Gaussian mixture model - does an improper uniform prior give a proper posterior?

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We draw $n$ i.i.d. points $x_1 , x_2 , ..., x_n$ from the following Gaussian mixture:



$$p(x|mu_1,mu_2) = frac12 textN (x|mu_1,1) + frac12 textN (x|mu_2,1).$$



The prior is the improper prior $p(mu_1, mu_2) propto 1$. Does the posterior have a finite integral (i.e., is it a proper distribution)?










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    We draw $n$ i.i.d. points $x_1 , x_2 , ..., x_n$ from the following Gaussian mixture:



    $$p(x|mu_1,mu_2) = frac12 textN (x|mu_1,1) + frac12 textN (x|mu_2,1).$$



    The prior is the improper prior $p(mu_1, mu_2) propto 1$. Does the posterior have a finite integral (i.e., is it a proper distribution)?










    share|cite|improve this question









    New contributor




    aastha agrrawal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      up vote
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      up vote
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      We draw $n$ i.i.d. points $x_1 , x_2 , ..., x_n$ from the following Gaussian mixture:



      $$p(x|mu_1,mu_2) = frac12 textN (x|mu_1,1) + frac12 textN (x|mu_2,1).$$



      The prior is the improper prior $p(mu_1, mu_2) propto 1$. Does the posterior have a finite integral (i.e., is it a proper distribution)?










      share|cite|improve this question









      New contributor




      aastha agrrawal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      We draw $n$ i.i.d. points $x_1 , x_2 , ..., x_n$ from the following Gaussian mixture:



      $$p(x|mu_1,mu_2) = frac12 textN (x|mu_1,1) + frac12 textN (x|mu_2,1).$$



      The prior is the improper prior $p(mu_1, mu_2) propto 1$. Does the posterior have a finite integral (i.e., is it a proper distribution)?







      prior posterior gaussian-mixture conjugate-prior






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      aastha agrrawal is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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      share|cite|improve this question









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      edited 2 hours ago









      Ben

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          From your specified model you have likelihood function:



          $$L_mathbfx(mu_1,mu_2) = (8 pi)^-n/2 prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg].$$



          To facilitate the analysis, let $mathscrB equiv 1,2 ^n$ be the set of all $n$-length binary vectors of values of the two means. For any vector $mathbfb in mathscrB$ let $mathcalS(mathbfb) equiv i=1,...,n $ be the set of index values that use the first mean. With the improper uniform prior $pi (mu_1, mu_2) propto 1$ you get the posterior:



          $$beginequation beginaligned
          pi (mu_1, mu_2 | mathbfx)
          &propto L_mathbfx(mu_1,mu_2) \[6pt]
          &propto prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg] \[6pt]
          &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i=1^n (x_i-mu_b_i)^2 Big) \[6pt]
          &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big) exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big). \[6pt]
          endaligned endequation$$



          Now, letting $barx_1^(k) (mathbfb) equiv sum_i in mathcalS(mathbfb) x_i^k$ and $barx_2^(k) (mathbfb) equiv sum_i notin mathcalS(mathbfb) x_i^k$ we have:



          $$beginequation beginaligned
          exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big)
          &= exp Big( -frac12 Big( mu_1^2 + 2 mu_1 sum_i in mathcalS(mathbfb) x_i + sum_i in mathcalS(mathbfb) x_i^2 Big) Big) \[6pt]
          &= exp Big( -frac12 ( mu_1^2 + 2 mu_1 barx_1^(1) (mathbfb) + barx_1^(2) (mathbfb) ) Big) \[10pt]
          &= exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big)
          exp Big( -frac12 (mu_1 - barx_1^(1) (mathbfb))^2 Big) \[10pt]
          &propto exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1), \[10pt]
          exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big) &propto exp Big( -frac12 ( barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2 ) Big) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[10pt]
          endaligned endequation$$



          So, letting $H(mathbfb) equiv barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 + barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2$ we have:



          $$beginequation beginaligned
          pi (mu_1, mu_2 | mathbfx)
          &propto sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[6pt]
          endaligned endequation$$



          This kernel has a finite integral, and yields the proper posterior density:



          $$beginequation beginaligned
          pi (mu_1, mu_2 | mathbfx)
          &= frac barx_2^(1) (mathbfb), 1)sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big). \[6pt]
          endaligned endequation$$






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            From your specified model you have likelihood function:



            $$L_mathbfx(mu_1,mu_2) = (8 pi)^-n/2 prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg].$$



            To facilitate the analysis, let $mathscrB equiv 1,2 ^n$ be the set of all $n$-length binary vectors of values of the two means. For any vector $mathbfb in mathscrB$ let $mathcalS(mathbfb) equiv i=1,...,n $ be the set of index values that use the first mean. With the improper uniform prior $pi (mu_1, mu_2) propto 1$ you get the posterior:



            $$beginequation beginaligned
            pi (mu_1, mu_2 | mathbfx)
            &propto L_mathbfx(mu_1,mu_2) \[6pt]
            &propto prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg] \[6pt]
            &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i=1^n (x_i-mu_b_i)^2 Big) \[6pt]
            &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big) exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big). \[6pt]
            endaligned endequation$$



