Analytical approximation for logit-normal-binomial distribution












1












$begingroup$


As I understand, there is no closed form expression for



$$f(x, mu, sigma) = int_0^1 p^{(x-1)}(1-p)^{n-x-1}expleft(-{(text{logit}(p) -mu)^2 over 2sigma^2}right)dp.$$



Is it possible to obtain an analytical approximation for this?






integration numerical-methods approximation approximate-integration







share|cite|improve this question









$endgroup$












  • $begingroup$
    Do you mean "asymptotic"? There are numerous analytic approximations to any continuous function.
    $endgroup$
    – user14717
    Dec 17 '18 at 7:28












  • $begingroup$
    I suppose there must be some that are more computationally efficient/reach small error with a smaller number of terms? I need to insert this as part of a statistical model so a tradeoff between accuracy and efficiency is needed.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 11:19










  • $begingroup$
    You could use quadrature to evaluate it at many arguments and then do interpolation.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:12










  • $begingroup$
    I suppose that can work. I've never attempted something similar so I had something like a multivariable Taylor series in mind. I guess that theoretical guarantees are irrelevant when there's enough computational power to brute force the approximation and ensure it works over the range of things I care about.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 16:23








  • 1




    $begingroup$
    The theoretical guarantees will be much stronger with an interpolator than with multivariate Taylor series.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:27
















1












$begingroup$


As I understand, there is no closed form expression for



$$f(x, mu, sigma) = int_0^1 p^{(x-1)}(1-p)^{n-x-1}expleft(-{(text{logit}(p) -mu)^2 over 2sigma^2}right)dp.$$



Is it possible to obtain an analytical approximation for this?






integration numerical-methods approximation approximate-integration







share|cite|improve this question









$endgroup$












  • $begingroup$
    Do you mean "asymptotic"? There are numerous analytic approximations to any continuous function.
    $endgroup$
    – user14717
    Dec 17 '18 at 7:28












  • $begingroup$
    I suppose there must be some that are more computationally efficient/reach small error with a smaller number of terms? I need to insert this as part of a statistical model so a tradeoff between accuracy and efficiency is needed.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 11:19










  • $begingroup$
    You could use quadrature to evaluate it at many arguments and then do interpolation.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:12










  • $begingroup$
    I suppose that can work. I've never attempted something similar so I had something like a multivariable Taylor series in mind. I guess that theoretical guarantees are irrelevant when there's enough computational power to brute force the approximation and ensure it works over the range of things I care about.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 16:23








  • 1




    $begingroup$
    The theoretical guarantees will be much stronger with an interpolator than with multivariate Taylor series.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:27














1












1








1





$begingroup$


As I understand, there is no closed form expression for



$$f(x, mu, sigma) = int_0^1 p^{(x-1)}(1-p)^{n-x-1}expleft(-{(text{logit}(p) -mu)^2 over 2sigma^2}right)dp.$$



Is it possible to obtain an analytical approximation for this?






integration numerical-methods approximation approximate-integration







share|cite|improve this question









$endgroup$




As I understand, there is no closed form expression for



$$f(x, mu, sigma) = int_0^1 p^{(x-1)}(1-p)^{n-x-1}expleft(-{(text{logit}(p) -mu)^2 over 2sigma^2}right)dp.$$



Is it possible to obtain an analytical approximation for this?






integration numerical-methods approximation approximate-integration




integration numerical-methods approximation approximate-integration






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 17 '18 at 5:00









zipzapboingzipzapboing

979




979












  • $begingroup$
    Do you mean "asymptotic"? There are numerous analytic approximations to any continuous function.
    $endgroup$
    – user14717
    Dec 17 '18 at 7:28












  • $begingroup$
    I suppose there must be some that are more computationally efficient/reach small error with a smaller number of terms? I need to insert this as part of a statistical model so a tradeoff between accuracy and efficiency is needed.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 11:19










  • $begingroup$
    You could use quadrature to evaluate it at many arguments and then do interpolation.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:12










  • $begingroup$
    I suppose that can work. I've never attempted something similar so I had something like a multivariable Taylor series in mind. I guess that theoretical guarantees are irrelevant when there's enough computational power to brute force the approximation and ensure it works over the range of things I care about.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 16:23








