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Learning Objectives

By the end of this section, you will be able to:

📚 Core Knowledge

• Explain why flat priors are NOT truly "non-informative"
• Define the Jeffreys prior using Fisher Information
• State the Jeffreys prior for common distributions
• Distinguish between proper and improper priors

🔧 Practical Skills

• Derive Jeffreys prior from Fisher Information
• Apply non-informative priors appropriately
• Recognize when improper priors yield proper posteriors
• Implement Jeffreys prior models in Python

🧠 Deep Learning Connections

• Weight Initialization: Xavier/He initialization has connections to "non-informative" principles
• Regularization ↔ Prior: L2 = Gaussian prior, L1 = Laplace prior, with Jeffreys prior on the regularization strength
• Automatic Relevance Determination: Hierarchical priors with log-uniform (Jeffreys) hyperpriors
• Natural Gradient: Uses Fisher Information - the same foundation as Jeffreys prior

Where You'll Apply This: Objective Bayesian analysis, meta-analysis combining studies without strong prior information, calibration of regularization hyperparameters, and any scenario where you want inference to be minimally influenced by subjective choices.

The Big Picture: The Quest for Objectivity

One of the most persistent criticisms of Bayesian statistics has been: "How do you choose the prior? Isn't it subjective?" While incorporating prior knowledge is often a feature, not a bug, there are legitimate scenarios where we genuinely lack prior information and want the data to dominate our conclusions.

The Central Question

Can we construct a prior that encodes "minimal prior information" in a principled, mathematically justified way?

Non-informative priors (also called "objective priors," "reference priors," or "vague priors") attempt to answer this question. The goal is to let the likelihood dominate the posterior, minimizing the impact of the prior on inference.

Historical Context

📜

Laplace (1812)

Pierre-Simon Laplace proposed the "Principle of Insufficient Reason": when there's no reason to prefer one value over another, assign equal probability. This led to the uniform (flat) prior.

🔬

Harold Jeffreys (1946)

British geophysicist Harold Jeffreys recognized that uniform priors fail to be invariant under reparameterization. He proposed using $\pi(\theta) \propto \sqrt{I(\theta)}$ where $I(\theta)$ is Fisher Information. This became the Jeffreys prior.

📊

Berger & Bernardo (1992)

Developed reference priors that maximize expected information gain from prior to posterior. These extend Jeffreys' ideas to handle multiparameter problems more elegantly, where the original Jeffreys prior can be problematic.

The Problem with Flat Priors

The most intuitive "non-informative" choice seems to be a flat (uniform) prior: assign equal probability to all parameter values. But this approach has a fundamental flaw.

The Parameterization Problem

Consider estimating the probability $p$ of a coin landing heads:

Parameterized by $p$ :

Flat prior: $\pi(p) = 1$ for $p \in (0, 1)$

This says all probabilities are equally likely.

Parameterized by odds $\theta = \frac{p}{1-p}$ :

Flat prior on odds: $\pi(\theta) = 1$

This implies a DIFFERENT prior on p!

The contradiction: A flat prior on p is NOT flat when transformed to odds (and vice versa). "Uniform" depends on your parameterization choice!

Mathematically, if $\theta = g(p)$ , and we place a flat prior on $p$ , the implied prior on $\theta$ is:

\pi(\theta) = \pi(p) \cdot \left|\frac{dp}{d\theta}\right| = \left|\frac{dp}{d\theta}\right|

This is NOT uniform unless $g$ is linear. The Jacobian term distorts the prior when we change parameterizations.

Interactive: The Parameterization Problem

See how a "uniform" prior transforms under reparameterization. Watch how the Jeffreys prior maintains its form while the flat prior changes shape.

Parameterization Invariance

Why Jeffreys priors are special: they give consistent results regardless of how you parameterize your model

Probability p = 0.500 → Odds = 1.000 → Logit = 0.000

Flat/Uniform Prior

In p-space: π(p) = 1 (constant)

In odds-space: π(odds) = 1/(1+odds)²

The shape CHANGES when you reparameterize! A "uniform" prior on p becomes non-uniform on odds.

