Risk-Aware Utility: Log-Normal Prediction Distributions

PyApprox Tutorial Library

Deriving the distribution and expected AVaR deviation of the standard deviation of a log-normal QoI using scaled log-normal random variables.

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Learning Objectives

After completing this tutorial, you will be able to:

Show that the standard deviation of $W = \exp(Q)$ is a scaled log-normal variable
Identify the parameters of its distribution in terms of the posterior mean and covariance
Derive the expected standard deviation using the log-normal mean formula
Apply the positive homogeneity of AVaR to compute the AVaR of the standard deviation

Prerequisites

Complete Risk-Aware Utility: Gaussian Posterior Expressions before this tutorial.

Setup: Log-Normal QoI

When the natural-scale QoI is $W = \exp(Q)$ and $Q \mid \mathbf{y} \sim \mathcal{N}(\tau, \sigma^2)$ with:

\[ \tau = \boldsymbol{\psi}\boldsymbol{\mu}_\star, \quad \sigma^2 = \boldsymbol{\psi}\boldsymbol{\Sigma}_\star\boldsymbol{\psi}^\top, \]

the variable $W \mid \mathbf{y}$ follows a log-normal distribution:

\[ W \mid \mathbf{y} \sim \mathcal{LN}(\tau, \sigma^2). \]

The standard deviation of $W \mid \mathbf{y}$ has the closed form:

\[ \mathrm{Std}(W \mid \mathbf{y}) = \underbrace{\exp\!\left(\tfrac{\sigma^2}{2}\right)\sqrt{\exp(\sigma^2) - 1}}_K \cdot \exp(\tau), \]

where $K$ is a deterministic constant (depending only on $\sigma^2$, which is observation-independent).

Distribution of the Standard Deviation

From the Gaussian analysis, the posterior mean $\boldsymbol{\mu}_\star$ is Gaussian, so:

\[ \tau = \boldsymbol{\psi}\boldsymbol{\mu}_\star \sim \mathcal{N}(\nu, \sigma_\tau^2), \quad \nu = \boldsymbol{\psi}\boldsymbol{\nu}, \quad \sigma_\tau^2 = \boldsymbol{\psi}\mathbf{C}\boldsymbol{\psi}^\top. \]

Because $K > 0$ is deterministic and $\exp(\tau)$ is the exponential of a Gaussian, the standard deviation of $W \mid \mathbf{y}$ is a scaled log-normal:

\[ \mathrm{Std}(W \mid \mathbf{y}) = K \cdot \exp(\tau) \sim \mathcal{LN}(\nu, \sigma_\tau^2) \quad \text{(scaled by } K \text{)}. \]

Expected Standard Deviation

The mean of a log-normal $X \sim \mathcal{LN}(\mu_X, \sigma_X^2)$ is $\mathbb{E}[X] = \exp(\mu_X + \sigma_X^2/2)$. Applying this:

\[ \mathbb{E}\!\left[\mathrm{Std}(W \mid \mathbf{y})\right] = K \cdot \exp\!\left(\nu + \tfrac{\sigma_\tau^2}{2}\right), \]

where all quantities are analytical:

$K = \exp(\sigma^2/2)\sqrt{\exp(\sigma^2) - 1}$,
$\nu = \boldsymbol{\psi}\boldsymbol{\nu}$,
$\sigma_\tau^2 = \boldsymbol{\psi}\mathbf{C}\boldsymbol{\psi}^\top$.

AVaR of the Standard Deviation

By positive homogeneity of AVaR — $\mathrm{AVaR}[c \cdot X] = c \cdot \mathrm{AVaR}[X]$ for $c > 0$ deterministic — we have:

\[ \mathrm{AVaR}_p\!\left[\mathrm{Std}(W \mid \mathbf{y})\right] = K \cdot \mathrm{AVaR}_p[\exp(\tau)]. \]

Since $\exp(\tau)$ is log-normally distributed with parameters $(\nu, \sigma_\tau^2)$, its AVaR at level $p$ is:

\[ \mathrm{AVaR}_p[\exp(\tau)] = \frac{\exp(\nu + \sigma_\tau^2/2)\,\Phi\!\left(\sigma_\tau - \Phi^{-1}(p)\right)}{1 - p}, \]

where $\Phi$ is the standard normal CDF. Combining:

\[ \mathrm{AVaR}_p\!\left[\mathrm{Std}(W \mid \mathbf{y})\right] = K \cdot \frac{\exp(\nu + \sigma_\tau^2/2)\,\Phi\!\left(\sigma_\tau - \Phi^{-1}(p)\right)}{1 - p}. \]

Numerical Verification

import numpy as np
from scipy.stats import norm

def lognormal_avar(nu: float, sigma_tau: float, sigma_sq: float, p: float) -> float:
    """
    AVaR of Std(W | y) where W = exp(Q) and Q|y ~ N(tau, sigma_sq),
    tau ~ N(nu, sigma_tau^2).

