Which of the following statements is TRUE? “A Bayesian observer puts more weight on the...”

Question

BlackTom AI · Accepted Answer

Bayes' rule multiplies prior and likelihood, so their contributions to the posterior are weighted according to their reliabilities (precisions). In the common Gaussian case this is explicit.

1) Relevant knowledge points and concepts
- Bayes' theorem: posterior ∝ likelihood × prior.
- Reliability is quantified by precision (τ = 1/variance). Higher precision = more reliable.
- For Gaussian prior N(μp, σp^2) and Gaussian likelihood N(μl, σl^2), the posterior mean is a precision-weighted average of μp and μl.

2) Complete reasoning process (step by step)
Start from the posterior mean formula for conjugate Gaussians:
posterior mean = (μp/σp^2 + μl/σl^2) / (1/σp^2 + 1/σl^2).

3) Intermediate calculation steps
Define precisions τp = 1/σp^2 and τl = 1/σl^2. Then:
posterior mean = (τp μp + τl μl) / (τp + τl).
This can be written as weighted sum:
posterior mean = wp μp + wl μl,
with weights wp = τp/(τp + τl) and wl = τl/(τp + τl).

4) Explain WHY the correct answer is correct
The weight on the likelihood is wl = τl/(τp + τl). If the likelihood is more reliable (τl larger, or σl^2 smaller), wl increases, so the Bayesian observer places more weight on the likelihood. Thus the correct statement is that a Bayesian observer puts more weight on the likelihood information if that is more reliable.

5) Why the other options are incorrect
- "prior information if that is less reliable." This reverses the dependence: a less reliable prior (low τp) gets less weight, not more.
- "likelihood information regardless of prior information." Incorrect because weighting depends on relative precision; if the prior is extremely informative it dominates.
- "prior information unless it has noise." Vague/incorrect: any noise changes its precision continuously; decision is based on relative precision, not a simple unless-clause.

6) Conclusion
Therefore the correct choice is "likelihood information if that is more reliable." This matches the provided answer.

Which of the following statements is TRUE? “A Bayesian observer puts more weight on the...”Single choice

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