Jeffreys Prior

In p-space: π_J(p) ∝ 1/√(p(1-p))

In odds-space: π_J(odds) ∝ 1/√(odds(1+odds))

The prior CORRECTLY transforms via the Jacobian! Same functional form derived either directly or by transformation.

Why This Matters

The Problem: When using a flat prior, your "non-informative" choice actually encodes information about which parameterization you happened to use. Scientists working with the same model but different parameterizations would get different answers!

The Solution: The Jeffreys prior is constructed using Fisher Information, which transforms correctly under reparameterization. If θ = g(φ), then:

I_φ(φ) = I_θ(g(φ)) × |g'(φ)|²

This ensures that √I(φ) picks up exactly the right Jacobian factor, making Jeffreys prior invariant under one-to-one transformations.

Practical Implications for ML

•When optimizing in log-space vs. linear space (common in neural networks), your "uniform" initialization has different meanings in each space
•Jeffreys-like initialization schemes can help ensure consistent behavior regardless of parameterization choices
•In variational inference, the choice of parameterization affects the implicit prior induced by your variational family

Jeffreys Prior: The Principled Solution

Harold Jeffreys recognized that a truly "non-informative" prior should give the same inference regardless of how we parameterize the problem. He proposed using the Fisher Information to construct such a prior.

The Jeffreys Prior

\pi_J(\theta) \propto \sqrt{I(\theta)}

where $I(\theta)$ is the Fisher Information

Component	Definition	Intuition
Fisher Information I(θ)	-E[∂²log f(X\|θ)/∂θ²]	Curvature of log-likelihood; measures information in data about θ
Jeffreys Prior π_J(θ)	∝ √I(θ)	Places more weight where data is more informative
Multivariate case	∝ √det(I(θ))	Use the square root of the Fisher Information matrix determinant

The Key Insight: Fisher Information transforms as

I_\phi(\phi) = I_\theta(g(\phi)) \cdot |g'(\phi)|^2

under reparameterization

\theta = g(\phi)

. Taking the square root, the Jeffreys prior picks up exactly the Jacobian factor needed to remain invariant.

Interactive: Deriving the Jeffreys Prior

Explore how the Jeffreys prior is derived from Fisher Information for different distributions. Watch how $\pi_J(\theta) \propto \sqrt{I(\theta)}$ varies with the parameter value.

Jeffreys Prior from Fisher Information

The Jeffreys prior is proportional to the square root of Fisher Information: π_J(θ) ∝ √I(θ)

Select Distribution:

Parameter value: 0.500

Fisher Information at θ = 0.500

I(θ) = 4.0000

Jeffreys Prior at θ = 0.500

π_J(θ) ∝ 0.6366

Fisher Information Formula:

I(p) = \frac{1}{p(1-p)}

Jeffreys Prior:

\pi_J(p) = \frac{1}{\pi \sqrt{p(1-p)}} \propto \text{Beta}(1/2, 1/2)

Interpretation: For Bernoulli, Jeffreys prior is Beta(1/2, 1/2) - the arcsine distribution. This gives more weight to extreme values (near 0 and 1).

Jeffreys Priors for Common Distributions

Let's derive the Jeffreys prior for several important distributions. These results are used constantly in Bayesian analysis.

Bernoulli/Binomial (p)

Fisher Info: $I(p) = \frac{n}{p(1-p)}$

Jeffreys: $\pi_J(p) \propto \frac{1}{\sqrt{p(1-p)}} = \text{Beta}(1/2, 1/2)$

The arcsine distribution. A proper prior that integrates to 1.

Normal (μ unknown, σ² known)

Fisher Info: $I(\mu) = \frac{1}{\sigma^2}$ (constant)

Jeffreys: $\pi_J(\mu) \propto 1$ (flat)

Improper (integrates to ∞), but matches intuition: no location is preferred.

Normal (σ² unknown, μ known)

Fisher Info: $I(\sigma^2) = \frac{1}{2\sigma^4}$

Jeffreys: $\pi_J(\sigma^2) \propto \frac{1}{\sigma^2}$

Log-uniform: uniform on log(σ²). Treats all orders of magnitude equally.

