Telecom Paris Dep. Informatique & Réseaux Nils Holzenberger ← Home page April 2024

Neuro-Symbolic Artificial Intelligence

with

Simon Coumes

and

Zacchary Sadeddine

➜            other AI courses

We saw how ProbLog programs can be used to express probabilistic models, then compute the probabilities of certain predicates. Sometimes, the structure of the underlying mechanism is known, but not the exact parameters of the mechanism. Let’s start with the example below.

Alice and Bob like to roll dice. They play this simple game: each have two dice. Alice starts, and rolls her pair of dice, then records the sum. Then it’s Bob’s turn, and Bob rolls his pair of dice, also recording the sum. Whoever rolls the largest sum wins the round. Bob is using biased dice. While he would like his dice to always land on 6, that would be too obvious, so the bias is more subtle, only slightly skewing the probabilities towards 6, and increasing the expected sum of both dice. Alice is suspecting this, and would like to estimate the bias of Bob’s dice, assuming they are equal.

If you have been outside of a building sometime in the past month, you know that weather can change quickly. Some changes are predictable, some aren’t. One simple way of modeling the weather is to use a Hidden Markov Model. Here is a simple summary of what an HMM is.
Assume that there is an automaton with \(n\) states. It can shift from one state to the next, with a certain probability, or stay in its current state. Those probabilities can be described by a matrix \(T\), where \(T_{i,j}\) is the probability that the automaton goes from state \(i\) to state \(j\). Naturally, \(\sum \limits_{j=1}^n T_{i,j} = 1\). In each state, the automaton can emit a symbol from a finite list of \(m\) symbols, with probability \(E_{i,j}\) where \(1 \leq i \leq n\) and \(1 \leq j \leq m\). \(E_{i,j}\) is the probability that the automaton will emit symbol number \(j\) when in state number \(i\). Of course, \(\sum \limits_{j=1}^n E_{i,j} = 1\). Last ingredient of this automaton, it starts from state \(i\) with probability \(s_i\).
An HMM starts in some state, then emits a symbol. Then, it moves on to the next state, and emits another symbol. Then, it moves to another state, and so on. This happens in discrete time, i.e. \(t=0, 1, 2, 3, ...\). We only observe the symbols that were emitted, without knowledge of the HMM’s state (hence the name Hidden Markov Model).

Let’s write an HMM that represents a fictitious weather model. In that fictitious model, there are 6 types of weather:

• 'cloudy-dry'
• 'cloudy-rainy'
• 'sunshine'
• 'fog'
• 'foggy-rainy'
• 'sunshine+rain'

Each of these is mutually exclusive; it cannot be both 'fog' and 'sunshine'. The HMM that generates these observable states of the weather has 3 hidden states (0, 1 and 2) respectively corresponding to dry, wet and funny weather.
First, the weather can start in state 0 or 1, each with probability .5: .5::start(0) ; .5::start(1). In the dry state, the weather can be ‘cloudy-dry’ with probability .3 or ‘sunshine’ with probability .7, but not both at the same time. We express this with the following ProbLog disjunction: .3::emit(Time, 0, 'cloudy-dry'); .7::emit(Time, 0, 'sunshine').. In the funny state, the weather is invariably 'sunshine+rain', which we encode with this deterministic rule: emit(_, 2, 'sunshine+rain').

The weather changes from one hidden state to the next. To say that from state 0 (dry), the weather stays in state 0 with probability .5, goes to state 1 (wet) with probability .3, and to state 2 (funny) with probability .2, we use this annotated disjunction:
.5::trans(Time, 0, 0); .3::trans(Time, 0, 1); .2::trans(Time, 0, 2).
To say that from state 1, the weather stays in state 1 with probability .5, goes to state 0 with probability .3, and to state 2 with probability .2, we write:
.3::trans(Time, 1, 0); .5::trans(Time, 1, 1); .2::trans(Time, 1, 2).

Combine the code you have written -- and that which was shown up to here about start, trans and emit -- with this implementation of HMMs in ProbLog. The query query(observe_sequence([X1,X2,X3])). will return all possible sequences of 3 weather types along with their probabilities. However, the number of possible sequences is \(m^t\), so that the query query(observe([X1,X2,X3,X4,X5,X6,X7,X8])). is unlikely to finish in a reasonable time on your laptop. However, it is possible to sample from this program by running
problog sample filename.pl -N 100
With this command, you can also estimate the probability of all 8-symbol sequences, using
problog sample filename.pl --estimate
and setting a maximum time of sampling or a maximum number of samples -- very convenient for intractable problems.

Here is a file containing samples from this weather program, formatted as evidence for parameter learning. Let’s see if we can recover the parameters of the HMM that served to produce them.

Our goal is to learn the parameters of an HMM, in order to fit the data we sampled from the previous weather model. Of course, this is not something you would do in practice, but this is a good exercise to get familiar with ProbLog’s parameter learning. Hopefully, ProbLog should be able to recover the parameters of the HMM model used to produce the data.

Now let’s learn those parameters. Use the code with the learnable parameters, and combine it with weather-base.pl. Use evidence.pl, containing samples from the weather program and run
problog lfi weather-learn.pl evidence.pl -O learned-model.pl -v
learned-model.pl contains the probabilities learned by ProbLog. Compare that to the original probabilities -- do they match?
To learn the parameters, ProbLog adjusts the parameters to maximize the observed evidence. This is the maximum likelihood estimation of the parameters. To achieve that, ProbLog uses expectation maximization.

We have worked with an HMM where transition probabilities are fixed, i.e. the matrices \(T\) and \(E\) are fixed across states. For an example of probabilities that change depending on the state, see this simplified model of sleep I coded.

• Inference and learning in probabilistic logic programs using weighted Boolean formulas or alternate link.
• Neural probabilistic logic programming in DeepProbLog (slightly more up to date than paper below)
• DeepProbLog: neural probabilistic logic programming

• ProbLog website
• ProbLog installation instructions
• ProbLog online interface
• ProbLog documentation
• Deep ProbLog repository

            

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