Evidence averaging under an interval prior ($Z=(b-a{)}^{-1}{\int }_a^b\mathcal{L}(\theta )d\theta$)

For a normalized uniform prior on $[a,b]$, Bayesian model evidence is $Z=(b-a{)}^{-1}{\int }_a^b\mathcal{L}(\theta )d\theta$. It averages the likelihood over the prior range. Extending a well-placed interval into low-likelihood regions reduces that average, but moving or widening a poorly placed interval can improve it. A model prior is separate: posterior model odds equal the evidence ratio multiplied by prior model odds.

If $\mathcal{L}(\theta )=C{e}^{-|\theta -\mu |/s}$, integrate the two exponential branches at $\mu$. An interval of half-width $w$ centred at $\mu$ has $Z/C=(s/w)(1-e^{-w/s})$. An interval entirely below the peak has $Z/C=s[e^{(b-\mu )/s}-e^{(a-\mu )/s}]/(b-a)$. These expressions show quantitatively how fit location and prior width combine.