Poisson (λ)

Fisher Info: $I(\lambda) = \frac{1}{\lambda}$

Jeffreys: $\pi_J(\lambda) \propto \frac{1}{\sqrt{\lambda}}$

Improper but yields proper posteriors with at least one observation.

Reference: Complete Gallery

Click on any distribution to see detailed information about its Jeffreys prior and applications in machine learning.

Jeffreys Priors for Common Distributions

Click on any row to see detailed notes and ML applications

Distribution	Parameter	Fisher Info I(θ)	Jeffreys Prior π_J(θ)	Proper?
Bernoulli / Binomial	p ∈ (0, 1)	n / [p(1-p)]	Beta(1/2, 1/2)	Yes
Poisson	λ > 0	1 / λ	π(λ) ∝ λ^(-1/2)	No
Exponential	λ > 0 (rate)	1 / λ²	π(λ) ∝ 1/λ	No
Normal (μ unknown, σ² known)	μ ∈ ℝ	1 / σ² (constant)	π(μ) ∝ 1 (flat)	No
Normal (σ² unknown, μ known)	σ² > 0	1 / (2σ⁴)	π(σ²) ∝ 1/σ²	No
Normal (μ and σ² unknown)	(μ, σ²) ∈ ℝ × ℝ⁺	diag(1/σ², 1/(2σ⁴))	π(μ, σ²) ∝ 1/σ³	No
Gamma	(α, β) shape and rate	Complex expression involving ψ'(α)	π(α, β) ∝ √(ψ'(α) - 1/α) / β	No
Beta	(α, β) > 0	Complex matrix with trigamma functions	π(α, β) ∝ √(ψ'(α) + ψ'(β) - ψ'(α+β)) × ...	No
Multinomial / Categorical	p = (p₁, ..., pₖ) on simplex	diag(n/pᵢ)	Dirichlet(1/2, ..., 1/2)	Yes
Uniform(0, θ)	θ > 0 (upper bound)	n / θ²	π(θ) ∝ 1/θ	No

Proper prior (integrates to finite value)

Improper prior (but yields proper posteriors with data)

Improper Priors: When Infinity is OK

Many Jeffreys priors are improper: they don't integrate to a finite value. For example, $\pi(\mu) \propto 1$ over $\mu \in (-\infty, +\infty)$ integrates to infinity.

Improper priors are NOT probability distributions! They don't satisfy the axioms of probability. However, they can still be useful if they yield proper posteriors.

When is an improper prior acceptable?

The posterior is proper: After combining with the likelihood, the posterior integrates to a finite value and is a valid probability distribution.
Inference makes sense: Point estimates (mean, mode), credible intervals, and predictions are all well-defined and finite.
Not used for model comparison: Improper priors cause Bayes factors to be undefined or infinite. Use only for inference within a single model.

Rule of thumb: For common exponential family distributions, the Jeffreys prior yields a proper posterior as long as you have at least one observation. The data provides enough "regularization" to make the posterior finite.

Reference Priors: Beyond Jeffreys

While the Jeffreys prior works well for single-parameter models, it can have issues in multiparameter settings. Reference priors (Berger & Bernardo, 1992) address these problems.

Reference Prior Philosophy

Choose the prior that maximizes the expected Kullback-Leibler divergencefrom prior to posterior. This means the prior is designed to be maximally updated by data.

\pi^{ref}(\theta) = \arg\max_\pi \mathbb{E}_X\left[D_{KL}(\pi(\theta|X) \| \pi(\theta))\right]

For single-parameter models, the reference prior often equals the Jeffreys prior. For multiparameter models with a "parameter of interest," reference priors give more sensible results than the joint Jeffreys prior.

Interactive: Comparing Non-Informative Priors

Compare how different "non-informative" priors lead to different posteriors. With small samples, the choice matters significantly; with large samples, they converge.

