Which of the following statements is true about Bayesian observers?

Question

BlackTom AI · Accepted Answer

Bayesian observers weight pieces of information according to their reliability: more precise (less noisy) sources influence the posterior estimate more than less precise ones. Key concepts are prior (existing belief), likelihood (sensory evidence), posterior (updated belief), and that Bayes’ rule combines prior and likelihood multiplicatively: posterior ∝ likelihood × prior. In continuous Gaussian cases, “reliability” is formalized as precision = 1/variance.

Step-by-step reasoning with a Gaussian example:
1. Represent prior as N(μ_p, σ_p^2) and sensory likelihood as N(μ_l, σ_l^2).
2. The posterior mean is the precision-weighted average:
   posterior mean = (μ_p/σ_p^2 + μ_l/σ_l^2) / (1/σ_p^2 + 1/σ_l^2).
   Equivalently, weight_prior = (1/σ_p^2)/(1/σ_p^2 + 1/σ_l^2), weight_likelihood = (1/σ_l^2)/(1/σ_p^2 + 1/σ_l^2).
3. Numerical example: let μ_p = 0, σ_p = 2 (variance 4), μ_l = 5, σ_l = 1 (variance 1).
   Compute precisions: 1/σ_p^2 = 0.25, 1/σ_l^2 = 1.
   Weights: weight_prior = 0.25/(0.25+1) = 0.2; weight_likelihood = 1/(1.25) = 0.8.
   Posterior mean = 0*0.2 + 5*0.8 = 4.
This shows the posterior leans toward the more reliable (lower-variance) source.

Why the correct option is correct:
- “They take into account how reliable a source of information.” matches Bayes’ rule and precision-weighted averaging: observers combine prior and sensory evidence while weighting each by reliability.

Why other options are incorrect:
- “They exclude prior information because it’s less reliable.” Wrong: Bayesians include priors and down-weight them if noisy, not exclude.
- “They exclude sensory information because it’s less reliable.” Wrong for the same reason: sensory evidence is included and weighted by reliability.
- “They only rely on prior information because it’s 100% reliable.” Incorrect: priors are rarely perfect and Bayesians combine both sources; they don’t assume 100% reliability.

Final confirmation: The correct answer is “They take into account how reliable a source of information.”

Which of the following statements is true about Bayesian observers?Single choice

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