Manuscript Psy03-C@1 · Psy03-C-v1.tex · theorem · d3380b90

Theorem: Repeated-measurement observation reference

Probe geometry and exact task references › Exact ordinary observation ceiling with repeated corruption · lines 1098–1140 · proof included or linked

Psy03-C@1#thm:measurementbayes, Psy03-C@1#eq:measurementbayes

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Suppose the same Gaussian latent point $X$ is repeated and the ordinary cell receives

u_{t} = g_{e} (X + ξ E_{t}) + τ G_{t}^{in},

with $g_{e} > 0$ and independent standard Gaussian noises. Let

v_{T} = s^{2} + \frac{ξ ^{2} + τ ^{2} / g _{e} ^{2}}{T} .

If $v_{T} > 0$ , the Bayes accuracy from the complete cell-input sequence is

Q^{*} (u_{1 : T}) = Φ_{G} (\frac{m}{v _{T}})^{2} .

If $v_{T} = 0$ , it is one.

Any source ordinary recurrent classifier has accuracy at most this value. The latent cluster uncertainty $s^{2}$ does not decrease with $T$ . If $g_{e} = 0$ , the cell-input sequence is label independent and its Bayes accuracy is $1/4$ .

Proof. Divide inputs by $g_{e}$ . Their average is

u ˉ / g_{e} = (m U_{1}, m U_{2}) + s Z + \frac{1}{T} t \sum (ξ E_{t} + τ G_{t}^{in} / g_{e}) .

Its conditional coordinate variance is $v_{T}$ . The sequence residuals around this average are jointly Gaussian, independent of the average and of the label. Thus the average is sufficient for the class likelihood. Apply Theorem thm:bayes. The recurrent output is processing of that sequence.

Displayed formulas in this passage

unlabeled · line 1102
$u_{t} = g_{e} (X + ξ E_{t}) + τ G_{t}^{in},$
unlabeled · line 1107
$v_{T} = s^{2} + \frac{ξ ^{2} + τ ^{2} / g _{e} ^{2}}{T} .$
Psy03-C@1#eq:measurementbayes · line 1112
$Q^{*} (u_{1 : T}) = Φ_{G} (\frac{m}{v _{T}})^{2} .$
unlabeled · line 1130
$u ˉ / g_{e} = (m U_{1}, m U_{2}) + s Z + \frac{1}{T} t \sum (ξ E_{t} + τ G_{t}^{in} / g_{e}) .$

Nearby in this section

section: Exact ordinary observation ceiling with repeated corruption