Responsibility-Weighted Update

• Responsibility-Weighted Update is an extension of Bayes’ rule that raises prior and likelihood to positive exponents, altering the distribution’s concentration and entropy.
• It enables controlled modulation of informational influence through parameters α and β, thereby amplifying or attenuating the impact of prior beliefs and new data.
• This flexible framework models both human and algorithmic biases, allowing systematic deviations from standard Bayesian rationality.

A Responsibility-Weighted Update is an information-theoretically motivated generalization of Bayes’ rule, where the prior and likelihood functions are each raised to positive real exponents before normalization. These exponents serve as “responsibility weights,” allowing the decision maker to systematically upweight or downweight the influence of the prior or data. This scheme, as formalized by Zinn, modifies the Shannon entropy of the resulting posterior in a monotonic fashion: weights greater than one yield distributions with reduced entropy (greater concentration), while weights less than one yield more diffuse (higher entropy) posteriors. The approach provides a flexible modeling tool for capturing human or algorithmic biases away from standard Bayesian rationality by allowing explicit control over the informativeness attributed to each component of the update (Zinn, 2016).

1. Formal Definition and Responsibility Weights

The Responsibility-Weighted Update operates over a parameter space $\Theta$ with prior density $\pi(\theta)$ and likelihood $f(x|\theta)$. The classical Bayesian posterior is given by $\pi(\theta|x) \propto \pi(\theta) f(x|\theta)$. The responsibility-weighted posterior, in contrast, is defined as: $$\pi_w(\theta|x) = \frac{\pi(\theta)^\alpha f(x|\theta)^\beta}{Z(\alpha,\beta;x)}$$ where the normalization constant is

$$Z(\alpha, \beta; x) = \int_\Theta \pi(\theta)^\alpha f(x|\theta)^\beta d\theta$$

with weights $\alpha, \beta > 0$ denoting the responsibility coefficients for the prior and likelihood, respectively. Here, $\alpha > 1$ or $\beta > 1$ increases the concentration (influence) of the respective component, while values less than $1$ render the associated information less influential in the posterior construction. $\alpha$ encodes responsibility towards the prior; $\beta$ encodes responsibility towards the data (Zinn, 2016).

2. Entropy Shifts Induced by Weighted Updating

The entropy of a distribution $g$ over $\Theta$ is quantified by Shannon entropy: $$H(g) = -\int_\Theta g(\theta) \log g(\theta) d\theta$$ For the responsibility-weighted posterior, the entropy is: $$H(\pi_w) = -\alpha\, \mathbb{E}_w[\log \pi] - \beta\, \mathbb{E}_w[\log f] + \log Z$$ where $\mathbb{E}_w[\cdot]$ denotes expectation under $\pi_w$. The shift in entropy due to responsibility weighting, relative to the original prior, is: $$H(\pi_w) - H(\pi) = (1-\alpha) \mathbb{E}_w[\log \pi] - \beta\, \mathbb{E}_w[\log f] + \log Z$$ This expression reveals that increasing $\alpha$ (holding others fixed) generally decreases the entropy of $\pi_w$ relative to $\pi$, leading to more concentrated posteriors. Similarly, increasing $\beta$ decreases the entropy contributed by the data. This implies $\alpha, \beta > 1$ enforce stronger concentration than Bayesian updating, while $\alpha, \beta < 1$ yield greater dispersion (Zinn, 2016).

3. Theoretical Guarantees: Monotonicity and Concentration/Dispersion

A key result formalizes the monotonic entropy implications of exponentiating and normalizing a density [(Zinn, 2016), Corollary 5]:

• For any density $g$ on support $\Omega$, define $$g_\gamma(\omega) = g(\omega)^\gamma / \int_\Omega g(\omega)^\gamma d\omega$$.
  • If $\gamma>1$, $g_\gamma$ is a monotone concentration of $g$ and $H(g_\gamma)<H(g)$.
  • If $\gamma<1$, $g_\gamma$ is a monotone dispersion and $H(g_\gamma)>H(g)$.
  • Proofs are obtained via demonstrating preservation of mode orderings (mode-preserving) and the contraction/expansion of density ratios, with monotonicity certified using Gibbs’ (Kullback–Leibler) inequality. This applies directly to both $\pi(\theta)$ and $f(x|\theta)$, verifying that increasing (decreasing) responsibility parameters sharpens (flattens) the posterior.

4. Modeling Scenarios with Responsibility Weights

Distinct agent attitudes or modeler assumptions can be instantiated by specific choices of $\alpha$ and $\beta$:

• Overweighting the likelihood ($\beta>1$): models agents treating observed data as exceptionally informative, leading to sharply concentrated posteriors on parameter values for which $x$ is highly likely.
• Underweighting the prior ($\alpha<1$): represents agents who discount prior knowledge, allowing the data to play a more prominent role, thereby increasing posterior dispersion.
• Mixed biases: combinations such as $\alpha>1, \beta<1$ or vice versa encode over- or under-reliance on different sources, capturing nuanced attitudes toward information sources.

A plausible implication is that the framework operationalizes a spectrum between strict Bayesian rationality and systematically biased or trust-modulated inference, useful for representing both human cognitive biases and algorithmic heuristics (Zinn, 2016).

5. Algorithmic Implementation

Implementation proceeds as follows:

1. Compute the un-normalized weighted density:

$$u(\theta) \leftarrow \pi(\theta)^\alpha \, f(x|\theta)^\beta$$

2. Compute the normalizing constant:

$$Z \leftarrow \int_\Theta u(\theta) d\theta = \int_\Theta \pi(\theta)^\alpha f(x|\theta)^\beta d\theta$$

3. Form the responsibility-weighted posterior:

$\pi_w(\theta|x) \leftarrow u(\theta)/Z$

4. (Optional) Compute posterior entropy:

$$H_w \leftarrow - \int_\Theta \pi_w(\theta|x) \log \pi_w(\theta|x) d\theta = -\alpha\, \mathbb{E}_w[\log \pi] - \beta\, \mathbb{E}_w[\log f] + \log Z$$

Interpretation of the update process and entropy computation follows directly from the weighting scheme. For $\alpha>1$ or $\beta>1$, the entropy of the corresponding component drops; for $\alpha<1$ or $\beta<1$, the entropy increases relative to the Bayesian benchmark.

6. Information-Theoretic Rationale and Proof Structure

The information-theoretic foundation establishes that exponential reweighting and normalization monotonically transforms entropy. Key ingredients in the proof include:

• Verification of order preservation and contraction/expansion of density ratios from properties of $x \mapsto x^\gamma$.
• Application of Kullback–Leibler divergence or Gibbs’ inequality, showing that $H(g_\gamma)<H(g)$ for $\gamma>1$ and $H(g_\gamma)>H(g)$ for $\gamma<1$. This underpins the control that responsibility weights exert over the informativeness encoded in the posterior distribution [(Zinn, 2016), Appendix].

7. Relation to Bayesian and Non-Bayesian Inference

Responsibility-Weighted Updating encompasses Bayes’ rule as the special case $\alpha = \beta = 1$. Departures from unity yield systematically biased posteriors:

• Bayesian updating treats all information at “face value.”
• Responsibility-weighted updating allows flexible specification of trust or skepticism with respect to either prior or data. In empirical and behavioral modeling, this suggests broad utility for modeling agent heterogeneity, bias, and non-standard rationality—capturing cases where individuals or systems systematically overweight or underweight particular information sources (Zinn, 2016).