) = 0.Source: §B.3 · p. 43

Practical limitations

• Normal approximations can yield poorly performing confidence intervals and hypothesis tests when n is small, especially for decisions involving hypothesis rejection.The approximation is justified only when n is sufficiently large.Source: §B.3 · p. 44

Cramér-Fréchet-Rao information lower bound

• For unbiased estimators at finite n, the Cramér–Fréchet–Rao bound requires nVar(T_n) ≥ I_X^-1(θ*), showing Fisher information also matters beyond large samples.Unbiasedness means E(T_n) = θ* for every true value and fixed n.Source: §B.3 · p. 44
• The finite-sample bound is restrictive because unbiased estimators generally exclude the MLE and cannot attain the bound for multidimensional parameters.For vector-valued parameters, the bound therefore does not establish that searching for a better estimator should stop.Source: §B.3 · p. 44

Practical limitations

• The MLE’s superiority requires the data-generating relationship f to be correctly specified, so model misspecification can undermine its performance and motivate empirical specification tests.The paper illustrates this risk by contrasting normally assumed data with data actually generated from a Cauchy distribution.Source: §B.3 · p. 44

Bayesian use of the Fisher-Rao Metric: The Jeffreys’s Prior

• Jeffreys’s prior is interpreted as a uniform prior on the model M_Θ.Source: Appendix C · p. 44
• The explanation characterizes dm_θ(X) as a displacement in model space along an interval of probability mass functions.Source: Appendix C · p. 44
• The derivation connects IX(θ)dθ in parameter space to arc-length computations using integration by substitution.Source: Appendix C · p. 45

C.1. Tangent vectors

• The functional ν uniquely maps each pmf in MΘ to a parameter in Θ, allowing integration over model-space intervals to be replaced by parameter-space intervals.Distinct pmfs receive distinct parameter values, so the endpoints remain separated under this mapping.Source: §C.1 · p. 45
• Tangent vectors approximate pmf displacements by summing component arrows obtained from Taylor expansions of the parameterization at a fixed parameter value.For small parameter distance, the Taylor intercept is practically zero, so the displacement is approximated by a straight line; Fig. 12 depicts the resulting tangent arrow.Source: Figure 12 · p. 45, §C.1 · pp. 45–46
• The tangent-vector length is the root of the sum of squared component displacements, matching the square root of Fisher information multiplied by the parameter distance.For the φ example, component displacements of -0.14 and 0.17 produce a length of 0.22.Source: §C.1 · p. 47
• More generally, the root of Fisher information converts a small parameter distance at θa into a displacement in model space at mθa(X).This provides the local link between parameter-space distance and the length of curves through pmfs.Source: §C.1 · p. 47

C.2. The Fisher-Rao metric

• Using dm_θ(X) as the metric measures distance in model space and interprets V as the total length of M_Θ.Source: §C.2 · p. 47
• Choosing J_θ = M_Θ yields the normalizing constant V = ∫ I_X(θ)dθ.Source: §C.2 · p. 47
• The metric is named the Fisher-Rao metric after Calyampudi Radhakrishna Rao’s 1945 contribution [e.g., Amari et al., 1987; Atkinson and Mitchell, 1981; Burbea, 1984; Burbea and Rao, 1982, 1984; Dawid, 1977; Efron, 1975; Kass and Vos, 2011].Source: §C.2 · p. 48

C.3. Fisher-Rao metric for vector-valued parameters

• The Fisher-Rao treatment represents every w-outcome pmf by a unique β ∈ R^(w−1), making the parameter space B isomorphic to the complete pmf set P.The pmfs satisfy nonnegativity and sum-to-one constraints.Source: §C.3.1 · p. 48

C.3.1. The parameter functional ν ∶P →B and the categorical distribution

• For three categories, ν selects β1 = p(L) and β2 = p(M), with B = [0,1]×[0,β1], while pβ(X) = [β1,β2,1−β1−β2].The inverse parameterization is restricted by β1,β2 ≥ 0 and β1+β2 ≤ 1.Source: §C.3.1 · p. 48
• The three-category model generalizes Bernoulli sampling, with β1 and β2 interpreted as propensities for choosing L and M across iid trials.The outcome-count representation uses xL, xM, and xR responses.Source: §C.3.1 · p. 48
• The square-root pmf representation forms the positive part of a radius-two sphere, whose extreme points correspond to the pure outcomes L, M, and R.The extreme parameter values are β=(1,0), β=(0,1), and β=(0,0), respectively.Source: §C.3.1 · p. 48

C.3.2. The stick-breaking parameterization of the categorical distribution

• The alternative stick-breaking functional uses γ1 = pL and γ2 = pM/(1−pL), with parameter space Γ = [0,1]×[0,1].The γ coordinates provide a functional relationship for the categorical model.Source: §C.3.2 · p. 49
• The stick-breaking parameterization spans the full model, so Γ and the pmf set are isomorphic, and B and Γ are equivalent through their shared model representation despite B being a strict subset of Γ.This equivalence proceeds through β ↦ mβ(X) and the functional ˜ν.Source: §C.3.2 · p. 49

C.3.3. Multidimensional Jeffreys’s prior via the Fisher information matrix and orthogonal parameters