Comparing Non-Informative Priors

See how different "non-informative" priors lead to different posteriors

Select Priors to Compare:

Number of observations: 10

Number of successes: 5

Flat/Uniform

Prior: π(θ) ∝ 1

Posterior: Beta(6.00, 6.00)

Mean: 0.5000

Mode: 0.5000

Jeffreys

Prior: π(θ) ∝ √I(θ)

Posterior: Beta(5.50, 5.50)

Mean: 0.5000

Mode: 0.5000

Key Insight

With small samples, the choice of "non-informative" prior matters significantly! As n → ∞, all priors converge to the same posterior (dominated by likelihood). This is why the prior choice is most critical when data is scarce.

Try setting n=5 vs n=100 to see this convergence effect.

Real-World Examples

AI/ML Applications

Non-informative priors and their principles underpin many modern ML techniques. Understanding these connections deepens your grasp of regularization, initialization, and Bayesian deep learning.

⚖️ Regularization as Prior

L2 regularization = Gaussian prior on weights. The regularization strength $\lambda$ corresponds to prior precision $1/(2\sigma^2)$ . Jeffreys prior on $\sigma^2$ gives automatic relevance determination.

🎲 Weight Initialization

Xavier initialization samples weights to preserve gradient magnitude. This can be viewed as choosing a "non-informative" initialization that doesn't favor any particular scale of activations.

📈 Natural Gradient Descent

Natural gradient uses the Fisher Information matrix - the same foundation as Jeffreys prior! It accounts for the geometry of parameter space, making optimization invariant to reparameterization.

🔍 Empirical Bayes

Estimate hyperparameters from data rather than specifying them. Uses marginal likelihood, which implicitly integrates over a non-informative prior on parameters of interest.

Interactive: Priors and Regularization

See the direct connection between Gaussian priors on weights and L2 regularization. Adjust the regularization strength to see how it corresponds to prior tightness.

Non-Informative Priors in Deep Learning

How Bayesian principles connect to regularization in neural networks

Regularization strength (λ) = 1.00 → Prior std (σ) = 1.00

Higher λ = tighter Gaussian prior = stronger regularization = smaller weights

The Equivalence

MAP Estimation with Gaussian prior N(0, σ²):

argmax[log L(w|X) + log π(w)]

Equals minimizing:

-log L(w|X) + (1/2σ²)||w||²

Where λ = 1/(2σ²) is the L2 regularization weight!

Jeffreys Prior for σ

If σ (the prior std) is itself unknown, the Jeffreys prior is:

π_J(σ) ∝ 1/σ (log-uniform)

This corresponds to placing a uniform prior on log(σ), which treats all orders of magnitude equally.

Used in automatic relevance determination (ARD) and hierarchical Bayes.

Modern ML Applications

Weight Initialization

Xavier/He initialization draws from distributions derived from "non-informative" principles to preserve gradient magnitude.

Bayesian Neural Networks

Scale-mixture priors (like Horseshoe) extend Jeffreys prior ideas for sparse weight estimation.

Hyperprior Learning

Empirical Bayes methods estimate regularization strength from data, implicitly using non-informative priors on hyperparameters.

Practical Note: In practice, truly non-informative priors are rarely used directly in neural networks. However, understanding them helps explain why regularization works and guides the choice of hyperpriors in hierarchical Bayesian models.

Python Implementation

Jeffreys Priors in Practice

🐍python

Explanation(11)

Code(142)

1Imports

Import NumPy for numerical computations and SciPy for statistical functions.

6Fisher Information Function

Computes Fisher Information I(θ) = -E[∂²log L/∂θ²] for the Bernoulli distribution. This measures how much information a single observation carries about the parameter p.

EXAMPLE

fisher_info_bernoulli(0.5) = 4.0

10Jeffreys Prior Derivation

The Jeffreys prior is proportional to √I(θ). For Bernoulli, this gives π(p) ∝ 1/√(p(1-p)), which equals the Beta(0.5, 0.5) distribution.

15Beta PDF Evaluation

Using scipy.stats.beta to compute the probability density. The Jeffreys prior Beta(0.5, 0.5) is the arcsine distribution - note how it gives more weight to extreme values.