    Parameters
    ----------
    nu       : mean of tau (= psi @ nu_vec)
    sigma_tau: std dev of tau (= sqrt(psi @ C @ psi^T))
    sigma_sq : posterior variance of Q|y (= psi @ Sigma_star @ psi^T)
    p        : AVaR level in (0, 1)
    """
    # Deterministic scale factor K
    K = np.exp(sigma_sq / 2) * np.sqrt(np.exp(sigma_sq) - 1)
    # AVaR of exp(tau) where tau ~ N(nu, sigma_tau^2)
    avar_exp_tau = (
        np.exp(nu + sigma_tau**2 / 2)
        * norm.cdf(sigma_tau - norm.ppf(p))
        / (1 - p)
    )
    return K * avar_exp_tau


def lognormal_expected_std(nu: float, sigma_tau: float, sigma_sq: float) -> float:
    """Expected value of Std(W | y)."""
    K = np.exp(sigma_sq / 2) * np.sqrt(np.exp(sigma_sq) - 1)
    return K * np.exp(nu + sigma_tau**2 / 2)

We verify against Monte Carlo:

np.random.seed(0)
nu, sigma_tau, sigma_sq, p = 0.5, 0.3, 0.2, 0.9

# Analytical
avar_analytical   = lognormal_avar(nu, sigma_tau, sigma_sq, p)
estd_analytical   = lognormal_expected_std(nu, sigma_tau, sigma_sq)

# Monte Carlo reference
N = 2_000_000
tau_samples = np.random.normal(nu, sigma_tau, N)
K = np.exp(sigma_sq / 2) * np.sqrt(np.exp(sigma_sq) - 1)
std_samples = K * np.exp(tau_samples)
avar_mc   = np.mean(std_samples[std_samples >= np.quantile(std_samples, p)])
estd_mc   = np.mean(std_samples)

print(f"E[Std(W|y)]  — analytical: {estd_analytical:.6f},  MC: {estd_mc:.6f}")
print(f"AVaR_{p}     — analytical: {avar_analytical:.6f},  MC: {avar_mc:.6f}")

E[Std(W|y)]  — analytical: 0.896833,  MC: 0.897025
AVaR_0.9     — analytical: 1.463276,  MC: 1.462718

Key Takeaways

The standard deviation of $W = \exp(Q)$ is a scaled log-normal random variable; the scale factor $K$ depends only on the (deterministic) posterior variance $\sigma^2$.
The randomness in $\mathrm{Std}(W \mid \mathbf{y})$ comes entirely from the Gaussian posterior mean $\tau = \boldsymbol{\psi}\boldsymbol{\mu}_\star$.
Expected standard deviation and AVaR both have closed forms involving the log-normal mean formula and the log-normal AVaR formula, respectively.
Positive homogeneity of AVaR allows the scale factor $K$ to be factored out, reducing the computation to the AVaR of $\exp(\tau)$.

Exercises

Verify that as $\sigma_\tau \to 0$ (i.e., $\mathbf{C} \to 0$, meaning the prior is very tight), $\mathbb{E}[\mathrm{Std}(W \mid \mathbf{y})]$ reduces to a deterministic value. What is it?
Plot $\mathrm{AVaR}_p[\mathrm{Std}(W \mid \mathbf{y})]$ as a function of $p$ for $p \in [0.5, 0.99]$. Compare to $\mathbb{E}[\mathrm{Std}(W \mid \mathbf{y})]$. At what level $p$ does AVaR exceed the mean by a factor of 2?
The AVaR formula for the log-normal distribution uses $\Phi(\sigma_\tau - \Phi^{-1}(p))$. Verify this formula by deriving it from the definition: $\mathrm{AVaR}_p[X] = \frac{1}{1-p}\int_p^1 F_X^{-1}(u)\,\mathrm{d}u$.