• Orthogonal γ parameters make the categorical model easier to describe because the Fisher information matrix has zero off-diagonal entries.The β parameterization has nonzero coupling, whereas γ parameters can be treated independently in the relevant calculations.Source: §C.3.3 · pp. 49, 51
• The γ parameterization turns the local geometry at p(X) = [1/3,1/3,1/3] into orthogonal tangent vectors spanning a rectangle, unlike the β parameterization’s diamond.Figure 13 visualizes the contrast between the two parameterizations at the corresponding points in model space.Source: §C.3.3 · p. 50, Figure 13 · p. 50
• Orthogonality decouples the multidimensional Jeffreys-prior integral, allowing the model volume to be calculated through independent parameter contributions.The paper verifies the resulting volume geometrically and notes that the construction extends beyond the three-outcome visualization under regularity conditions.Source: §C.3.3 · p. 51

D.1. Coding theory, code length and log-loss

• Coding systems translate outcomes into code words whose lengths behave like logarithms of a pmf, motivating log-loss in the MDL paradigm.The coding-theoretic setup is illustrated with a three-outcome random variable.Source: §D.1 · p. 51

D.1.1. Kraft-McMillan inequality: From code lengths of a specific coding system to a pmf

• Prefix coding permits unequal code lengths while preserving unique decipherability, and shorter strings can reduce communication load.The example contrasts a fixed 16-bit encoding with an 11-bit encoding for n = 8 responses.Source: §D.1.1 · p. 52
• Kraft and McMillan showed that uniquely decipherable coding systems satisfy an inequality, so code lengths behave like logarithms of a pmf.For D = 2, the example uses lengths 2, 1, and 2 bits, yielding 2^-2 + 2^-1 + 2^-2 = 1.Source: §D.1.1 · p. 52

D.1.2. Shannon-Fano algorithm: From a pmf to a coding system with specific code lengths

• The Shannon-Fano algorithm assigns outcome x a code length of −log2 p*(x) bits, giving the most probable outcomes the shortest codes.For p*(X) = [0.25,0.5,0.25], the resulting lengths for [L,M,R] are 2, 1, and 2 bits.Source: §D.1.2 · pp. 52–53
• Kraft-McMillan and Shannon-Fano coding establish an equivalence between pmfs and uniquely decipherable coding systems, so subsequent MDL discussion can focus on code lengths.The notation f(X ∣β) denotes a pmf viewed as a coding system, whereas pβ(X) denotes it as a data-generating device.Source: §D.1.2 · p. 53

• H(p∗(X)) is Shannon entropy under base-2 logarithms, representing average code length; for p∗(X) = [0.25,0.5,0.25], it is 1.5 bits per trial.For n = 8 trials, this requires 12 bits using the true coding system.Source: §D.1.3 · p. 53
• 2.97 bits per trial are required when f(X ∣β) = [0.01,0.18,0.81] encodes data generated by p∗(X) = [0.25,0.5,0.25], versus 1.5 bits with f(X ∣β∗).Source: §D.1.3 · pp. 53–54
• Cross entropy H(p∗(X)∥β) is the average code length under a postulated coding system and cannot be smaller than entropy: H(p∗(X)) ≤ H(p∗(X)∥β).The difference is the nonnegative Kullback-Leibler divergence, which is zero only when f(X ∣β) = p∗(X).Source: §D.1.3 · pp. 53–54
• Because p∗(X) is unknown, practice replaces it with the sample’s empirical pmf and approximates population cross entropy by a sample average for fixed β.Source: §D.1.3 · p. 54

D.2. Data compression and statistical inference

• Shortest average coding is equivalent to finding the true data-generating process, and the MLE-associated code minimizes log-loss.This follows from the entropy inequality and the equivalence between minimizing negative log-likelihood and maximizing likelihood.Source: §D.2 · p. 54
• The MLE plug-in is the model probability mass function closest to the observations, with KL-divergence providing a nonnegative measure of remaining discrepancy.A positive divergence indicates that the empirical distribution of the observations does not lie on the model.Source: §D.2 · p. 55
• The second derivative of KL-divergence with respect to β yields the Fisher-Rao metric, linking divergence geometry to Fisher information.This connection supports generalizing Jeffreys’s prior to nonparametric models.Source: §D.2 · p. 55

Appendix E: Regularity conditions

• Statistical inference seeks a probability measure within a constructed candidate set PΘ that explains both observed and unseen data, rather than treating parameters as the ultimate goal.Models are well-specified when P ∗∈PΘ and misspecified when P ∗∉PΘ.Source: Appendix E · pp. 55–56
• Normal models offer tractable inference but impose stringent symmetry and tail assumptions, creating a high risk of misspecification and making outliers practically unexpected.The appendix presents this as a limitation of using an overly idealized normal distribution.Source: Appendix E · p. 55
• A regular parametric model requires an open parameter domain, differentiability in function space, nonsingular Fisher information, and continuous variation of tangent functions.These conditions support local geometric analysis and prevent the model from collapsing to a lower dimension.Source: Appendix E · pp. 56–58
• A more readily checked regularity formulation assumes continuous differentiability in θ, finite quadratic-mean score norms, and Fisher information that is nonsingular and continuous in θ.The lemma presents these conditions as stronger but less general alternatives to Fréchet differentiability.Source: Appendix E · p. 58