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This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $f(x)$ such that:

$$ g(y) = \int_{\text{supp}(X)} p(y\mid x)\, f(x)\, \text{d}x,\quad \forall y\in \text{supp}(Y) $$

In general, this is an ill-posed inverse problem, and thus it is challenging for both analytical and numerical approaches. As @Glen_b mentions, a characterization of the support of $X$ is needed, as some approaches start off with its discretization.

For instance, notice that:

$$ f(x) = \int q(x\mid y)\,g(y)\,\text{d}y = \int \frac{p(y\mid x)\, f(x)}{\int p(y\mid \tilde{x})\, f(\tilde{x})\,\text{d}\tilde{x}}\,g(y)\,\text{d}y $$

This motivates a simple fixed-point iteration to solve for $f(x)$ in the case of $X$ being discrete with support $\{c_1,\cdots, c_d\}$, $d\geq 2$, and $p(y\mid x)$ being available for direct evaluation:

1. Initialize $p_0(x)=1/d$, $\forall x\in\{c_1,\cdots, c_d\}$
2. For each iteration step $n$, draw $m$ i.i.d. samples from $g(y)$, and let:

$$ f_{n}(x) = \frac{1}{m}\sum_{j=1}^m \frac{p(y_i\mid x)\, f_{n-1}(x)}{\sum_{i=1}^d p(y_j\mid c_i)\, f_{n-1}(c_i)},\quad \forall x\in\{c_1,\cdots, c_d\} $$

The motivation is that, for a sufficiently large $N$, $f_N(x)\approx f(x)$ on its support. This is based on:

Kondor (1983). Method of convergent weights — An iterative procedure for solving Fredholm's integral equations of the first kind. Nuclear Instruments and Methods in Physics Research, Volume 216, Issues 1–2, 1983, Pages 177-181, ISSN 0167-5087, https://doi.org/10.1016/0167-5087(83)90348-4.

There are more refined approaches including:

• Expectation Maximization Smoothing (EMS): Silverman et al (1990). A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography. Journal of the Royal Statistical Society. Series B (Methodological), 52(2), 271–324. http://www.jstor.org/stable/2345438
• Iterative Bayes (IB): Ma (2011). Indirect density estimation using the iterative Bayes algorithm. Computational Statistics & Data Analysis, Volume 55, Issue 3, 2011, Pages 1180-1195, ISSN 0167 9473, https://doi.org/10.1016/j.csda.2010.09.018.
• Sequential Monte Carlo (SMC): Crucinio et al (2023). A Particle Method for Solving Fredholm Equations of the First Kind. Journal of the American Statistical Association, 118(542), 937–947. https://doi.org/10.1080/01621459.2021.1962328

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $p(x)$ such that:

$$ p(y) = \int_{\text{supp}(X)} p(y\mid x)\, p(x)\, \text{d}x,\quad \forall y\in \text{supp}(Y) $$

In general, this is an ill-posed inverse problem, and thus it is challenging for both analytical and numerical approaches. As @Glen_b mentions, a characterization of the support of $X$ is needed, as some approaches start off with its discretization.

For instance, notice that:

$$ p(x) = \int p(x\mid y)\,p(y)\,\text{d}y = \int \frac{p(y\mid x)\, p(x)}{\int p(y\mid \tilde{x})\, p(\tilde{x})\,\text{d}\tilde{x}}\,p(y)\,\text{d}y $$

This motivates a simple fixed-point iteration to solve for $p(x)$ in the case of $X$ being discrete with support $\{c_1,\cdots, c_d\}$, $d\geq 2$, and $p(y\mid x)$ being available for direct evaluation:

1. Initialize $p_0(x)=1/d$, $\forall x\in\{c_1,\cdots, c_d\}$
2. For each iteration step $n$, draw $m$ i.i.d. samples from $p(y)$, and let:

$$ p_{n}(x) = \frac{1}{m}\sum_{j=1}^m \frac{p(y_i\mid x)\, p_{n-1}(x)}{\sum_{i=1}^d p(y_j\mid c_i)\, p_{n-1}(c_i)},\quad \forall x\in\{c_1,\cdots, c_d\} $$

The motivation is that, for a sufficiently large $N$, $p_N(x)\approx p(x)$ in its support. This is based on:

Kondor (1983). Method of convergent weights — An iterative procedure for solving Fredholm's integral equations of the first kind. Nuclear Instruments and Methods in Physics Research, Volume 216, Issues 1–2, 1983, Pages 177-181, ISSN 0167-5087, https://doi.org/10.1016/0167-5087(83)90348-4.