Next Steps

Goal-Oriented OED: Convergence Verification — using PredOEDDiagnostics to numerically validate these analytical expressions

--- title: "Risk-Aware Utility: Log-Normal Prediction Distributions" subtitle: "PyApprox Tutorial Library" description: "Deriving the distribution and expected AVaR deviation of the standard deviation of a log-normal QoI using scaled log-normal random variables." tutorial_type: analysis topic: experimental_design difficulty: advanced estimated_time: 15 render_time: 6 prerequisites: - boed_pred_gaussian_analysis tags: - experimental-design - lognormal - risk-measures - avar format: html: code-fold: false code-tools: true toc: true execute: echo: true warning: false jupyter: python3 --- ::: {.callout-tip collapse="true"} ## Download Notebook [Download as Jupyter Notebook](notebooks/boed_pred_lognormal_analysis.ipynb) ::: ## Learning Objectives After completing this tutorial, you will be able to: - Show that the standard deviation of $W = \exp(Q)$ is a scaled log-normal variable - Identify the parameters of its distribution in terms of the posterior mean and covariance - Derive the expected standard deviation using the log-normal mean formula - Apply the positive homogeneity of AVaR to compute the AVaR of the standard deviation ## Prerequisites Complete [Risk-Aware Utility: Gaussian Posterior Expressions](boed_pred_gaussian_analysis.qmd) before this tutorial. ## Setup: Log-Normal QoI When the natural-scale QoI is $W = \exp(Q)$ and $Q \mid \mathbf{y} \sim \mathcal{N}(\tau, \sigma^2)$ with: $$ \tau = \boldsymbol{\psi}\boldsymbol{\mu}_\star, \quad \sigma^2 = \boldsymbol{\psi}\boldsymbol{\Sigma}_\star\boldsymbol{\psi}^\top, $$ the variable $W \mid \mathbf{y}$ follows a log-normal distribution: $$ W \mid \mathbf{y} \sim \mathcal{LN}(\tau, \sigma^2). $$ The standard deviation of $W \mid \mathbf{y}$ has the closed form: $$ \mathrm{Std}(W \mid \mathbf{y}) = \underbrace{\exp\!\left(\tfrac{\sigma^2}{2}\right)\sqrt{\exp(\sigma^2) - 1}}_K \cdot \exp(\tau), $$ where $K$ is a **deterministic** constant (depending only on $\sigma^2$, which is observation-independent). ## Distribution of the Standard Deviation From the [Gaussian analysis](boed_pred_gaussian_analysis.qmd), the posterior mean $\boldsymbol{\mu}_\star$ is Gaussian, so: $$ \tau = \boldsymbol{\psi}\boldsymbol{\mu}_\star \sim \mathcal{N}(\nu, \sigma_\tau^2), \quad \nu = \boldsymbol{\psi}\boldsymbol{\nu}, \quad \sigma_\tau^2 = \boldsymbol{\psi}\mathbf{C}\boldsymbol{\psi}^\top. $$ Because $K > 0$ is deterministic and $\exp(\tau)$ is the exponential of a Gaussian, the standard deviation of $W \mid \mathbf{y}$ is a scaled log-normal: $$ \mathrm{Std}(W \mid \mathbf{y}) = K \cdot \exp(\tau) \sim \mathcal{LN}(\nu, \sigma_\tau^2) \quad \text{(scaled by } K \text{)}. $$ ## Expected Standard Deviation The mean of a log-normal $X \sim \mathcal{LN}(\mu_X, \sigma_X^2)$ is $\mathbb{E}[X] = \exp(\mu_X + \sigma_X^2/2)$. Applying this: $$ \mathbb{E}\!\left[\mathrm{Std}(W \mid \mathbf{y})\right] = K \cdot \exp\!\left(\nu + \tfrac{\sigma_\tau^2}{2}\right), $$ where all quantities are analytical: - $K = \exp(\sigma^2/2)\sqrt{\exp(\sigma^2) - 1}$, - $\nu = \boldsymbol{\psi}\boldsymbol{\nu}$, - $\sigma_\tau^2 = \boldsymbol{\psi}\mathbf{C}\boldsymbol{\psi}^\top$. ## AVaR of the Standard Deviation By **positive homogeneity** of AVaR — $\mathrm{AVaR}[c \cdot X] = c \cdot \mathrm{AVaR}[X]$ for $c > 0$ deterministic — we have: $$ \mathrm{AVaR}_p\!