Introduction to Bayesian Inference

We’ve implemented our quantum Bayesian network. It represents a passenger’s overall chance to survive the Titanic shipwreck. It considers two features, the passenger’s sex and whether the passenger was a child.

It’s time to use this network. We want to infer something we don’t already know. We perform inference.

Generally, statistical inference is the process of deducing properties about a population or probability distribution from data. This is why we build the entire network. We want to be able to make predictions about some new data from the data we already know.

Specifically, Bayesian inference is the process of deducing properties about a population or probability distribution from data using Bayes’ Theorem.

There are various questions we can answer with inference. We’ve already performed one type of inference, marginal inference. We calculated the overall probability of survival. Given our network with three variables, we tried to find the probability of one variable, Survival.

The posterior inference is the second type of inference. It aims to find the posterior distribution $P(H | E=e)$ for a hidden variable $H$, given some evidence $E=e$. Simply put, we’ll infer the posterior probabilities by applying the Bayes rule. For example, given that we know the passenger’s sex and age, what was their chance to survive? We perform this type of inference when we use our Bayesian network to predict the survival of a single passenger.

The maximum-a-posteriori (MAP) inference is the third type of inference. It’s a variational approach for fitting model parameters to training data. We can use it to estimate the distribution of a hidden variable that best explains an observed dataset. In general, variational methods approximate the distribution of a hidden variable analytically. Based on a mathematical expression of the distribution of interest, these methods solve alternate expressions close to the original distribution.

Let’s do it.