            Now, letting $barx_1^(k) (mathbfb) equiv sum_i in mathcalS(mathbfb) x_i^k$ and $barx_2^(k) (mathbfb) equiv sum_i notin mathcalS(mathbfb) x_i^k$ we have:



            $$beginequation beginaligned
            exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big)
            &= exp Big( -frac12 Big( mu_1^2 + 2 mu_1 sum_i in mathcalS(mathbfb) x_i + sum_i in mathcalS(mathbfb) x_i^2 Big) Big) \[6pt]
            &= exp Big( -frac12 ( mu_1^2 + 2 mu_1 barx_1^(1) (mathbfb) + barx_1^(2) (mathbfb) ) Big) \[10pt]
            &= exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big)
            exp Big( -frac12 (mu_1 - barx_1^(1) (mathbfb))^2 Big) \[10pt]
            &propto exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1), \[10pt]
            exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big) &propto exp Big( -frac12 ( barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2 ) Big) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[10pt]
            endaligned endequation$$



            So, letting $H(mathbfb) equiv barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 + barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2$ we have:



            $$beginequation beginaligned
            pi (mu_1, mu_2 | mathbfx)
            &propto sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[6pt]
            endaligned endequation$$



            This kernel has a finite integral, and yields the proper posterior density:



            $$beginequation beginaligned
            pi (mu_1, mu_2 | mathbfx)
            &= frac barx_2^(1) (mathbfb), 1)sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big). \[6pt]
            endaligned endequation$$






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              From your specified model you have likelihood function:



              $$L_mathbfx(mu_1,mu_2) = (8 pi)^-n/2 prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg].$$



              To facilitate the analysis, let $mathscrB equiv 1,2 ^n$ be the set of all $n$-length binary vectors of values of the two means. For any vector $mathbfb in mathscrB$ let $mathcalS(mathbfb) equiv i=1,...,n $ be the set of index values that use the first mean. With the improper uniform prior $pi (mu_1, mu_2) propto 1$ you get the posterior:



              $$beginequation beginaligned
              pi (mu_1, mu_2 | mathbfx)
              &propto L_mathbfx(mu_1,mu_2) \[6pt]
              &propto prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg] \[6pt]
              &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i=1^n (x_i-mu_b_i)^2 Big) \[6pt]
              &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big) exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big). \[6pt]
              endaligned endequation$$



              Now, letting $barx_1^(k) (mathbfb) equiv sum_i in mathcalS(mathbfb) x_i^k$ and $barx_2^(k) (mathbfb) equiv sum_i notin mathcalS(mathbfb) x_i^k$ we have:



              $$beginequation beginaligned
              exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big)
              &= exp Big( -frac12 Big( mu_1^2 + 2 mu_1 sum_i in mathcalS(mathbfb) x_i + sum_i in mathcalS(mathbfb) x_i^2 Big) Big) \[6pt]
              &= exp Big( -frac12 ( mu_1^2 + 2 mu_1 barx_1^(1) (mathbfb) + barx_1^(2) (mathbfb) ) Big) \[10pt]
              &= exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big)
              exp Big( -frac12 (mu_1 - barx_1^(1) (mathbfb))^2 Big) \[10pt]
              &propto exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1), \[10pt]
              exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big) &propto exp Big( -frac12 ( barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2 ) Big) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[10pt]
              endaligned endequation$$



              So, letting $H(mathbfb) equiv barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 + barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2$ we have:



              $$beginequation beginaligned
              pi (mu_1, mu_2 | mathbfx)
              &propto sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[6pt]
              endaligned endequation$$



              This kernel has a finite integral, and yields the proper posterior density:



              $$beginequation beginaligned
              pi (mu_1, mu_2 | mathbfx)
              &= frac barx_2^(1) (mathbfb), 1)sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big). \[6pt]
              endaligned endequation$$






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                up vote
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                From your specified model you have likelihood function:



                $$L_mathbfx(mu_1,mu_2) = (8 pi)^-n/2 prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg].$$



                To facilitate the analysis, let $mathscrB equiv 1,2 ^n$ be the set of all $n$-length binary vectors of values of the two means. For any vector $mathbfb in mathscrB$ let $mathcalS(mathbfb) equiv i=1,...,n $ be the set of index values that use the first mean. With the improper uniform prior $pi (mu_1, mu_2) propto 1$ you get the posterior:



                $$beginequation beginaligned
                pi (mu_1, mu_2 | mathbfx)
                &propto L_mathbfx(mu_1,mu_2) \[6pt]
                &propto prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg] \[6pt]
                &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i=1^n (x_i-mu_b_i)^2 Big) \[6pt]
                &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big) exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big). \[6pt]
                endaligned endequation$$



