Which of the following statements is INCORRECT? A Bayesian observer weighs in both the prior and the likelihood information depending on their relative…

Question

BlackTom AI · Accepted Answer

Bayesian inference combines the prior and the likelihood into a posterior by weighting each source of information according to its uncertainty—specifically its precision (the inverse of variance). For a Gaussian prior and Gaussian likelihood this is explicit.

1) Relevant concepts
- Prior: belief before seeing data, with variance σ0^2 (precision τ0 = 1/σ0^2).
- Likelihood: data evidence, with variance σL^2 (precision τL = 1/σL^2).
- Posterior mean is a precision-weighted average of prior mean μ0 and observed data x.

2) Step-by-step derivation (Gaussian case)
Prior: N(μ0, σ0^2), Likelihood: N(x, σL^2).
Posterior precision: τ_post = τ0 + τL = 1/σ0^2 + 1/σL^2.
Posterior mean:
μ_post = (τ0 μ0 + τL x) / (τ0 + τL).
Write τ0 = 1/σ0^2 and τL = 1/σL^2:
μ_post = (μ0/σ0^2 + x/σL^2) / (1/σ0^2 + 1/σL^2).
Thus the weight on the prior is τ0/(τ0+τL) and on the data is τL/(τ0+τL).

3) Why the correct answer is “recency.”
- Precision and reliability: These are the correct terms—precision (inverse variance) and reliability (quality of a signal) directly determine the weights in Bayesian updating. Higher precision (more reliable, lower noise) yields larger weight.
- Noise: Increased noise raises variance (lowers precision), so relative noise between sources determines weighting; thus “noise” is a valid way to describe the factor that changes weights.
- Recency: Temporal recency (how recent an observation is) is not a general principle used by a Bayesian observer to weight prior vs. likelihood. Unless the model explicitly encodes time-dependent priors or discounting of old data, Bayesian weighting depends on uncertainty (precision/reliability/noise), not merely whether information is recent. Therefore “recency” is incorrect in this general statement.

4) Why other options are incorrect choices for “INCORRECT”
- “Noise”: correct concept because higher noise reduces weight.
- “Reliability”: correct synonym for low noise/high precision.
- “Precision”: the exact mathematical determinant of weights.

Conclusion: The incorrect statement is “recency.” This matches the provided answer.

Which of the following statements is INCORRECT? A Bayesian observer weighs in both the prior and the likelihood information depending on their relative…Single choice

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