  • 1




    $begingroup$
    The theoretical guarantees will be much stronger with an interpolator than with multivariate Taylor series.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:27


















  • $begingroup$
    Do you mean "asymptotic"? There are numerous analytic approximations to any continuous function.
    $endgroup$
    – user14717
    Dec 17 '18 at 7:28












  • $begingroup$
    I suppose there must be some that are more computationally efficient/reach small error with a smaller number of terms? I need to insert this as part of a statistical model so a tradeoff between accuracy and efficiency is needed.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 11:19










  • $begingroup$
    You could use quadrature to evaluate it at many arguments and then do interpolation.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:12










  • $begingroup$
    I suppose that can work. I've never attempted something similar so I had something like a multivariable Taylor series in mind. I guess that theoretical guarantees are irrelevant when there's enough computational power to brute force the approximation and ensure it works over the range of things I care about.
    $endgroup$
    – zipzapboing
    Dec 17 '18 at 16:23








  • 1




    $begingroup$
    The theoretical guarantees will be much stronger with an interpolator than with multivariate Taylor series.
    $endgroup$
    – user14717
    Dec 17 '18 at 16:27
















$begingroup$
Do you mean "asymptotic"? There are numerous analytic approximations to any continuous function.
$endgroup$
– user14717
Dec 17 '18 at 7:28






$begingroup$
Do you mean "asymptotic"? There are numerous analytic approximations to any continuous function.
$endgroup$
– user14717
Dec 17 '18 at 7:28














$begingroup$
I suppose there must be some that are more computationally efficient/reach small error with a smaller number of terms? I need to insert this as part of a statistical model so a tradeoff between accuracy and efficiency is needed.
$endgroup$
– zipzapboing
Dec 17 '18 at 11:19




$begingroup$
I suppose there must be some that are more computationally efficient/reach small error with a smaller number of terms? I need to insert this as part of a statistical model so a tradeoff between accuracy and efficiency is needed.
$endgroup$
– zipzapboing
Dec 17 '18 at 11:19












$begingroup$
You could use quadrature to evaluate it at many arguments and then do interpolation.
$endgroup$
– user14717
Dec 17 '18 at 16:12




$begingroup$
You could use quadrature to evaluate it at many arguments and then do interpolation.
$endgroup$
– user14717
Dec 17 '18 at 16:12












$begingroup$
I suppose that can work. I've never attempted something similar so I had something like a multivariable Taylor series in mind. I guess that theoretical guarantees are irrelevant when there's enough computational power to brute force the approximation and ensure it works over the range of things I care about.
$endgroup$
– zipzapboing
Dec 17 '18 at 16:23






$begingroup$
I suppose that can work. I've never attempted something similar so I had something like a multivariable Taylor series in mind. I guess that theoretical guarantees are irrelevant when there's enough computational power to brute force the approximation and ensure it works over the range of things I care about.
$endgroup$
– zipzapboing
Dec 17 '18 at 16:23






1




1




$begingroup$
The theoretical guarantees will be much stronger with an interpolator than with multivariate Taylor series.
$endgroup$
– user14717
Dec 17 '18 at 16:27




$begingroup$
The theoretical guarantees will be much stronger with an interpolator than with multivariate Taylor series.
$endgroup$
– user14717
Dec 17 '18 at 16:27










1 Answer
1






active

oldest

votes


















1












$begingroup$

Here's what you need to do:




  • Decide on an interpolator. I suggest a tricubic b-spline, but finding software for this is going to be painful. To understand this interpolant, start in Rainer Kress's Numerical Analysis which introduces it in 1D, learn about the bicubic b-splines in 2D, and then you'll be able to understand the tricubic. If you don't like tricubic b-splines, as an alternative, you might also be able to use multivariate Chebyshev series.


  • Interpolators require data at a particular geometry of points; figure out what those points are for your given interpolator and then evaluate the integral by quadrature at each point. (For tricubic b-splines it's easy: A uniform grid.) It looks like tanh-sinh quadrature is probably the best for this integral but Gaussian or Gauss-Kronrod will also work fine.



Another alternative is just to use quadrature to evaluate $f$ at any point $(x, mu, sigma)$, and ditch the interpolator. This will reduce the speed by a factor of 10 to 100, but since a quadrature takes about 500ns-1$mu$s, you might not really care.