21Posterior Computation

Conjugate update: posterior = Beta(α + successes, β + failures). Starting with Jeffreys prior Beta(0.5, 0.5), we add 7 successes and 3 failures.

EXAMPLE

Posterior = Beta(7.5, 3.5)

27Posterior Summaries

Computing key summaries: the mean (expected value), mode (peak of distribution), and 95% highest posterior density interval.

36Normal Mean Jeffreys Prior

For the mean of a Normal distribution with known variance, Fisher Information is constant (1/σ²), so the Jeffreys prior is flat: π(μ) ∝ 1.

44Normal Variance Jeffreys Prior

For variance parameter σ², Fisher Information is 1/(2σ⁴), so Jeffreys prior is π(σ²) ∝ 1/σ². This is log-uniform, treating all orders of magnitude equally.

EXAMPLE

Same prior density at σ²=0.1 and σ²=10

52Inverse Gamma Posterior

With Jeffreys prior on σ² (equivalent to InverseGamma(0, 0) limit), the posterior is InverseGamma(n/2, sum((x-x̄)²)/2).

61Parameterization Invariance Check

Demonstrating that Jeffreys prior gives consistent inference regardless of whether we parameterize by p or by odds = p/(1-p). The transformed prior has the correct form!

74Regularization Connection

Showing the fundamental connection: L2 regularization with strength λ equals placing a Gaussian prior N(0, 1/(2λ)) on weights and computing MAP. The regularization strength is the prior precision!

