Stochastic kernels vs. conditional probability distributions
Larry Wasserman‘s recent post about misinterpretation of p-values is a good reminder about a fundamental distinction anyone working in information theory, control or machine learning should be aware of — namely, the distinction between stochastic kernels and conditional probability distributions.
Roughly speaking, stochastic kernels are building blocks, objects that have to be interconnected in order to instantiate stochastic systems. Conditional probability distributions, on the other hand, arise only when we apply Bayes’ theorem to joint probability distributions induced by these interconnections.
At a very high level of abstraction, we may imagine a space of observations or outcomes and a space of states or inputs . Each possible state induces a probability distribution over — let’s denote it by . The interpretation is that if the state is , then the probability that we observe an outcome in some set is . Notice that this stipulation has the flavor of a conditional statement: if A then B. Mathematical statisticians (going back to Abraham Wald, and greatly elaborated by Lucien Le Cam and his followers) like to think of the collection as an experiment that reveals something about the state in through a random observation in . Note that is not a set of probability distributions — two distinct ‘s may carry two identical ‘s (which would indicate that these two ‘s are statistically indistinguishable on the basis of observations); or, in the simplest case of a binary state space , the experiment that has and is different from the one with and , where and are two fixed probability distributions on . So perhaps it is better to think about the experiment as a function from into , the space of all probability distributions on . When we impose a measurable structure on the state space as well and then require this function to be sufficiently well-behaved, so that the mapping is nice (read: measurable) for any , then we have a stochastic kernel.
Larry’s point about p-values is as follows: a binary hypothesis testing problem is a binary experiment , where and can be thought of as two fixed distributions on some observation space that “explain” the observed outcomes given each hypothesis. If we compute a test statistic and let be the realized value of , then the p-value (for a two-sided test) is
It’s not a conditional probability of anything, but rather the probability a certain event would have if the state were . In order to have conditioning, all relevant quantities must be instantiated as random variables. Let’s consider an example any information theorist should relate to: a binary symmetric channel.
The ingredients are: the input space , the output space , and the experiment
where is the channel’s crossover probability. We often write the channel transition probabilities suggestively as etc., but that is, strictly speaking, incorrect. Until we specify something about the input to the channel, all we have is the experiment , together with a list of possible statements like
if the input is , then the probability of observing at the output is
and the like. If we now say that the input is a random variable taking values in according to a given distribution , then we may make conditional probability statements and compute conditional probability distributions and . Of course, in this case it so happens that the conditional probability distribution is already given in terms of the original experiment. But, properly speaking, this conditional object does not exist until we fix . Even more dramatically, the posterior does not exist at all until we specify , interconnect the source of to the kernel corresponding to the experiment , and start doing what Bayesians would call inverse inference. This distinction between kernel specifications and conditional probability distributions may seem purely notational, but it matters a great deal as soon as feedback enters into the picture.
When the control laws have been selected and instrumented, and only then, the control variables (and the state and output variables, and the cost) become random variables, that is, become functionally related to the given random variables (noise, initial state) and therefore become functionally related to the underlying probability space. But to the designer who is still seeking for good control laws and has not made a selection yet, the realizations of control are not even random variables. They are just “random variables to be” of yet uncertain status.
I also recommend a recent paper by Jan Willems on what he terms “open stochastic systems.” It is best to illustrate the main idea of this paper through another example any information theorist should be familiar with: additive white Gaussian noise (AWGN) channel. We are all used to writing it down as
where is the input, is the additive noise independent of , and is the output. Of course, this expression tacitly assumes that we have already specified a distribution of the input . One may fix things by writing
with the understanding that is some input. But even this is not quite right in view of the above quote from Witsenhausen: is just a placeholder, something waiting to be assigned. Properly speaking, the only “legitimate” random variable here is the Gaussian noise . So Willems suggests thinking instead of the set of all pairs such that
If is now realized as a random input from some distribution , we get back our usual model from (1). However, this new viewpoint is a lot more interesting because it has room for things like causality. Consider, for example, a more complicated arrangement, in which the input taking values in some space is first “modulated” by some nonlinear transformation , and then the noise is added to the output of this nonlinear transformation. Then we can represent the overall open system as the set of all pairs , such that
or that is somewhere in the preimage, under , of a Gaussian random variable with mean and variance . Unless we are in an exceptional situation (e.g., if is one-to-one), it is reasonable to say that (3) expresses a “more complex” statement compared to (2). From this viewpoint, it is more reasonable to say that is the input that causes , and not the other way around. (This way of thinking about causality has, in fact, already found its way into machine learning literature.) The paper of Willems contains a lot more insightful examples and thought-provoking discussion.
So, to summarize: the misunderstanding about conditioning that Larry Wasserman has sought to clear up persists not only in statistics, but also in all other fields involving stochastic systems, such as communications, control, machine learning, etc., and we should always be careful not to fall into this trap.