If you've never done anything like this get ready for some effort shock.






share|cite|improve this answer











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    1 Answer
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    1 Answer
    1






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    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    Here's what you need to do:




    • Decide on an interpolator. I suggest a tricubic b-spline, but finding software for this is going to be painful. To understand this interpolant, start in Rainer Kress's Numerical Analysis which introduces it in 1D, learn about the bicubic b-splines in 2D, and then you'll be able to understand the tricubic. If you don't like tricubic b-splines, as an alternative, you might also be able to use multivariate Chebyshev series.


    • Interpolators require data at a particular geometry of points; figure out what those points are for your given interpolator and then evaluate the integral by quadrature at each point. (For tricubic b-splines it's easy: A uniform grid.) It looks like tanh-sinh quadrature is probably the best for this integral but Gaussian or Gauss-Kronrod will also work fine.



    Another alternative is just to use quadrature to evaluate $f$ at any point $(x, mu, sigma)$, and ditch the interpolator. This will reduce the speed by a factor of 10 to 100, but since a quadrature takes about 500ns-1$mu$s, you might not really care.



    If you've never done anything like this get ready for some effort shock.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Here's what you need to do:




      • Decide on an interpolator. I suggest a tricubic b-spline, but finding software for this is going to be painful. To understand this interpolant, start in Rainer Kress's Numerical Analysis which introduces it in 1D, learn about the bicubic b-splines in 2D, and then you'll be able to understand the tricubic. If you don't like tricubic b-splines, as an alternative, you might also be able to use multivariate Chebyshev series.


      • Interpolators require data at a particular geometry of points; figure out what those points are for your given interpolator and then evaluate the integral by quadrature at each point. (For tricubic b-splines it's easy: A uniform grid.) It looks like tanh-sinh quadrature is probably the best for this integral but Gaussian or Gauss-Kronrod will also work fine.



      Another alternative is just to use quadrature to evaluate $f$ at any point $(x, mu, sigma)$, and ditch the interpolator. This will reduce the speed by a factor of 10 to 100, but since a quadrature takes about 500ns-1$mu$s, you might not really care.



      If you've never done anything like this get ready for some effort shock.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Here's what you need to do:




        • Decide on an interpolator. I suggest a tricubic b-spline, but finding software for this is going to be painful. To understand this interpolant, start in Rainer Kress's Numerical Analysis which introduces it in 1D, learn about the bicubic b-splines in 2D, and then you'll be able to understand the tricubic. If you don't like tricubic b-splines, as an alternative, you might also be able to use multivariate Chebyshev series.


        • Interpolators require data at a particular geometry of points; figure out what those points are for your given interpolator and then evaluate the integral by quadrature at each point. (For tricubic b-splines it's easy: A uniform grid.) It looks like tanh-sinh quadrature is probably the best for this integral but Gaussian or Gauss-Kronrod will also work fine.



        Another alternative is just to use quadrature to evaluate $f$ at any point $(x, mu, sigma)$, and ditch the interpolator. This will reduce the speed by a factor of 10 to 100, but since a quadrature takes about 500ns-1$mu$s, you might not really care.



        If you've never done anything like this get ready for some effort shock.






        share|cite|improve this answer











        $endgroup$



        Here's what you need to do:




        • Decide on an interpolator. I suggest a tricubic b-spline, but finding software for this is going to be painful. To understand this interpolant, start in Rainer Kress's Numerical Analysis which introduces it in 1D, learn about the bicubic b-splines in 2D, and then you'll be able to understand the tricubic. If you don't like tricubic b-splines, as an alternative, you might also be able to use multivariate Chebyshev series.


        • Interpolators require data at a particular geometry of points; figure out what those points are for your given interpolator and then evaluate the integral by quadrature at each point. (For tricubic b-splines it's easy: A uniform grid.) It looks like tanh-sinh quadrature is probably the best for this integral but Gaussian or Gauss-Kronrod will also work fine.



        Another alternative is just to use quadrature to evaluate $f$ at any point $(x, mu, sigma)$, and ditch the interpolator. This will reduce the speed by a factor of 10 to 100, but since a quadrature takes about 500ns-1$mu$s, you might not really care.



        If you've never done anything like this get ready for some effort shock.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 17 '18 at 17:04

























        answered Dec 17 '18 at 16:57









        user14717user14717

        3,8631120




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