131 lines without explanation

1import numpy as np
2from scipy import stats
3from scipy.special import digamma, polygamma
4import matplotlib.pyplot as plt
5
6# ============================================
7# 1. Jeffreys Prior for Bernoulli/Binomial
8# ============================================
9
10def fisher_info_bernoulli(p):
11    """Fisher Information for Bernoulli: I(p) = 1/(p(1-p))"""
12    return 1 / (p * (1 - p))
13
14def jeffreys_prior_bernoulli(p):
15    """Jeffreys prior: π(p) ∝ sqrt(I(p)) = Beta(1/2, 1/2)"""
16    return stats.beta.pdf(p, 0.5, 0.5)
17
18# Example: Coin flip analysis
19p_values = np.linspace(0.01, 0.99, 100)
20jeffreys_density = [jeffreys_prior_bernoulli(p) for p in p_values]
21
22print("Jeffreys Prior for Bernoulli: Beta(0.5, 0.5)")
23print(f"Mean = {stats.beta.mean(0.5, 0.5):.3f}")
24print(f"Prior puts more weight near 0 and 1 (arcsine distribution)")
25
26# Posterior with Jeffreys prior
27def bernoulli_posterior(alpha_prior, beta_prior, successes, failures):
28    """Conjugate update: Beta(α + k, β + n-k)"""
29    return stats.beta(alpha_prior + successes, beta_prior + failures)
30
31posterior = bernoulli_posterior(0.5, 0.5, successes=7, failures=3)
32print(f"\nPosterior after 7/10 successes: Beta({posterior.args[0]}, {posterior.args[1]})")
33print(f"Posterior mean: {posterior.mean():.4f}")
34print(f"95% Credible Interval: ({posterior.ppf(0.025):.4f}, {posterior.ppf(0.975):.4f})")
35
36
37# ============================================
38# 2. Jeffreys Prior for Normal Mean (known σ²)
39# ============================================
40
41def jeffreys_prior_normal_mean():
42    """For Normal with known variance, Jeffreys prior is flat: π(μ) ∝ 1"""
43    return "π(μ) ∝ 1 (improper flat prior)"
44
45print(f"\nJeffreys Prior for Normal Mean: {jeffreys_prior_normal_mean()}")
46
47
48# ============================================
49# 3. Jeffreys Prior for Normal Variance
50# ============================================
51
52def jeffreys_prior_normal_variance(sigma2):
53    """Jeffreys prior: π(σ²) ∝ 1/σ² (log-uniform)"""
54    return 1 / sigma2 if sigma2 > 0 else np.inf
55
56# Posterior for variance (Jeffreys prior = InverseGamma(0,0) limit)
57def normal_variance_posterior(data, known_mean=None):
58    """
59    With Jeffreys prior π(σ²) ∝ 1/σ², posterior is InverseGamma.
60    If mean is unknown and estimated, use Bessel-corrected formula.
61    """
62    n = len(data)
63    if known_mean is not None:
64        ss = np.sum((data - known_mean)**2)
65        alpha = n / 2
66        beta = ss / 2
67    else:
68        x_bar = np.mean(data)
69        ss = np.sum((data - x_bar)**2)
70        alpha = (n - 1) / 2
71        beta = ss / 2
72    return stats.invgamma(alpha, scale=beta)
73
74
75# ============================================
76# 4. Parameterization Invariance Check
77# ============================================
78
79def verify_invariance():
80    """Demonstrate that Jeffreys prior is invariant under reparameterization"""
81    p_values = np.linspace(0.1, 0.9, 5)
82
83    print("\nVerifying Parameterization Invariance:")
84    print("Jeffreys prior in p-space: π(p) ∝ 1/sqrt(p(1-p))")
85    print("Transformation: θ = p/(1-p) (odds)")
86    print("\n{:>8} {:>12} {:>12} {:>12}".format("p", "π_J(p)", "odds", "π_J(odds)"))
87
88    for p in p_values:
89        odds = p / (1 - p)
90        # Jeffreys prior in p-space
91        pi_p = 1 / np.sqrt(p * (1 - p))
92        # Jeffreys prior in odds-space (derived directly)
93        pi_odds = 1 / np.sqrt(odds * (1 + odds))
94        print(f"{p:8.3f} {pi_p:12.4f} {odds:12.4f} {pi_odds:12.4f}")
95
96verify_invariance()
97
98
99# ============================================
100# 5. Connection to Regularization
101# ============================================
102
103def regularization_prior_equivalence():
104    """Show L2 regularization = Gaussian prior equivalence"""
105    print("\n" + "="*50)
106    print("L2 REGULARIZATION = GAUSSIAN PRIOR")
107    print("="*50)
108
109    # L2 loss: Loss(w) + λ||w||²
110    # MAP with N(0, σ²) prior: -log L(w) + w²/(2σ²)
111    # Equivalence: λ = 1/(2σ²), so σ = 1/sqrt(2λ)
112
113    lambdas = [0.01, 0.1, 1.0, 10.0]
114    print("\n{:>10} {:>15} {:>15}".format("λ", "Prior σ", "Prior σ²"))
115    for lam in lambdas:
116        sigma = 1 / np.sqrt(2 * lam)
117        sigma2 = 1 / (2 * lam)
118        print(f"{lam:10.2f} {sigma:15.4f} {sigma2:15.4f}")
119
120    print("\nKey insight: Larger λ = smaller σ = tighter prior = stronger regularization")
121
122regularization_prior_equivalence()
123
124
125# ============================================
126# 6. Multinomial Jeffreys Prior
127# ============================================
128
129def dirichlet_jeffreys_prior(k):
130    """
131    Jeffreys prior for Multinomial: Dirichlet(1/2, ..., 1/2)
132    k = number of categories
133    """
134    alpha = np.full(k, 0.5)
135    return stats.dirichlet(alpha)
136
137# Example: 4-category classification
138k = 4
139jeffreys_dirichlet = dirichlet_jeffreys_prior(k)
140print(f"\nJeffreys Prior for {k}-category Multinomial:")
141print(f"Dirichlet({', '.join(['1/2'] * k)})")
142print(f"Expected probability for each category: {1/k:.4f}")

Common Misconceptions

❌

"Non-informative priors contain no information"

Reality: Every prior encodes some information. "Non-informative" means the prior is chosen to minimize its influence relative to the likelihood, not that it's information-free. Even a flat prior says "all values are equally plausible."

❌

"Improper priors are wrong and should never be used"

Reality: Improper priors are perfectly acceptable if they yield proper posteriors. The posterior is what matters for inference. However, they cannot be used for Bayes factors or model comparison.

