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Let $y$ follow $f(y|\theta,x)$ with $\theta$ parameters with prior $\pi(\theta)$ and $x$ covariates with distribution $p(x)$. Let $p(\theta| D)$ be the posterior distribution of the parameters, where data $D=(y,x)$. For a new value $\tilde{x}$, the posterior predictive distribution is $$f(\tilde{y}|\tilde{x},D) = \int_\theta f(\tilde{y}|\theta, \tilde{x}) p(\theta| D) d \theta$$

This is the setting of the ppd in regression analysis, for example. Now what I am interested in is posterior inference about the cumulative distribution of $\tilde{y}$: $$F(\tilde{y} | \theta ) = \int_x F(\tilde{y}|\theta,\tilde{x}) p(\tilde{x}) d\tilde{x}. $$

One way to sample from the posterior predictive distribution of $F$ evaluated at $y$ seems to be $$ \int_x \int_\theta F(\tilde{y}|\tilde{x},\theta) p(\theta|D) p(\tilde{x}) d\theta d\tilde{x}.$$ So if we can sample from $p(\theta|D)$, and we know / can calculate $F$ we would

  1. Sample $\theta_k \sim p(\theta|D) $
  2. Sample $\tilde{x}_k \sim p(\tilde{x})$
  3. Evaluate $F(\tilde{y}|\tilde{x}_k,\theta_k)$ to get $\hat{F}_K$
  4. Repeat many times. Then $\hat{F}_1,...,\hat{F}_K$ are the posterior samples of $F$ at $\tilde{y}$ integrated across $\theta, x$
  5. To get the posterior mean estimate of $F(\tilde{y}|\theta)$, for example, we could take the average of $F_k$ across $k$

Although this procedure seems valid, an alternative seems to be based on

$$f(\tilde{y}|D) = \int_x \int_\theta f(\tilde{y}|\theta, \tilde{x}) p(\theta| D) p(\tilde{x}) d \theta d\tilde{x},$$ yielding procedure

  1. Sample $\theta_k \sim p(\theta|D) $
  2. Sample $\tilde{x}_k \sim p(\tilde{x})$
  3. Sample $\tilde{y}_k \sim f(\tilde{y}|\theta_k,\tilde{x}_k) $. These are the samples from the posterior predictive distribution integrated across $\theta, \tilde{x}$
  4. Evaluate the empirical cumulative distribution of the samples $y_1,...,y_K$

At first I thought the two procedures are equivalent because they both seem to yield $F(\tilde{y}|D)$. However they yield different results. In particular the second approach does not seem to yield valid inference on $F(\tilde{y}|\theta)$. Am I right, and if so, why?


Edit After the reply by Xi'an and some more own thought I have come up with a third, and potentially superior approach which is a modification of the non-parametric sampling algorithm 2 above. Would welcome thoughts.

Let $q_\alpha(\tilde{y}| \theta)$ denote the $\alpha$ quantile of $\tilde{y}$ given $\theta$. Then

  1. Sample $\theta_k \sim p(\theta|D) $
  2. Now we estimate $q_\alpha(\tilde{y}| \theta_k)$ by monte carlo. For this we sample J times, j=1,...,J, from $\tilde{x}_j \sim p(\tilde{x})$.
  3. Sample $\tilde{y}_j \sim f(\tilde{y}|\theta_k,\tilde{x}_j), j=1,...,J $.
  4. Estimate $q_\alpha(\tilde{y}| \theta_k)$ as empirical quantile of $\tilde{y}_1,...,\tilde{y}_J$

We now have the posterior distribution of $q_\alpha(\tilde{y}| \theta_1),...,q_\alpha(\tilde{y}| \theta_K)$. Let the posterior median be $\bar{q}$, then $\alpha$ is a posterior estimate for $F(\tilde{y}|\theta)$ at $\tilde{y}=\bar{q}$. A credible interval can also be obtained.

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The two quantities $$ F(\tilde y|D) = \int_\Theta \int_{\mathcal X} F(\tilde{y}|\tilde{x},\theta) p(\theta|D) p(\tilde{x})\, d\tilde{x}\, d \theta \tag{1} $$ and $$ f(\tilde{y}|D) = \int_\Theta \int_{\mathcal X} f(\tilde{y}|\theta, \tilde{x}) p(\theta| D) p(\tilde{x}) \, d\tilde{x}\, d \theta \tag{2} $$ are identical in that one is the cdf and the other one the pdf. The difference in the simulation schemes is that (1) produces a (parametric) Rao-Blackwellised approximation to the marginal cdf, $$ \frac{1}{N}\sum_{i=1}^n F(\tilde y|\theta_i,\tilde x_i) $$ while (2) leads to a (non-parametric) empirical approximation, hence less efficient.

Note that both aim at approximating the marginal predictive distribution on $\tilde Y$, which does not bring information about $F(\tilde y|\theta)$ since $\theta$ is integrated out. To find the posterior of the quantity $F(\tilde y|\theta)$ (for a fixed $\tilde y$) simply requires to find the push-forward transform of $p(\theta|D)$ by $F(\tilde y|\cdot)$. In that sense, version 1 of the algorithm does produce a Monte Carlo sample from that distribution.

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  • $\begingroup$ It seems that with algorithm 2 (push forward) I can also get a credible interval by evaluating the percentiles of $F_1,...,F_k$ at $\tilde{y}$, whereas with algorithm 1 I only obtain something like an approximation of the cumulative distribution of $\tilde{Y}$.. $\endgroup$
    – tomka
    Nov 18 '20 at 21:21
  • $\begingroup$ And on the same note, I could perhaps also use the median of $F_1,...,F_k$ at $\tilde{y}$ instead of the mean; in other words I can draw posterior inference on $F$ as usual. Is that correct? $\endgroup$
    – tomka
    Nov 18 '20 at 22:36
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    $\begingroup$ The sample of $F_i$'s produced by the first algorithm in your question is an iid sample from the posterior distribution of $F(\tilde y|\theta)$, hence can be used in any way you wish. The second algorithm in your question does not produce information about $F(\tilde y|\theta)$. $\endgroup$
    – Xi'an
    Nov 19 '20 at 6:36
  • $\begingroup$ Thanks. I have come up with another method to estimate the marginal cdf, please see above. Would welcome comments. This also works when $F$ does not have a known parametrical form and gives a posterior interval at each quantile. $\endgroup$
    – tomka
    Nov 19 '20 at 14:46

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