Parameter learning

The task of learning the parameters of a model means that given a set of data cases or observations $\boldsymbol{X}$ and a model structure $\mathcal{H}$, one can infer the distribution over the model parameters $\boldsymbol{\theta}$, found for instance in the conditional probability table in Table 3.1, in the clique potentials $\psi$ in Equation (3.13), in the mapping $\mathbf{A}$ in Equation (3.14), or the transition probabilities in Figure 3.3. Parameter learning does not differ from inference in Bayesian probability theory, so the reasons for studying them separately are mostly practical. For instance in the EM algorithm, the updates of parameters and latent variables are done separately and in different ways. Also, local update rules work efficiently in parameter learning, whereas explicit propagation of information is important in state inference of dynamic systems (see Publication V).

Parameter learning can be really simple. Consider learning the values in the conditional probability table for neighbour 1 calling about the alarm in case there is an alarm or not, $P(N_1\mid A)$ in Table 3.1. Given data samples where we observe whether there was an alarm and whether the neighbour called or not, let us settle for a point estimate: the most likely set of parameters. The ML solution is simply to count how many times each of the four cases appear in the data and turn them into probabilities by normalising each row to sum to one.