Compute the posterior density for r












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I'm trying to understand the posterior distribution. For a simple example if i consider the beta function with parameters $alpha=1=beta$. Then the prior would be uniform in the range of 0 to 1 so $p(r) = 1$. If the likelihood is binomial, since we consider an experiment with two possible outcomes, would it be correct to say that the posterior density is given by the following?
$$p(r|y) = P(y|r) p(r) = binom{n}{y}r^y(1-r)^{n-y}cdot 1$$
Also if anyone could point me to some good resources regarding this it would be greatly appreciated, as i find a lot of confusing stuff online regarding this topic.






probability statistics statistical-inference machine-learning bayesian







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    0












    $begingroup$


    I'm trying to understand the posterior distribution. For a simple example if i consider the beta function with parameters $alpha=1=beta$. Then the prior would be uniform in the range of 0 to 1 so $p(r) = 1$. If the likelihood is binomial, since we consider an experiment with two possible outcomes, would it be correct to say that the posterior density is given by the following?
    $$p(r|y) = P(y|r) p(r) = binom{n}{y}r^y(1-r)^{n-y}cdot 1$$
    Also if anyone could point me to some good resources regarding this it would be greatly appreciated, as i find a lot of confusing stuff online regarding this topic.






    probability statistics statistical-inference machine-learning bayesian







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      I'm trying to understand the posterior distribution. For a simple example if i consider the beta function with parameters $alpha=1=beta$. Then the prior would be uniform in the range of 0 to 1 so $p(r) = 1$. If the likelihood is binomial, since we consider an experiment with two possible outcomes, would it be correct to say that the posterior density is given by the following?
      $$p(r|y) = P(y|r) p(r) = binom{n}{y}r^y(1-r)^{n-y}cdot 1$$
      Also if anyone could point me to some good resources regarding this it would be greatly appreciated, as i find a lot of confusing stuff online regarding this topic.






      probability statistics statistical-inference machine-learning bayesian







      share|cite|improve this question











      $endgroup$




      I'm trying to understand the posterior distribution. For a simple example if i consider the beta function with parameters $alpha=1=beta$. Then the prior would be uniform in the range of 0 to 1 so $p(r) = 1$. If the likelihood is binomial, since we consider an experiment with two possible outcomes, would it be correct to say that the posterior density is given by the following?
      $$p(r|y) = P(y|r) p(r) = binom{n}{y}r^y(1-r)^{n-y}cdot 1$$
      Also if anyone could point me to some good resources regarding this it would be greatly appreciated, as i find a lot of confusing stuff online regarding this topic.






      probability statistics statistical-inference machine-learning bayesian




      probability statistics statistical-inference machine-learning bayesian






      share|cite|improve this question















      share|cite|improve this question













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      share|cite|improve this question








      edited Jan 7 at 21:33







      sn3jd3r

















      asked Jan 7 at 21:26









      sn3jd3rsn3jd3r

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      376






















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          $begingroup$

          Not quite:




          • $binom{n}{y}r^y(1-r)^{n-y}$ may sum over $y$ to $1$ but it does not integrate over $r$ to $1$ so it is not a probability density


          • more generally the posterior density is proportional to the prior density multiplied by the likelihood rather than being equal to it


          • so the answer should be $dfrac{binom{n}{y}r^y(1-r)^{n-y}cdot 1}{intlimits _{r=0}^1 binom{n}{y}r^y(1-r)^{n-y}cdot 1, dr}$ with some obvious cancellations; this is the density of a Beta distribution with parameters $alpha=1+y,beta=1+n-y$







          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
            $endgroup$
            – sn3jd3r
            Jan 7 at 23:09






          • 1




            $begingroup$
            @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
            $endgroup$
            – Henry
            Jan 7 at 23:33














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          active

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          1












          $begingroup$

          Not quite:




          • $binom{n}{y}r^y(1-r)^{n-y}$ may sum over $y$ to $1$ but it does not integrate over $r$ to $1$ so it is not a probability density


          • more generally the posterior density is proportional to the prior density multiplied by the likelihood rather than being equal to it