There are more refined approaches including:

• Expectation Maximization Smoothing (EMS): Silverman et al (1990). A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography. Journal of the Royal Statistical Society. Series B (Methodological), 52(2), 271–324. http://www.jstor.org/stable/2345438
• Iterative Bayes (IB): Ma (2011). Indirect density estimation using the iterative Bayes algorithm. Computational Statistics & Data Analysis, Volume 55, Issue 3, 2011, Pages 1180-1195, ISSN 0167 9473, https://doi.org/10.1016/j.csda.2010.09.018.
• Sequential Monte Carlo (SMC): Crucinio et al (2023). A Particle Method for Solving Fredholm Equations of the First Kind. Journal of the American Statistical Association, 118(542), 937–947. https://doi.org/10.1080/01621459.2021.1962328

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $p(x)$ such that:

$$ p(y) = \int_{\text{supp}(X)} p(y\mid x)\, p(x)\, \text{d}x,\quad \forall y\in \text{supp}(Y) $$

In general, this is an ill-posed inverse problem, and thus it is challenging for both analytical and numerical approaches. As @Glen_b mentions, a characterization of the support of $X$ is needed, as some approaches start off with its discretization.

For instance, notice that:

$$ p(x) = \int p(x\mid y)\,p(y)\,\text{d}y = \int \frac{p(y\mid x)\, p(x)}{\int p(y\mid \tilde{x})\, p(\tilde{x})\,\text{d}\tilde{x}}\,p(y)\,\text{d}y $$

This motivates a simple fixed-point iteration to solve for $p(x)$ in the case of $X$ being discrete with support $\{c_1,\cdots, c_d\}$, $d\geq 2$, and $p(y\mid x)$ being available for direct evaluation:

1. Initialize $p_0(x)=1/d$, $\forall x\in\{c_1,\cdots, c_d\}$
2. For each iteration step $n$, draw $m$ i.i.d. samples from $p(y)$, and let:

$$ p_{n}(x) = \frac{1}{m}\sum_{j=1}^m \frac{p(y_i\mid x)\, p_{n-1}(x)}{\sum_{i=1}^d p(y_j\mid c_i)\, p_{n-1}(c_i)},\quad \forall x\in\{c_1,\cdots, c_d\} $$

The motivation is that, for a sufficiently large $N$, $p_N(x)\approx p(x)$ in its support. This is based on:

Kondor (1983). Method of convergent weights — An iterative procedure for solving Fredholm's integral equations of the first kind. Nuclear Instruments and Methods in Physics Research, Volume 216, Issues 1–2, 1983, Pages 177-181, ISSN 0167-5087, https://doi.org/10.1016/0167-5087(83)90348-4.

There are more refined approaches including:

• Expectation Maximization Smoothing (EMS): Silverman et al (1990). A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography. Journal of the Royal Statistical Society. Series B (Methodological), 52(2), 271–324. http://www.jstor.org/stable/2345438
• Iterative Bayes (IB): Ma (2011). Indirect density estimation using the iterative Bayes algorithm. Computational Statistics & Data Analysis, Volume 55, Issue 3, 2011, Pages 1180-1195, ISSN 0167 9473, https://doi.org/10.1016/j.csda.2010.09.018.
• Sequential Monte Carlo (SMC): Crucinio et al (2023). A Particle Method for Solving Fredholm Equations of the First Kind. Journal of the American Statistical Association, 118(542), 937–947. https://doi.org/10.1080/01621459.2021.1962328

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $f(x)$ such that:

$$ g(y) = \int_{\text{supp}(X)} p(y\mid x)\, f(x)\, \text{d}x,\quad \forall y\in \text{supp}(Y) $$

In general, this is an ill-posed inverse problem, and thus it is challenging for both analytical and numerical approaches. As @Glen_b mentions, a characterization of the support of $X$ is needed, as some approaches start off with its discretization.

