Exact inferential targets and approximation chains

Fix a measurable parameter space $(\mathrm{\Theta },\mathcal{T})$, a prior probability measure ${\mathrm{\Pi }}_0$ on it, observed data $x$, and an explicitly supplied $\mathcal{T}$-measurable likelihood version ${\mathrm{\ell }}_x:\mathrm{\Theta }\to [0,\mathrm{\infty })$. Define the evidence $Z_x:={\int }_{\mathrm{\Theta }}{\mathrm{\ell }}_{x}d{\mathrm{\Pi }}_0\mathrm{.}$ When $0<Z_x<\mathrm{\infty }$, the exact inferential target is the probability measure ${\mathrm{\Pi }}_x(A):=\frac{1}{Z_x}{\int }_A{\mathrm{\ell }}_{x}d{\mathrm{\Pi }}_0(A\in \mathcal{T})\mathrm{.}$ The measure is well-defined by the indefinite-integral construction The indefinite integral of a nonnegative measurable function is a measure. This is a fixed-data definition: Likelihood densities selects, under AC, a measurable data-space density representative $p_{\theta }$ for each parameter $\theta$, but does not guarantee measurability of $\theta \mapsto p_{\theta }(x)$. Here the fixed-data likelihood version and its parameter measurability are supplied explicitly; no posterior is assigned when the evidence is zero or infinite.

For approximation accounting, declare a metric space $(\mathcal{M},d)$ and a finite chain of elements $T_0,T_1,\dots ,T_m,\widehat{T}\in \mathcal{M}$, where $m\ge 1$ and $T_1={\mathrm{\Pi }}_x$ whenever the exact target is included. The first distance $d(T_0,T_1)$ is the declared reference-to-target discrepancy; later adjacent distances and $d(T_m,\widehat{T})$ are approximation and computation discrepancies on the same metric scale. For probability measures on a common measurable space, total variation may be used with the event-supremum convention of Total variation distance for probability laws. For scalar random quantities on one probability space, the statistical and computational stages are the same-scale signed differences defined in Successive statistical and computational error targets. This notation does not assert independence, unbiasedness, convergence, or that any numerical method attains a global optimum.