                Now, letting $barx_1^(k) (mathbfb) equiv sum_i in mathcalS(mathbfb) x_i^k$ and $barx_2^(k) (mathbfb) equiv sum_i notin mathcalS(mathbfb) x_i^k$ we have:



                $$beginequation beginaligned
                exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big)
                &= exp Big( -frac12 Big( mu_1^2 + 2 mu_1 sum_i in mathcalS(mathbfb) x_i + sum_i in mathcalS(mathbfb) x_i^2 Big) Big) \[6pt]
                &= exp Big( -frac12 ( mu_1^2 + 2 mu_1 barx_1^(1) (mathbfb) + barx_1^(2) (mathbfb) ) Big) \[10pt]
                &= exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big)
                exp Big( -frac12 (mu_1 - barx_1^(1) (mathbfb))^2 Big) \[10pt]
                &propto exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1), \[10pt]
                exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big) &propto exp Big( -frac12 ( barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2 ) Big) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[10pt]
                endaligned endequation$$



                So, letting $H(mathbfb) equiv barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 + barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2$ we have:



                $$beginequation beginaligned
                pi (mu_1, mu_2 | mathbfx)
                &propto sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[6pt]
                endaligned endequation$$



                This kernel has a finite integral, and yields the proper posterior density:



                $$beginequation beginaligned
                pi (mu_1, mu_2 | mathbfx)
                &= frac barx_2^(1) (mathbfb), 1)sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big). \[6pt]
                endaligned endequation$$






                share|cite|improve this answer












                From your specified model you have likelihood function:



                $$L_mathbfx(mu_1,mu_2) = (8 pi)^-n/2 prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg].$$



                To facilitate the analysis, let $mathscrB equiv 1,2 ^n$ be the set of all $n$-length binary vectors of values of the two means. For any vector $mathbfb in mathscrB$ let $mathcalS(mathbfb) equiv i=1,...,n $ be the set of index values that use the first mean. With the improper uniform prior $pi (mu_1, mu_2) propto 1$ you get the posterior:



                $$beginequation beginaligned
                pi (mu_1, mu_2 | mathbfx)
                &propto L_mathbfx(mu_1,mu_2) \[6pt]
                &propto prod_i=1^n Bigg[ exp Big( -frac12 (x_i-mu_1)^2 Big) + exp Big( -frac12 (x_i-mu_2)^2 Big) Bigg] \[6pt]
                &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i=1^n (x_i-mu_b_i)^2 Big) \[6pt]
                &= sum_mathbfb in mathscrB exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big) exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big). \[6pt]
                endaligned endequation$$



                Now, letting $barx_1^(k) (mathbfb) equiv sum_i in mathcalS(mathbfb) x_i^k$ and $barx_2^(k) (mathbfb) equiv sum_i notin mathcalS(mathbfb) x_i^k$ we have:



                $$beginequation beginaligned
                exp Big( -frac12 sum_i in mathcalS(mathbfb) (x_i-mu_1)^2 Big)
                &= exp Big( -frac12 Big( mu_1^2 + 2 mu_1 sum_i in mathcalS(mathbfb) x_i + sum_i in mathcalS(mathbfb) x_i^2 Big) Big) \[6pt]
                &= exp Big( -frac12 ( mu_1^2 + 2 mu_1 barx_1^(1) (mathbfb) + barx_1^(2) (mathbfb) ) Big) \[10pt]
                &= exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big)
                exp Big( -frac12 (mu_1 - barx_1^(1) (mathbfb))^2 Big) \[10pt]
                &propto exp Big( -frac12 ( barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 ) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1), \[10pt]
                exp Big( -frac12 sum_i notin mathcalS(mathbfb) (x_i-mu_2)^2 Big) &propto exp Big( -frac12 ( barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2 ) Big) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[10pt]
                endaligned endequation$$



                So, letting $H(mathbfb) equiv barx_1^(2) (mathbfb) - barx_1^(1) (mathbfb)^2 + barx_2^(2) (mathbfb) - barx_2^(1) (mathbfb)^2$ we have:



                $$beginequation beginaligned
                pi (mu_1, mu_2 | mathbfx)
                &propto sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big) cdot textN(mu_1 | barx_1^(1) (mathbfb), 1) cdot textN(mu_2 | barx_2^(1) (mathbfb), 1). \[6pt]
                endaligned endequation$$



                This kernel has a finite integral, and yields the proper posterior density:



                $$beginequation beginaligned
                pi (mu_1, mu_2 | mathbfx)
                &= frac barx_2^(1) (mathbfb), 1)sum_mathbfb in mathscrB exp Big( -frac12 H(mathbfb) Big). \[6pt]
                endaligned endequation$$







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