\left[\mathrm{Std}(W \mid \mathbf{y})\right] = K \cdot \mathrm{AVaR}_p[\exp(\tau)]. $$ Since $\exp(\tau)$ is log-normally distributed with parameters $(\nu, \sigma_\tau^2)$, its AVaR at level $p$ is: $$ \mathrm{AVaR}_p[\exp(\tau)] = \frac{\exp(\nu + \sigma_\tau^2/2)\,\Phi\!\left(\sigma_\tau - \Phi^{-1}(p)\right)}{1 - p}, $$ where $\Phi$ is the standard normal CDF. Combining: $$ \mathrm{AVaR}_p\!\left[\mathrm{Std}(W \mid \mathbf{y})\right] = K \cdot \frac{\exp(\nu + \sigma_\tau^2/2)\,\Phi\!\left(\sigma_\tau - \Phi^{-1}(p)\right)}{1 - p}. $$ ## Numerical Verification ```{python} import numpy as np from scipy.stats import norm def lognormal_avar(nu: float, sigma_tau: float, sigma_sq: float, p: float) -> float: """ AVaR of Std(W | y) where W = exp(Q) and Q|y ~ N(tau, sigma_sq), tau ~ N(nu, sigma_tau^2). Parameters ---------- nu : mean of tau (= psi @ nu_vec) sigma_tau: std dev of tau (= sqrt(psi @ C @ psi^T)) sigma_sq : posterior variance of Q|y (= psi @ Sigma_star @ psi^T) p : AVaR level in (0, 1) """ # Deterministic scale factor K K = np.exp(sigma_sq / 2) * np.sqrt(np.exp(sigma_sq) - 1) # AVaR of exp(tau) where tau ~ N(nu, sigma_tau^2) avar_exp_tau = ( np.exp(nu + sigma_tau**2 / 2) * norm.cdf(sigma_tau - norm.ppf(p)) / (1 - p) ) return K * avar_exp_tau def lognormal_expected_std(nu: float, sigma_tau: float, sigma_sq: float) -> float: """Expected value of Std(W | y).""" K = np.exp(sigma_sq / 2) * np.sqrt(np.exp(sigma_sq) - 1) return K * np.exp(nu + sigma_tau**2 / 2) ``` We verify against Monte Carlo: ```{python} np.random.seed(0) nu, sigma_tau, sigma_sq, p = 0.5, 0.3, 0.2, 0.9 # Analytical avar_analytical = lognormal_avar(nu, sigma_tau, sigma_sq, p) estd_analytical = lognormal_expected_std(nu, sigma_tau, sigma_sq) # Monte Carlo reference N = 2_000_000 tau_samples = np.random.normal(nu, sigma_tau, N) K = np.exp(sigma_sq / 2) * np.sqrt(np.exp(sigma_sq) - 1) std_samples = K * np.exp(tau_samples) avar_mc = np.mean(std_samples[std_samples >= np.quantile(std_samples, p)]) estd_mc = np.mean(std_samples) print(f"E[Std(W|y)] — analytical: {estd_analytical:.6f}, MC: {estd_mc:.6f}") print(f"AVaR_{p} — analytical: {avar_analytical:.6f}, MC: {avar_mc:.6f}") ``` ## Key Takeaways - The standard deviation of $W = \exp(Q)$ is a scaled log-normal random variable; the scale factor $K$ depends only on the (deterministic) posterior variance $\sigma^2$. - The randomness in $\mathrm{Std}(W \mid \mathbf{y})$ comes entirely from the Gaussian posterior mean $\tau = \boldsymbol{\psi}\boldsymbol{\mu}_\star$. - Expected standard deviation and AVaR both have closed forms involving the log-normal mean formula and the log-normal AVaR formula, respectively. - Positive homogeneity of AVaR allows the scale factor $K$ to be factored out, reducing the computation to the AVaR of $\exp(\tau)$. ## Exercises 1. Verify that as $\sigma_\tau \to 0$ (i.e., $\mathbf{C} \to 0$, meaning the prior is very tight), $\mathbb{E}[\mathrm{Std}(W \mid \mathbf{y})]$ reduces to a deterministic value. What is it? 2. Plot $\mathrm{AVaR}_p[\mathrm{Std}(W \mid \mathbf{y})]$ as a function of $p$ for $p \in [0.5, 0.99]$. Compare to $\mathbb{E}[\mathrm{Std}(W \mid \mathbf{y})]$. At what level $p$ does AVaR exceed the mean by a factor of 2? 3. The AVaR formula for the log-normal distribution uses $\Phi(\sigma_\tau - \Phi^{-1}(p))$. Verify this formula by deriving it from the definition: $\mathrm{AVaR}_p[X] = \frac{1}{1-p}\int_p^1 F_X^{-1}(u)\,\mathrm{d}u$. ## Next Steps - [Goal-Oriented OED: Convergence Verification](boed_pred_usage.qmd) — using `PredOEDDiagnostics` to numerically validate these analytical expressions