This module contains a simple (and musically rather naive) probabilistic model of protovoice derivations. This model can be used to sample a derivation, evaluate a derivations probability, or infer posterior distributions of the model parmeters from given derivations (i.e., "learn" the model's probabilities).

This model is a locally conjugate model: It samples a derivation using a sequence of random decisions with certain probabilities. These probabilities are generally unknown, so they are themselves modeled as random variables with prior distributions. The full model \(p(d, \theta)\) thus splits into \[p(D, \theta) = p(d \mid \theta) \cdot p(\theta),\] the prior over the probability variables \[p(\theta) = \prod_i p(\theta_i),\] and the likelihood of the derivation(s) given these probabilities \[p(D \mid \theta) = \prod_{d \in D} p(d \mid \theta) = \prod_{d \in D} \prod_i p(d_i \mid \theta, d_0, \ldots, d_{i-1}).\] Given all prior decisions, the likelihood of a decision \(d_i\) based on some parameter \(\theta_a\) \[p(d_i \mid \theta, d_{<i})\] is conjugate with the prior of that parameter \(p(\theta_a)\), which means that the posterior of the parameters given one (or several) derivation(s) \(p(\theta \mid D)\) can be computed analytically.

The parameters \(\theta\) and their prior distributions are represented by the higher-kinded type PVParams. Different instantiations of this type (using Hyper or Probs) results in concrete record types that represent prior or posterior distributions or concrete values (probabilities) for the parameters. PVParams also supports jeffreysPrior and uniformPrior as default priors, as well as sampleProbs for sampling from a prior (see Inferenc.Conjugate).

The likelihood \(p(d \mid \theta)\) of a derivation is represented by sampleDerivation. It can be executed under different "modes" (probability monads) for sampling, inference, or tracing (see Inference.Conjugate). The decisions during the derivation are represented by a Trace (here Trace PVParams). In order to learn from a given derivation, the corresponding trace can be obtained using observeDerivation. A combination of getting a trace and learning from it is provided by trainSinglePiece.

A higher-kinded type that represents the global parameters (probabilities) of the model. Use it as 'Hyper PVParams' to represent hyperparameters (priors and posteriors) or as 'Probs PVParams' to represent actual probabilites. Each record field corresponds to one parameter that influences a specific type of decision in the generation process.

sampleDerivation represents a probabilistic program that samples a derivation. that can be interpreted in various modes for

• sampling (sampleTrace, sampleResult),
• inference (evalTraceLogP, getPosterior),
• tracing (showTrace, traceTrace).

observeDerivation takes and existing derivation and returns the corresponding trace.

We need these specialized functions because of a dependency across steps: During a double step (elaborating two transitions), the process must decide whether to elaborate the left or right transition. To normalize the derivation order, we can't elaborate the left transition after the right one. That means that the model *sometimes* has to make the decision to go right, and sometimes not (i.e., after a right split). During parsing, we don't know whether this decision had to be made or not, since we don't know the previous derivation step yet. Therefore, we don't include the decision in the current step, but at the end of the previous one (in generation order), where the context is known. As a consequence, the result of this decision (if made) needs to be passed to the functions scoring the previous step. When parsing, make sure to maintain this information.