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Given $n$ probability density functions ($p_1$, ..., $p_n$) with known distributions, what are the ways of estimating the weights ($w_1$, ..., $w_n$) of these component distributions given a sample from a distribution $f$ which is some mixture of these probability density functions, i.e.

$f = \sum^n_{i=1} w_i p_i$.

I've found many methods looking at the seemingly harder problem, where the component distributions are unknown, but haven't had much luck with this simpler case.

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The EM algorithm applies most straightforwardly to this simpler case, with the $Q(\cdot;\cdot)$ function defined by $$Q(\mathbf p,\mathbf p^{(t)})=\sum_{i=1}^N\sum_{j=1}^n \mathbb P_{\mathbf p^{(t)}}(Z_i=j|x_i)\log p_j(x_i)$$ As does a Bayesian approach putting a prior on $(p_1,\ldots,p_n)$, since the Gibbs sampler step are quite similar to the EM steps.

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Here's the Bayesian solution I ended up using for a mixed discrete distribution. I believe this method is called the Richardson-Lucy algorithm. My original question was more general, but hopefully this will be helpful to someone.

The discrete probability density at a position $k$ in the mixed distribution $f$ given that it is made up of a mix of discrete density functions $p_i$, is given by

$$P(f_k) = \sum_i P(f_k|p_i)P(p_i),\tag{1}\label{eq1}$$

where $P(p_i) = w_i$ (the mixture weight from the original question). Baye's theorem can be stated as

$$P(f_k|p_i)w_i = P(p_i|f_k)P(f_k).\tag{2}\label{eq2}$$

From here we'd like to exclude the term $(p_i|f_k)$ and set up an iterative procedure to solve for $w_i$.

Similar to equation 1

$$w_i = \sum_k P(p_i|f_k)P(f_k),\tag{3}\label{eq3},$$

therefore, substituting equation 2 into 3, we get

$$w_i = \sum_k \frac{P(f_k|p_i)P(f_k)w_i}{P(f_k)}\tag{4}\label{eq4}.$$

Then, substituting equation 1 into 4, we get

$$w_i = \sum_k \frac{P(f_k|p_i)P(f_k)w_i}{\sum_j P(f_k|p_j)P(p_j)}\tag{5}\label{eq5}.$$

Given that the desired solution ($w_i$) appears on both sides of the equation we can set up an iterative process to estimate $w_i$ so that equation 5 becomes

$$w_{r+1,i} = w_{r,i} \sum_k \frac{P(f_k|p_i)P(f_k)}{\sum_j P(f_k|p_j)w_{r,i}}\tag{6}\label{eq6},$$

$$r=\{0,1,..\}.$$

In my particular case I could directly measure $P(f_k|p_i)$ for each $p_i$ (the component distributions were known) so equation 6 was my solution.

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