For instance, notice that:

$$ f(x) = \int q(x\mid y)\,g(y)\,\text{d}y = \int \frac{p(y\mid x)\, f(x)}{\int p(y\mid \tilde{x})\, f(\tilde{x})\,\text{d}\tilde{x}}\,g(y)\,\text{d}y $$

This motivates a simple fixed-point iteration to solve for $f(x)$ in the case of $X$ being discrete with support $\{c_1,\cdots, c_d\}$, $d\geq 2$, and $p(y\mid x)$ being available for direct evaluation:

1. Initialize $p_0(x)=1/d$, $\forall x\in\{c_1,\cdots, c_d\}$
2. For each iteration step $n$, draw $m$ i.i.d. samples from $g(y)$, and let:

$$ f_{n}(x) = \frac{1}{m}\sum_{j=1}^m \frac{p(y_i\mid x)\, f_{n-1}(x)}{\sum_{i=1}^d p(y_j\mid c_i)\, f_{n-1}(c_i)},\quad \forall x\in\{c_1,\cdots, c_d\} $$

The motivation is that, for a sufficiently large $N$, $f_N(x)\approx f(x)$ on its support. This is based on:

Kondor (1983). Method of convergent weights — An iterative procedure for solving Fredholm's integral equations of the first kind. Nuclear Instruments and Methods in Physics Research, Volume 216, Issues 1–2, 1983, Pages 177-181, ISSN 0167-5087, https://doi.org/10.1016/0167-5087(83)90348-4.

There are more refined approaches including:

• Expectation Maximization Smoothing (EMS): Silverman et al (1990). A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography. Journal of the Royal Statistical Society. Series B (Methodological), 52(2), 271–324. http://www.jstor.org/stable/2345438
• Iterative Bayes (IB): Ma (2011). Indirect density estimation using the iterative Bayes algorithm. Computational Statistics & Data Analysis, Volume 55, Issue 3, 2011, Pages 1180-1195, ISSN 0167 9473, https://doi.org/10.1016/j.csda.2010.09.018.
• Sequential Monte Carlo (SMC): Crucinio et al (2023). A Particle Method for Solving Fredholm Equations of the First Kind. Journal of the American Statistical Association, 118(542), 937–947. https://doi.org/10.1080/01621459.2021.1962328

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $p(x)$ such that:

$$ p(y) = \int_{\text{supp}(X)} p(y\mid x)\, p(x)\, \text{d}x,\quad \forall y\in \text{supp}(Y) $$

In general, this is an ill-posed inverse problem, and thus it is challenging for both analytical and numerical approaches. As @Glen_b mentions, a characterization of the support of $X$ is needed, as some approaches start off with its discretization.

For instance, notice that:

$$ p(x) = \int p(x\mid y)\,p(y)\,\text{d}y = \int \frac{p(y\mid x)\, p(x)}{\int p(y\mid \tilde{x})\, p(\tilde{x})\,\text{d}\tilde{x}}\,p(y)\,\text{d}y $$

A simple fixed-point iteration can be implemented to solve for $p(x)$ in the case of $X$ being discrete with support $\{c_1,\cdots, c_d\}$, $d\geq 2$, and $p(y\mid x)$ being available for direct evaluation:

1. Initialize $p_0(x)=1/d$, $\forall x\in\{c_1,\cdots, c_d\}$
2. For each iteration step $n$, draw $m$ i.i.d. samples from $p(y)$, and let:

$$ p_{n}(x) = \sum_{j=1}^m \frac{p(y_i\mid x)\, p_{n-1}(x)}{\sum_{i=1}^d p(y_j\mid c_i)\, p_{n-1}(c_i)},\quad \forall x\in\{c_1,\cdots, c_d\} $$

The motivation is that, for a sufficiently large $N$, $p_N(x)\approx p(x)$ in its support. This is based on:

Kondor (1983). Method of convergent weights — An iterative procedure for solving Fredholm's integral equations of the first kind. Nuclear Instruments and Methods in Physics Research, Volume 216, Issues 1–2, 1983, Pages 177-181, ISSN 0167-5087, https://doi.org/10.1016/0167-5087(83)90348-4.