❌

"Jeffreys prior is always the best objective choice"

Reality: For multiparameter problems, the full Jeffreys prior (using det(I(θ))) can have undesirable properties. Reference priors or ordered Jeffreys priors are often better. The "best" choice depends on context.

❌

"Non-informative priors make Bayesian analysis objective"

Reality: The choice of "which non-informative prior" is itself a choice. Different non-informative priors (flat, Jeffreys, reference) give different answers. The goal is to minimize, not eliminate, subjectivity.

Knowledge Check

Test your understanding of non-informative priors and the Jeffreys prior with this comprehensive quiz.

Knowledge Check

Question 1 of 10

What is the key advantage of Jeffreys prior over a flat/uniform prior?

Current score: 0 / 0

Summary

Key Takeaways

Flat priors aren't truly non-informative: A uniform prior on θ becomes non-uniform when you transform to φ = g(θ). "Non-informative" depends on parameterization.
Jeffreys prior solves this: By defining $\pi_J(\theta) \propto \sqrt{I(\theta)}$ , the prior is invariant under one-to-one reparameterizations. Scientists using different parameterizations will reach the same conclusions.
Improper priors can be useful: Many Jeffreys priors don't integrate to a finite value, but they still yield proper posteriors and valid inference when combined with data.
Common Jeffreys priors: Bernoulli → Beta(1/2, 1/2); Normal mean → flat; Normal variance → 1/σ²; Poisson → λ^(-1/2); Multinomial → Dirichlet(1/2, ..., 1/2).
Reference priors extend these ideas: For multiparameter problems, reference priors maximize expected information gain and can handle ordered parameters of interest.
Deep learning connection: Regularization strength relates to prior precision. Jeffreys prior on the regularization hyperparameter gives automatic relevance determination. Natural gradient uses Fisher Information - the same foundation as Jeffreys prior.

Completed Chapter 17! You've now mastered the foundations of Bayesian statistics: the philosophical paradigm, prior and posterior distributions, conjugate priors, and non-informative priors. In Chapter 18, we'll dive deeper into Bayesian Inference with point estimation, MAP, credible intervals, and model comparison using Bayes factors.

Learning Objectives

📚 Core Knowledge

🔧 Practical Skills

🧠 Deep Learning Connections

The Big Picture: The Quest for Objectivity

Historical Context

Laplace (1812)

Harold Jeffreys (1946)

Berger & Bernardo (1992)

The Problem with Flat Priors

The Parameterization Problem

Interactive: The Parameterization Problem

Parameterization Invariance

Flat/Uniform Prior

Jeffreys Prior

Why This Matters

Practical Implications for ML

Jeffreys Prior: The Principled Solution

The Jeffreys Prior

Interactive: Deriving the Jeffreys Prior

Jeffreys Prior from Fisher Information

Jeffreys Priors for Common Distributions

Bernoulli/Binomial (p)

Normal (μ unknown, σ² known)

Normal (σ² unknown, μ known)

Poisson (λ)

Reference: Complete Gallery

Jeffreys Priors for Common Distributions

Improper Priors: When Infinity is OK

When is an improper prior acceptable?

Reference Priors: Beyond Jeffreys

Reference Prior Philosophy

Interactive: Comparing Non-Informative Priors

Comparing Non-Informative Priors

Key Insight

Real-World Examples

📊Meta-Analysis: Combining Studies Without Bias

⚙️Sensor Calibration: Unknown Scale and Offset

🎯The German Tank Problem: Estimating an Unknown Maximum

AI/ML Applications

⚖️ Regularization as Prior

🎲 Weight Initialization

📈 Natural Gradient Descent

🔍 Empirical Bayes

Interactive: Priors and Regularization

Non-Informative Priors in Deep Learning

The Equivalence

Jeffreys Prior for σ

Modern ML Applications

Weight Initialization

Bayesian Neural Networks

Hyperprior Learning

Python Implementation

Common Misconceptions

Knowledge Check

Knowledge Check

Summary

Key Takeaways