Setup for all variations

  • True model (complete data):
    We have $N = 100$ weighings $y_i \sim N(\theta, 1), \quad i = 1, \dots, 100$ where θ is the unknown true weight.
  • Prior: noninformative (flat) prior on θ.
  • In every version, we end up observing only 91 measurements.
  • Let $\bar y_{\text{obs}}$​ be the mean of the 91 observed values.

The whole point: the 91 numbers can be the same, but depending on why the other 9 are missing (random, censored, truncated, unknown point…) the correct likelihood changes, so the posterior for θ changes.

1. Missing Completely at Random (MCAR), probability known

Data rule: each of the 100 measurements is reported with probability 0.9, independently of its value. So 9 are missing just by chance.

  • Inclusion model:
    • $I_i \sim \text{Bernoulli}(0.9), \quad I_i \perp y_i$
  • Because missingness is independent of y, the design is ignorable.
  • Posterior for θ is just the usual normal–normal result with 91 observations:
    • $p(\theta \mid y_{\text{obs}}, I) = p(\theta \mid y_{\text{obs}}) = N\!\left(\bar y_{\text{obs}}, \frac{1}{91}\right)$

So: here, even though we planned 100, observing 91 is as if we had just planned 91.

2. MCAR but missingness probability unknown

Now same situation (91 seen, 9 missing) but the reporting probability is unknown. Call it $\pi$.

  • Observation model:
    • $I_i \mid \pi \sim \text{Bernoulli}(\pi), \quad I_i \perp y_i$
  • Complete posterior:
    • $p(\theta, \pi \mid y_{\text{obs}}, I) \propto p(\theta, \pi)\, N\!\left(\bar y_{\text{obs}} \mid \theta, \frac{1}{91}\right)\, \text{Bin}(n=91 \mid 100, \pi)$
  • Key point:
    • if the prior factorizes, $p(\theta, \pi) = p(\theta)\,p(\pi),$ then θ and π are distinct parameters, and the posterior for θ is the same as in case 1:
      • $\theta \mid y_{\text{obs}} \sim N\!\left(\bar y_{\text{obs}}, \frac{1}{91}\right)$
  • But if π is functionally tied to θ (e.g. $\pi = \theta/(1+\theta)$), then the fact that you saw 91 out of 100 also contains information about θ, so the posterior for θ is no longer the simple normal—it’s tilted by the binomial term.

So: unknown missingness prob affects θ only when θ and the missingness are linked.

3. Censored data (upper limit known)

Now the scale always reports something, but:

  • if $y_i \le 200$: we see the number
  • if $y_i > 200$: we see “too heavy”

Say we have 91 actual numbers and 9 “too heavy”.

  • For the 91 numbers: likelihood pieces are $N(y_i \mid \theta, 1)$
  • For each “too heavy”:
    • that means $y_i > 200 \quad \Rightarrow \quad \Pr(y_i > 200 \mid \theta) = 1 – \Phi(200 – \theta) = \Phi(\theta – 200)$
    • (they write it as $\Phi(\theta – 200)$).
  • So total posterior:
    • $p(\theta \mid y_{\text{obs}}, I) \propto p(\theta)\, \Bigg[\prod_{i: \text{observed}} N(y_i \mid \theta, 1)\Bigg]\, \big[\Phi(\theta – 200)\big]^9$
  • With flat prior and 91 obs, the first part is $N(\bar y_{\text{obs}} \mid \theta, 1/91)$, so: $p(\theta \mid y_{\text{obs}}, I) \propto N\!\left(\bar y_{\text{obs}} \mid \theta, \frac{1}{91}\right) \big[\Phi(\theta – 200)\big]^9$

This is not the same as case 1 or 2—now the 9 “too heavy” measurements are telling us θ is likely above 200.

4. Censored data, censoring point unknown

Now instead of censoring at 200, we censor at an unknown point $\varphi$.

