Expected output of the diffusion objective estimator

Conditional on $I=i,J=t$, the sampled state has the forward marginal in [F3]. Every returned value is finite and measurable: the predictor is finite-valued Borel, all schedule denominators and variances are positive, and the decoder density is strictly positive. In simple mode, the stated finite squared-error expectations make $Y$ integrable, and independence and uniform sampling give $\mathbb{E}[Y]=(NT{)}^{-1}{\sum }_{i=1}^N{\sum }_{t=1}^T{\mathbb{E}}_{ϵ}\mathrm{\parallel }ϵ-{ϵ}_{\theta }(\sqrt{{\bar{\alpha }}_t}{x}_0^{(i)}+\sqrt{1-{\bar{\alpha }}_t}ϵ,t){\mathrm{\parallel }}_2^2=N^{-1}{\sum }_i{L}_{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}}(\theta ;x_0^{(i)})$.

In weighted-elbo mode, [F3] and [F4] with $v=1-{\bar{\alpha }}_T$, $w=1$, $m=\sqrt{{\bar{\alpha }}_T}{x}_0^{(i)}$ and $n=0$ identify the algorithm's $K_T(x_0^{(i)})$ as the exact terminal KL. The mode's hypotheses are those of [F2], so the decoder log-density and the interior weighted squared errors are integrable. Finite sums over $i,t$ therefore make $Y$ integrable, even though a decoder contribution may be negative. For fixed $i$, averaging the branch values over $J$ and the noise cancels each factor $T$ and counts $K_T$ once, giving $\mathbb{E}[Y\mid I=i]=K_T(x_0^{(i)})+{\mathbb{E}}_{q(x_1\mid x_0^{(i)})}[-\log p_{\theta }(x_0^{(i)}\mid x_1)]+{\sum }_{t=2}^T{\mathbb{E}}_{ϵ}[C_t+w_t\mathrm{\parallel }ϵ-{ϵ}_{\theta }(x_t,t){\mathrm{\parallel }}_2^2]=-{\mathrm{ELBO}}_{\theta }(x_0^{(i)})$.

Averaging step 1.2 over the uniform index $I$ proves the weighted identity. When $N=1$ the data average has one term, and when $T=1$ only the decoder branch is selected and the interior sum is empty; the terminal term is still counted once. Thus both identities hold throughout the admitted domain.