Active Learning
نویسنده
چکیده
Suppose X ∼ N(μ, σ) ∈ R. We want to test if μ = 0 or μ > 0, and say we know σ. Then noting that Y = X/σ ∼ N(0, 1) under the null, we suggest the test I(X/σ > zα) which has error rate P1(X/σ ≤ zα) = Φ(zα − μ/σ) Hence everything depends on SNR μ/σ. Getting an observation from X ∼ N(μ, σ) is exactly like getting an observation Y ∼ N(μ/σ, 1). Similarly, suppose we get M measurements, calculating X̄ ∼ N(μ, σ/M) just scales the variance down by M , and so is like getting one observation Y ∼ N( √ Mμ/σ, 1).
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