          • so the answer should be $dfrac{binom{n}{y}r^y(1-r)^{n-y}cdot 1}{intlimits _{r=0}^1 binom{n}{y}r^y(1-r)^{n-y}cdot 1, dr}$ with some obvious cancellations; this is the density of a Beta distribution with parameters $alpha=1+y,beta=1+n-y$







          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
            $endgroup$
            – sn3jd3r
            Jan 7 at 23:09






          • 1




            $begingroup$
            @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
            $endgroup$
            – Henry
            Jan 7 at 23:33


















          1












          $begingroup$

          Not quite:




          • $binom{n}{y}r^y(1-r)^{n-y}$ may sum over $y$ to $1$ but it does not integrate over $r$ to $1$ so it is not a probability density


          • more generally the posterior density is proportional to the prior density multiplied by the likelihood rather than being equal to it


          • so the answer should be $dfrac{binom{n}{y}r^y(1-r)^{n-y}cdot 1}{intlimits _{r=0}^1 binom{n}{y}r^y(1-r)^{n-y}cdot 1, dr}$ with some obvious cancellations; this is the density of a Beta distribution with parameters $alpha=1+y,beta=1+n-y$







          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
            $endgroup$
            – sn3jd3r
            Jan 7 at 23:09






          • 1




            $begingroup$
            @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
            $endgroup$
            – Henry
            Jan 7 at 23:33
















          1












          1








          1





          $begingroup$

          Not quite:




          • $binom{n}{y}r^y(1-r)^{n-y}$ may sum over $y$ to $1$ but it does not integrate over $r$ to $1$ so it is not a probability density


          • more generally the posterior density is proportional to the prior density multiplied by the likelihood rather than being equal to it


          • so the answer should be $dfrac{binom{n}{y}r^y(1-r)^{n-y}cdot 1}{intlimits _{r=0}^1 binom{n}{y}r^y(1-r)^{n-y}cdot 1, dr}$ with some obvious cancellations; this is the density of a Beta distribution with parameters $alpha=1+y,beta=1+n-y$







          share|cite|improve this answer









          $endgroup$



          Not quite:




          • $binom{n}{y}r^y(1-r)^{n-y}$ may sum over $y$ to $1$ but it does not integrate over $r$ to $1$ so it is not a probability density


          • more generally the posterior density is proportional to the prior density multiplied by the likelihood rather than being equal to it


          • so the answer should be $dfrac{binom{n}{y}r^y(1-r)^{n-y}cdot 1}{intlimits _{r=0}^1 binom{n}{y}r^y(1-r)^{n-y}cdot 1, dr}$ with some obvious cancellations; this is the density of a Beta distribution with parameters $alpha=1+y,beta=1+n-y$








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 7 at 22:42









          HenryHenry

          101k482170




          101k482170












          • $begingroup$
            Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
            $endgroup$
            – sn3jd3r
            Jan 7 at 23:09






          • 1




            $begingroup$
            @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
            $endgroup$
            – Henry
            Jan 7 at 23:33




















          • $begingroup$
            Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
            $endgroup$
            – sn3jd3r
            Jan 7 at 23:09






          • 1




            $begingroup$
            @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
            $endgroup$
            – Henry
            Jan 7 at 23:33


















          $begingroup$
          Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
          $endgroup$
          – sn3jd3r
          Jan 7 at 23:09




          $begingroup$
          Are you not allowed to omit the denominator in the answer in your post? I was under the assumption that when the prior is a beta distribution and the likelihood is a binomal distribution, then the beta is conjugate to the binomial.
          $endgroup$
          – sn3jd3r
          Jan 7 at 23:09




          1




          1




          $begingroup$
          @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
          $endgroup$
          – Henry
          Jan 7 at 23:33






          $begingroup$
          @sn3jd3r The posterior probability density function is in fact $dfrac{r^y(1-r)^{n-y}}{B(1+y,1+n-y)}$ when $0 le r le 1$ where the denominator is a Beta function which does not vary with $r$; this is the density of a Beta distribution. If you omit this denominator then you do not have a probability density function as it will not integrate to $1$, merely something proportional to the density.
          $endgroup$
          – Henry
          Jan 7 at 23:33




















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