Multinomial-dirichlet with fractional counts

Question

Suppose a lepidopterologist wants to estimate the relative proportions of three different species of butterfly. They go out into the field and count $N$ butterflies and record the number of each species $(N_A,N_B,N_C)$. This is a Bayesian lepidopterologist, however, and so they want to use the Dirichlet-Multinomial conjugate pair to obtain a Dirichlet posterior on the probabilities $(p_A,p_B,p_C)$ that an observed butterfly is of each species. If the counts are integer and the prior hyperparameters of the Dirichlet are $(\alpha_A,\alpha_B,\alpha_C)$, then the posterior hyperparameters are simply $(\alpha_A+N_A,\alpha_B+N_B,\alpha_C+N_C)$.

However, the lepidopterologist is not sure of the species of some of the butterflies and so assigns a probability $(p_A^i,p_B^i,p_C^i)$ to each individual butterfly $i$ that it is of each species. This probability is calculated quantitatively and under the assumption that all species are equally likely. These probabilities capture genuine uncertainty in the species determination of a butterly, perhaps because some of members of species A can look like members of species B. These probabilities are also unbiased, i.e. if we observed an infinite number of butterflies with $p_A^i = 0.54$, then 54% of those butterflies would be in species A.

Naively, the posterior hyperparameters would be of the same form but with $N_A$ now being the sum of the probabilities that each butterfly is of species $A$, etc.

Is this correct? If incorrect, are there limits where this is a good approximation?

TLDR: Can we generalise the dirichlet-multinomial conjugate prior-likelihood to fractional count data?

Edit for clarity and to add example data:

We observe $N=4$ butterflies and record that their probability of being a member of each species is:

Butterfly 1 - $p_A = 0.34,p_B = 0.42,p_C = 0.24$

Butterfly 2 - $p_A = 0.14,p_B = 0.23,p_C = 0.63$

Butterfly 3 - $p_A = 0.97,p_B = 0.01,p_C = 0.02$

Butterfly 4 - $p_A = 0.00,p_B = 0.67,p_C = 0.33$

Answer 1

I would change the problem slightly:

$i$ = butterfly observation

$j$ = butterfly class (A,B,C)

$X_{i,j}$ = subjective probability that a butterfly is in class $j$ that is recorded by the observer $i$

$\rho$ = vector of true proportion of butterflies in the observation area of each class

$\alpha_i$ vector of dirichlet parameters for butterfly $i$

Assume that $X_i \sim dirichlet(\alpha_i)$ so that $\alpha_i$ describes the underlying distribution of the recorded observations. Each butterfly might have a different $\alpha_i$ describing the difficulty of classifying that particular butterfly. In other words, if multiple people tried to classify that same butterfly and they all recorded probabilities of classes then their observations would form a dirichlet distribution. We only see one set of probabilities per butterfly.

Now, to simplify, assume that all the $\alpha_i$ are coming from the same underlying distribution which is centered on the $\rho$.

As a Bayesian, you can put a prior on $\alpha$ and then update using MCMC $X_i \sim dirichlet(\alpha)$

As a frequentist, you can get the dirichlet distribution estimates directly.

$\bar{X_j} = \alpha_j / \sum \alpha_j$

$s^2_{X_1} = \frac{z_1 (1 - z_1)}{1 + \sum \alpha_j}$ where $z_1 = \alpha_1 / \sum \alpha_j$

Note: you only need the second equation for one of the classes $j$.