  • Data rule:
    • if $y_i \le \varphi$: we see $y_i$
    • if $y_i > \varphi$: we see “too heavy”
  • We observed 91 numbers and 9 censored.
  • Likelihood now has $\Phi(\theta – \varphi)$ terms, not $\Phi(\theta – 200)$.
  • Even if prior is independent, $p(\theta, \varphi) = p(\theta)\,p(\varphi),$ the posterior is not independent because those 9 censored values couple θ and $\varphi$:
    • $p(\theta, \varphi \mid y_{\text{obs}}, I) \propto p(\theta, \varphi)\, N\!\left(\bar y_{\text{obs}} \mid \theta, \frac{1}{91}\right) \big[\Phi(\theta – \varphi)\big]^9 \quad \text{with } \varphi > \max(y_{\text{obs}})$
  • To get the marginal for θ you have to integrate over $\varphi$, usually on a 2D grid or with MCMC.

Important lesson: when the censoring rule contains unknown parameters, you cannot ignore the censoring mechanism—otherwise you drop the $\Phi(\theta – \varphi)$ factors and get the wrong posterior.

5. Truncated data (upper limit known, but N unknown)

Now: someone else did the weighing and only gave you the 91 values that are $\le 200$. You don’t even know how many >200 there were. That’s truncation, not censoring.

  • Censoring: you know how many are above the limit.
  • Truncation: you don’t even know they existed.

Here:

  • true model still $y_i \sim N(\theta, 1)$
  • observed model = normal truncated above at 200:
    • $f_{\text{trunc}}(y \mid \theta) = \frac{N(y \mid \theta, 1)}{\Phi(200 – \theta)}, \quad y \le 200$
  • They show you can also derive it by treating N as unknown and summing over N; if you put $p(N) \propto 1/N$, you end up with a posterior for θ:
    • $p(\theta \mid y_{\text{obs}}, I) \propto N\!\left(\bar y_{\text{obs}} \mid \theta, \frac{1}{91}\right) \big[1 – \Phi(\theta – 200)\big]^{91}$
    • (since $1 – \Phi(\theta – 200) = \Phi(200 – \theta)$).
  • Intuition: if you only ever see values ≤200, that pushes the posterior downward compared to the uncensored case.

So: truncation = like censoring but you lose the count of how many were cut off, so you have less information about how far the tail goes.

6. Truncated data with unknown truncation point

Last variation: now even the truncation point $\varphi$ is unknown.

  • Observed data: 91 values, all ≤ $\varphi$, but we don’t know $\varphi$.
  • Joint posterior:
    • $p(\theta, \varphi, N \mid y_{\text{obs}}, I) \propto p(\theta, \varphi, N)\, N\!\left(\bar y_{\text{obs}} \mid \theta, \frac{1}{91}\right) [\Phi(\theta – \varphi)]^{N-91}$
    • and after summing over N with
      • $p(N) \propto 1/N$,
    • they get
      • $p(\theta, \varphi \mid y_{\text{obs}}, I) \propto p(\theta, \varphi)\, N\!\left(\bar y_{\text{obs}} \mid \theta, \frac{1}{91}\right) [1 – \Phi(\theta – \varphi)]^{91}$
  • But with a flat prior on $\varphi$, the posterior in $\varphi$ can actually be improper (it doesn’t die off as $\varphi \to \infty$).
  • In that case, the marginal for θ falls back to the simple:
    • $\theta \mid y_{\text{obs}} \sim N\!\left(\bar y_{\text{obs}}, \frac{1}{91}\right)$ just like the MCAR case.

So: sometimes, when the truncation point is totally unidentified, adding that unknown doesn’t change the information about θ.

Key Lessons from All 6 Variations

  1. Same observed numbers ≠ same posterior. The missing-data mechanism matters.
  2. Ignorable mechanism (MCAR, parameters distinct) → you can safely ignore I and just use the observed likelihood.
  3. Nonignorable mechanism (censoring with unknown point, truncation with unknown N/point, dependence between θ and missingness) → you must model the observation process; otherwise posterior is wrong.
  4. Censoring vs truncation:
    • censoring: you know how many got cut off → extra likelihood factors like $[\Phi(\theta – \varphi)]^{\text{censored}}$
    • truncation: you don’t even see them → less information, different likelihood.
  5. All of this is just the general Bayesian rule: condition on everything you observed (the values and the fact that some values were not fully observed).