Automatic conditioning

Given a joint distribution $p(\mathrm{d}x,\mathrm{d}y) = p(\mathrm{d}x) p(\mathrm{d}y\mid x)$, conditioning is the computation:

$$p(\mathrm{d}x\mid y)=\frac{p(y\mid x)p(\mathrm{dx})}{p(y)}.$$

That is, we condition on the value of $y$ to obtain the conditional distribution of $x$ given $y$. We may also refer to this as Bayesian updating, insofar as we interpret $p(\mathrm{d}x)$ as a prior distribution that we update to a posterior distribution $p(\mathrm{d}x\mid y)$.

Consider:

x ~ Gamma(2.0, 1.0);
y ~ Poisson(x);

Conditioning is triggered when y obtains a value. This can occur in several circumstances:

• If, in the above code, y already has a value, or if the second line is replaced with #!birch y ~> Poisson(x);, then the conditioning is triggered immediately.
• If, in the above code, y does not already have a value, but one is requested by replacing the second line with #!birch y <~ Poisson(x);, then the conditioning is triggered immediately.
• If the above code remains the same, but y.value() is later used to obtain a value for y, then the conditioning is triggered at that time.

In all of these cases x remains marginalized out, but the distribution associated with it is updated to the conditional distribution of x given y. If x.value() is later used to obtain a value for x, it will be drawn from that conditional distribution. In this way random variables are simulated consistently from the joint distribution.

Conditioning can occur multiple times:

x ~ Gamma(2.0, 1.0);
y <~ Poisson(x);
z <~ Poisson(x);

Here, the distribution associated with x is updated twice, to the conditional distribution given both y and z.