Our empirical analysis course is really cool. Here’s what we covered last week:

The setup is quite clean. There is one agents. Four objects:

ObjectSymbolRole
Likelihood$\ell(x\mid\theta)$family of probability models for data $x$, indexed by parameter $\theta\in\Theta$
Prior$\pi(\theta)$your subjective beliefs over which $\theta$ is true
Utility$U(\gamma,\theta)$what you care about; $\gamma$ is a “prize rule” mapping data into an outcome
Decision rule$\delta(z)$what you actually choose, as a function of observed $z$

Split the data: $x=(w,z)$, where $z$ is observed before act and $w$ is realized after—the decision $\delta$ may only use $z$. The prize is then $\gamma_\delta(x)=\Psi(\delta(z),x)$.

The Bayesian benchmark

Define Wald’s risk function:

$$ \overline{U}(\gamma\mid\theta)=\int_\mathcal{X} U[\gamma(x),\theta]\,\ell(x\mid\theta)\,d\tau(x) $$
note: $\tau$ is the reference measure on the data space $\mathcal X$. For different $\mathcal X$ — continuous, discrete, $\tau$ corresponds to Lebesgue measure of counting measure. Or for simplicity just think $\text{d}\tau(x)$ as $\text{d}x$.

Then we can define the full-confidence benchmark. Assumes
(i) $\ell(\cdot\mid\theta)$ is correct for some $\theta$, and
(ii) $\pi$ is the right prior, and —

$$ \max_{\gamma\in\Gamma(\Delta)}\;\int_\Theta \overline{U}(\gamma\mid\theta)\,d\pi(\theta) $$

Now let’s relaxes both.

Doubting the prior → ambiguity

Write any alternative prior as a reweighting of the baseline:

$$ d\tilde\pi(\theta)=n(\theta)\,d\pi(\theta),\qquad n\geq 0,\;\int n\,d\pi=1 $$

So $n$ is the density of the new prior with respect to the old. Measure its distance from $n\equiv 1$ using a convex $\phi_1$ with $\phi_1(1)=0$:

$$ D_{\phi_1}(n)=\int_\Theta \phi_1[n(\theta)]\,d\pi(\theta). $$

Note: KL is the choice $\phi_1(n)=n\log n$.

Standard KL of $\tilde\pi$ relative to $\pi$ is
$$D_{\text{KL}}(\tilde\pi\|\pi) = \int_\Theta \log\!\left(\frac{d\tilde\pi}{d\pi}\right)d\tilde\pi$$
Substitute $d\tilde\pi = n\,d\pi$ and $\frac{d\tilde\pi}{d\pi} = n$:
$$D_{\text{KL}}(\tilde\pi\|\pi) = \int_\Theta \log[n(\theta)]\cdot n(\theta)\,d\pi(\theta) = \int_\Theta n(\theta)\log n(\theta)\,d\pi(\theta)$$

The robust Bayesian objective replaces $\int \overline{U}\,d\pi$ with a worst-case-plus-penalty:

$$ \max_{\gamma\in\Gamma(\Delta)}\;\min_{n\in\mathcal{N}}\;\int_\Theta \overline{U}(\gamma\mid\theta)\,n(\theta)\,d\pi(\theta)+\xi_1\int_\Theta \phi_1[n(\theta)]\,d\pi(\theta) $$

The hyperparameter $\xi_1\geq 0$ governs trust in $\pi$. Denote as $n^* := \arg\min_{n\in \mathcal N} \text{(above)}$.

  • $\xi_1\to\infty$: $n^*\equiv 1$, back to standard Bayes.
  • $\xi_1\to 0$: adversarial choose $n^*$ free (classic max-min).

Note that the inner minimization problem is over $n$ so $U$ doesn’t have to be concave. As long as $\phi_1$ is convex we get nice properties. Eg:

For KL $n^*$ has a closed-form:

$$n^*(\theta)= \frac{\exp[-u(\theta)/\xi_1]}{\int_\Theta \exp[-u(\theta')/\xi_1]\,d\pi(\theta')}\propto \exp\!\left[-\frac{1}{\xi_1}\overline{U}(\gamma\mid\theta)\right]$$

Note that overweights the $\theta$ where $\gamma$ performs badly.

