Add mixed distribution support to TrueMeasure

[Laasya-73]

Summary

This PR adds support for finite mixed distributions through a new Mixture TrueMeasure.

Given component distributions with densities

$$\rho_0(x), \rho_1(x), \ldots, \rho_{s-1}(x)$$

and probabilities

$$p_0, p_1, \ldots, p_{s-1}, \qquad p_j > 0, \qquad \sum_{j=0}^{s-1} p_j = 1,$$

the resulting mixed distribution has density

$$\rho(x) = \sum_{j=0}^{s-1} p_j \rho_j(x).$$

To sample from the mixture, Mixture uses one additional uniform coordinate. For a component dimension d, the outer sampler therefore has dimension d + 1.

For

$$u = (u_0, u_1, \ldots, u_d) \in [0,1]^{d+1},$$

the first coordinate $u_0$ selects a component according to the cumulative probabilities, and the remaining coordinates $(u_1, \ldots, u_d)$ are passed through the selected component's existing TrueMeasure transform. The resulting sample is in dimension d.

Implementation

• Adds a general Mixture TrueMeasure that can combine different compatible TrueMeasure components.
• Requires all components to have the same output dimension.
• Validates component types, probabilities, and the required d + 1 sampler dimension.
• Uses each selected component's recursive transform, so composed TrueMeasures are preserved.
• Computes mixture weights as the probability-weighted sum of the component weights.
• Supports replicated samples and same-dimension spawning.
• Exposes Mixture through the public qmcpy API.

A new demo notebook shows the selection mechanism as well as mixtures involving different component types, including Gaussian, zero-inflated, bounded, heavy-tailed, composed, and multidimensional examples.

Tests

Added focused tests for component selection, cumulative-probability boundaries, validation, mixture weights, composed transforms, spawning, and replicated samples.

[fjhickernell]

Thanks for doing this Laasya.

We talked and I requested an example showing the advantage of mixture sampling over non-mixture sampling. Let me know when you have added that.

[Laasya-73]

Thanks for doing this Laasya.

We talked and I requested an example showing the advantage of mixture sampling over non-mixture sampling. Let me know when you have added that.

Sure!!