For Poisson noise, any regular increasing function can be realized as the conditional median of some prior, and linear medians are achievable by non-gamma priors for slopes below 1/e.
Poisson communication theory,
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Linearity-Inducing Priors for Poisson Parameter Estimation Under $L^{1}$ Loss
For Poisson noise, any regular increasing function can be realized as the conditional median of some prior, and linear medians are achievable by non-gamma priors for slopes below 1/e.