Bayesian parameter estimation with a pre-computed grid of function calls
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I am estimating the parameters of an observed galaxy based on simulations that I have run. The simulation is a function $f$ that takes arguments $x_1, x_2, ldots x_k$ (describing things like the luminosity and distance of the galaxy). My simulation produces $y$ as output: $f(mathbf x) = y$. The simulation takes a (very) long time to run, and so I have pre-computed a large grid (~25,000) of function calls for values of $x_1, x_2, ldots x_k$ that were randomly selected in a reasonable interval.
I have one measurement of the galaxy, $y pm sigma$, where $sigma$ is the uncertainty of the measurement. I want to find the posterior distribution of parameters $mathbf x$ from my pre-computed grid that best match the observed $y$ (and its uncertainty). Importantly, different inputs $mathbf x$ can produce the same output $y$.
I can define a likelihood as follows:
$$mathcal L (mathbf x | y) = frac{(f(mathbf x) - y)^2}{sigma}$$
and easily compute it for all the pre-computed models.
I can define a prior distribution, but for now we can say it is the uniform distribution according to the bounds in which I have generated the models.
How can I turn this information into the posterior distribution $P(mathbf x | y)$ to determine the input parameters $mathbf x = x_1, x_2, ldots x_k$ that correspond to the observation?
Normally MCMC or optimization methods would be used for this problem, but they rely on the ability to call $f$, whereas here I want to use a pre-computed grid.
statistics bayesian parameter-estimation
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
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add a comment |
$begingroup$
I am estimating the parameters of an observed galaxy based on simulations that I have run. The simulation is a function $f$ that takes arguments $x_1, x_2, ldots x_k$ (describing things like the luminosity and distance of the galaxy). My simulation produces $y$ as output: $f(mathbf x) = y$. The simulation takes a (very) long time to run, and so I have pre-computed a large grid (~25,000) of function calls for values of $x_1, x_2, ldots x_k$ that were randomly selected in a reasonable interval.
I have one measurement of the galaxy, $y pm sigma$, where $sigma$ is the uncertainty of the measurement. I want to find the posterior distribution of parameters $mathbf x$ from my pre-computed grid that best match the observed $y$ (and its uncertainty). Importantly, different inputs $mathbf x$ can produce the same output $y$.
I can define a likelihood as follows:
$$mathcal L (mathbf x | y) = frac{(f(mathbf x) - y)^2}{sigma}$$
and easily compute it for all the pre-computed models.
I can define a prior distribution, but for now we can say it is the uniform distribution according to the bounds in which I have generated the models.
How can I turn this information into the posterior distribution $P(mathbf x | y)$ to determine the input parameters $mathbf x = x_1, x_2, ldots x_k$ that correspond to the observation?
Normally MCMC or optimization methods would be used for this problem, but they rely on the ability to call $f$, whereas here I want to use a pre-computed grid.
statistics bayesian parameter-estimation
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
$endgroup$
add a comment |
$begingroup$
I am estimating the parameters of an observed galaxy based on simulations that I have run. The simulation is a function $f$ that takes arguments $x_1, x_2, ldots x_k$ (describing things like the luminosity and distance of the galaxy). My simulation produces $y$ as output: $f(mathbf x) = y$. The simulation takes a (very) long time to run, and so I have pre-computed a large grid (~25,000) of function calls for values of $x_1, x_2, ldots x_k$ that were randomly selected in a reasonable interval.
I have one measurement of the galaxy, $y pm sigma$, where $sigma$ is the uncertainty of the measurement. I want to find the posterior distribution of parameters $mathbf x$ from my pre-computed grid that best match the observed $y$ (and its uncertainty). Importantly, different inputs $mathbf x$ can produce the same output $y$.
I can define a likelihood as follows:
$$mathcal L (mathbf x | y) = frac{(f(mathbf x) - y)^2}{sigma}$$
and easily compute it for all the pre-computed models.
I can define a prior distribution, but for now we can say it is the uniform distribution according to the bounds in which I have generated the models.
How can I turn this information into the posterior distribution $P(mathbf x | y)$ to determine the input parameters $mathbf x = x_1, x_2, ldots x_k$ that correspond to the observation?
Normally MCMC or optimization methods would be used for this problem, but they rely on the ability to call $f$, whereas here I want to use a pre-computed grid.
statistics bayesian parameter-estimation
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
$endgroup$
I am estimating the parameters of an observed galaxy based on simulations that I have run. The simulation is a function $f$ that takes arguments $x_1, x_2, ldots x_k$ (describing things like the luminosity and distance of the galaxy). My simulation produces $y$ as output: $f(mathbf x) = y$. The simulation takes a (very) long time to run, and so I have pre-computed a large grid (~25,000) of function calls for values of $x_1, x_2, ldots x_k$ that were randomly selected in a reasonable interval.
I have one measurement of the galaxy, $y pm sigma$, where $sigma$ is the uncertainty of the measurement. I want to find the posterior distribution of parameters $mathbf x$ from my pre-computed grid that best match the observed $y$ (and its uncertainty). Importantly, different inputs $mathbf x$ can produce the same output $y$.
I can define a likelihood as follows:
$$mathcal L (mathbf x | y) = frac{(f(mathbf x) - y)^2}{sigma}$$
and easily compute it for all the pre-computed models.
I can define a prior distribution, but for now we can say it is the uniform distribution according to the bounds in which I have generated the models.
How can I turn this information into the posterior distribution $P(mathbf x | y)$ to determine the input parameters $mathbf x = x_1, x_2, ldots x_k$ that correspond to the observation?
Normally MCMC or optimization methods would be used for this problem, but they rely on the ability to call $f$, whereas here I want to use a pre-computed grid.
statistics bayesian parameter-estimation
statistics bayesian parameter-estimation
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
asked Dec 14 '18 at 11:19
rhombidodecahedronrhombidodecahedron
6281619
6281619
add a comment |
add a comment |
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