There are more refined approaches including:

• Expectation Maximization Smoothing (EMS): Silverman et al (1990). A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography. Journal of the Royal Statistical Society. Series B (Methodological), 52(2), 271–324. http://www.jstor.org/stable/2345438
• Iterative Bayes (IB): Ma (2011). Indirect density estimation using the iterative Bayes algorithm. Computational Statistics & Data Analysis, Volume 55, Issue 3, 2011, Pages 1180-1195, ISSN 0167 9473, https://doi.org/10.1016/j.csda.2010.09.018.
• Sequential Monte Carlo (SMC): Crucinio et al (2023). A Particle Method for Solving Fredholm Equations of the First Kind. Journal of the American Statistical Association, 118(542), 937–947. https://doi.org/10.1080/01621459.2021.1962328

This problem is equivalent to solving a Fredholm integral equation of the first kind. This is, solving for $p(x)$ such that:

$$ p(y) = \int_{\text{supp}(X)} p(y\mid x)\, p(x)\, \text{d}x,\quad \forall y\in \text{supp}(Y) $$

In general, this is an ill-posed inverse problem, and thus it is challenging for both analytical and numerical approaches. As @Glen_b mentions, a characterization of the support of $X$ is needed, as some approaches start off with its discretization.

For instance, notice that:

$$ p(x) = \int p(x\mid y)\,p(y)\,\text{d}y = \int \frac{p(y\mid x)\, p(x)}{\int p(y\mid \tilde{x})\, p(\tilde{x})\,\text{d}\tilde{x}}\,p(y)\,\text{d}y $$

This motivates a simple fixed-point iteration to solve for $p(x)$ in the case of $X$ being discrete with support $\{c_1,\cdots, c_d\}$, $d\geq 2$, and $p(y\mid x)$ being available for direct evaluation:

1. Initialize $p_0(x)=1/d$, $\forall x\in\{c_1,\cdots, c_d\}$
2. For each iteration step $n$, draw $m$ i.i.d. samples from $p(y)$, and let:

$$ p_{n}(x) = \frac{1}{m}\sum_{j=1}^m \frac{p(y_i\mid x)\, p_{n-1}(x)}{\sum_{i=1}^d p(y_j\mid c_i)\, p_{n-1}(c_i)},\quad \forall x\in\{c_1,\cdots, c_d\} $$

The motivation is that, for a sufficiently large $N$, $p_N(x)\approx p(x)$ in its support. This is based on:

Kondor (1983). Method of convergent weights — An iterative procedure for solving Fredholm's integral equations of the first kind. Nuclear Instruments and Methods in Physics Research, Volume 216, Issues 1–2, 1983, Pages 177-181, ISSN 0167-5087, https://doi.org/10.1016/0167-5087(83)90348-4.

There are more refined approaches including:

• Expectation Maximization Smoothing (EMS): Silverman et al (1990). A Smoothed EM Approach to Indirect Estimation Problems, with Particular, Reference to Stereology and Emission Tomography. Journal of the Royal Statistical Society. Series B (Methodological), 52(2), 271–324. http://www.jstor.org/stable/2345438
• Iterative Bayes (IB): Ma (2011). Indirect density estimation using the iterative Bayes algorithm. Computational Statistics & Data Analysis, Volume 55, Issue 3, 2011, Pages 1180-1195, ISSN 0167 9473, https://doi.org/10.1016/j.csda.2010.09.018.
• Sequential Monte Carlo (SMC): Crucinio et al (2023). A Particle Method for Solving Fredholm Equations of the First Kind. Journal of the American Statistical Association, 118(542), 937–947. https://doi.org/10.1080/01621459.2021.1962328