From Data to Insights

Jeremias Sulam

Trustworthy methods for modern biomedical imaging

ARISE Network Virtual Meeting

50 years ago ... 

first CT scan
ELECTRIC & MUSICAL INDUSTRIES

50 years ago ... 

imaging
diagnostics

complete hardware & software description

human expert diagnosis and recommendations

imaging was "simple"

... 50 years forward 

Data

Compute & Hardware

Sensors & Connectivity

Research & Engineering

... 50 years forward 

Data

Compute & Hardware

Sensors & Connectivity

Research & Engineering

data-driven  imaging
automatic analysis and rec.
societal implications
data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

Demographic fairness

hardware & protocol optimization

inverse problems

uncertainty quantification

model-agnostic interpretability

robustness

generalization

policy & regulation

Demographic fairness

hardware & protocol optimization

data-driven  imaging
automatic analysis and rec.
societal implications

Problems in trustworthy biomedical imaging

Demographic fairness

Inputs (features):            \(X\in\mathcal X \subset \mathbb R^d\)

Responses (labels):        \(Y\in\mathcal Y = \{0,1\}\)

Sensitive attributes        \(Z \in \mathcal Z \subseteq \mathbb R^k \)  (sex, race, age, etc)

                                          \((X,Y,Z) \sim \mathcal D\)

Eg: \(Z_1: \) biological sex, \(X_1: \) BMI, then

\( g(Z,X) = \boldsymbol{1}\{Z_1 = 1 \land X_1 > 35 \}: \) women with BMI > 35

Goal: ensure that \(f\) is fair w.r.t groups \(g \in \mathcal G\)

Demographic fairness

Group memberships      \( \mathcal G = \{ g:\mathcal X \times \mathcal Z \to \{0,1\} \} \)

Predictor     \( f(X) : \mathcal X \to [0,1]\)  (e.g. likelihood of X having disease Y)

  • Group/Associative Fairness
           Predictors should not have very different (error) rates among groups
         [Calders et al, '09][Zliobaite, '15][Hardt et al, '16]
  • Individual Fairness
           Similar individuals/patients should have similar outputs
           [Dwork et al, '12][Fleisher, '21][Petersen et al, '21]
  • Causal Fairness
           Predictors should be fair in a counter-factual world
          [Nabi & Shpitser, '18][Nabi et al, '19][Plecko & Bareinboim, '22]
  • Multiaccuracy/Multicalibration
           Predictors should be approximately unbiased/calibrated for every group
           [Kim et al, '20][Hebert-Johnson et al, '18][Globus-Harris et at, 22]

Demographic fairness

  • Multiaccuracy/Multicalibration
           Predictors should be approximately unbiased/calibrated for every group
           [Kim et al, '20][Hebert-Johnson et al, '18][Globus-Harris et at, 22]

Demographic fairness

Observation 1:
measuring (& correcting) for MA/MC requires samples over \((X,Y,Z)\)

Definition:            \(\text{MA} (f,g) = \big| \mathbb E [ g(X,Z) (f(X) - Y) ] \big|  \)

\(f\) is \((\mathcal G,\alpha)\)-multiaccurate if   \( \max_{g\in\mathcal G} \text{MA}(f,g) \leq \alpha \)

Definition:             \(\text{MC} (f,g) = \mathbb E\left[ \big| \mathbb E [ g(X,Z) (f(X) - Y) | f(X) = v] \big| \right]  \)

\(f\) is \((\mathcal G,\alpha)\)-multicalibrated if   \( \max_{g\in\mathcal G} \text{MC}(f,g) \leq \alpha \)

Observation 2: That's not always possible... 

Observation 2: That's not always possible... 

sex and race attributes missing

  • We might want to conceal \(Z\) on purpose, or might need to

We observe samples  over \((X,Y)\) to obtain \(\hat Y = f(X)\) for \(Y\) 

Fairness in partially observed regimes

\( \text{MSE}(f) = \mathbb E [(Y-f(X))^2 ] \)

A developer provides us with proxies  \( \color{Red} \hat{g} : \mathcal X \to \{0,1\} \)

\( \text{err}(\hat g) = \mathbb P [({\color{Red}\hat g(X)} \neq {\color{blue}g(X,Z)} ] \)

 [Awasti et al, '21][Kallus et al, '22][Zhu et al, '23][Bharti et al, '24]

Question

Can we (how) use \(\hat g\) to measure (and correct) \( (\mathcal G,\alpha)\)-MA/MC?

Fairness in partially observed regimes

Theorem [Bharti, Clemens-Sewall, Yi, S.]

With access to \((X,Y)\sim \mathcal D_{\mathcal{XY}}\), proxies \( \hat{\mathcal G}\) and predictor \(f\)

 

\[ \max_{\color{Blue}g\in\mathcal G} MC(f,{\color{blue}g}) \leq \max_{\color{red}\hat g\in \hat{\mathcal{G}} } B(f,{\color{red}\hat g}) + MC(f,{\color{red}\hat g}) \]

 

with \(B(f,\hat g) = \min \left( \text{err}(\hat g), \sqrt{MSE(f)\cdot \text{err}(\hat g)} \right) \)

  • Practical/computable upper bounds \(\)
  • Multicalibrating w.r.t \(\hat{\mathcal G}\) provably improves upper bound
    [Gopalan et al. (2022)][Roth (2022)]

true error

worst-case error

Fairness in partially observed regimes

CheXpert:  Predicting abnormal findings in chest X-rays
(not accessing race or biological sex)

\(f(X): \) likelihood of \(X\) having  \(\texttt{pleural effusion}\)

Demographic fairness

Take-home message

  • Proxies can be very useful in certifying max. fairness violations
  • Can allow for simple post-processing corrections

Thanks to the ARISE Community!