Substituting back collapses the objective to the smooth-ambiguity form

$$\max_{\gamma\in\Gamma(\Delta)}\{-\xi_1\log\int_\Theta \exp\!\left[-\frac{1}{\xi_1}\overline{U}(\gamma\mid\theta)\right]d\pi(\theta)\}$$

a certainty-equivalent operator across models. Ambiguity aversion looks formally like risk aversion, but applied across models rather than within one.

NOTE: The cancellation isn’t a coincidence — it’s the duality between KL-penalized expectation and the log-Laplace transform, a well-known fact (the Donsker–Varadhan variational formula):
$$-\xi\log\mathbb{E}\!\left[e^{-X/\xi}\right] = \min_{Q\ll P}\;\Big\{\mathbb{E}_Q[X] + \xi\,D_{\text{KL}}(Q\|P)\Big\}$$
Letting $P=\pi$, $Q=\tilde\pi$, $X=u(\theta)$.

Doubting the likelihood → misspecification

Same trick, different target. Hold $\theta$ fixed; allow $\ell$ itself to be wrong. Write any tilted likelihood as $\tilde\ell(x\mid\theta)=m(x\mid\theta)\,\ell(x\mid\theta)$ with $\int m\,\ell\,d\tau=1$. Penalty (with convex $\phi_2$, $\phi_2(1)=0$):

$$\int_\mathcal{X} \phi_2[m(x\mid\theta)]\,\ell(x\mid\theta)\,d\tau(x)$$

Inner min with KL gives the parallel formula

$$-\xi_2\log\int_\mathcal{X}\exp\!\left[-\frac{1}{\xi_2}U[\gamma(x),\theta]\right]\ell(x\mid\theta)\,d\tau(x)$$

— a certainty-equivalent operator now acting on data given $\theta$.

Note. No prior is placed over $m$’s. Misspecification is a constrained set, not a Bayesian unknown. If you had a credible prior over $m$’s, you’d just integrate them out and call the result a new $\ell$. Treating it as a set is what makes “misspecification” different from “another layer of parameter uncertainty.”

A unified view for robust Bayes and misspecification

Stack both layers and let the decision maker maximize, the adversary minimize on each axis:

$$\max_{\gamma\in\Gamma(\Delta)}\;\min_{n\in\mathcal{N}}\;\min_{m\in\mathcal{M}}\;\;\underbrace{\int_\Theta\!\int_\mathcal{X} U[\gamma(x),\theta]\,m(x\mid\theta)\,\ell(x\mid\theta)\,d\tau(x)\,n(\theta)\,d\pi(\theta)}_{\text{worst-case expected utility}}\\ +\;\xi_2\underbrace{\int_\Theta\!\int_\mathcal{X}\phi_2[m]\,\ell\,d\tau\,n\,d\pi}_{\text{likelihood-misspec penalty}}\;+\;\xi_1\underbrace{\int_\Theta\phi_1[n]\,d\pi}_{\text{prior-ambiguity penalty}}$$

Two independent hyperparameters:

  • $\xi_1$ — how much you trust the prior $\pi$.
  • $\xi_2$ — how much you trust the likelihood family $\ell(\cdot\mid\theta)$.

Setting $\xi_1=\xi_2=\infty$ recovers vanilla Bayes.

The minimizers $n^*(\theta)$ and $m^*(x\mid\theta)$ tell us where uncertainty bites for this decision. Same prior, same likelihood, different $\gamma$ → generally different worst cases. So we don’t just have error bars, but error bars weighted by decision relevance.

Reference

Risk, Ambiguity, and Misspecification. Lars Peter Hansen (University of Chicago) and Thomas Sargent (NYU) https://lphansen.github.io/QuantMFR/book/10_risk_ambiguity_misspecification.html