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$begingroup$

I'm trying to prove a result that claims the following function is concave. Let $p$ be an input probability distribution and $Q$ be a transition matrix. $q = Qp$ is a valid output probability distribution. The components of $p$ and $q$ are given as $p_i$ and $q_i$ respectively.

For any $p, p'$, define

$$J(p, p') = -D(p||p') + sum_ip_ileft(Q_ijlog(Q_ij) - sum_jq_jlog q'_jright),$$

where $q'_j = sum_k Q_jkp'_k$ and $D(p||p') = sum_i p_ilog fracp_ip'_i$, the divergence.

The function $J$ should be concave in both $p$ and $p'$. I can see this for $p$ quite easily and numerically, it seems to be true for $p'$ but how can I prove it? Somehow, I need to combine the concave term $-D(p||p')$ and the convex $-log$ term together to make a concave function but I don't know how.

probability-distributions logarithms convex-analysis entropy

share|cite|improve this question

edited 16 hours ago

user1936752

asked 17 hours ago

user1936752user1936752

5791513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are $lambda$ and $lambda’$?
  $endgroup$
  – David M.
  16 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DavidM. sorry, my mistake with the notation. It was p and p'
  $endgroup$
  – user1936752
  16 hours ago
  

  
  
  
  

add a comment | 

1

$begingroup$

I'm trying to prove a result that claims the following function is concave. Let $p$ be an input probability distribution and $Q$ be a transition matrix. $q = Qp$ is a valid output probability distribution. The components of $p$ and $q$ are given as $p_i$ and $q_i$ respectively.

For any $p, p'$, define

$$J(p, p') = -D(p||p') + sum_ip_ileft(Q_ijlog(Q_ij) - sum_jq_jlog q'_jright),$$

where $q'_j = sum_k Q_jkp'_k$ and $D(p||p') = sum_i p_ilog fracp_ip'_i$, the divergence.

The function $J$ should be concave in both $p$ and $p'$. I can see this for $p$ quite easily and numerically, it seems to be true for $p'$ but how can I prove it? Somehow, I need to combine the concave term $-D(p||p')$ and the convex $-log$ term together to make a concave function but I don't know how.

probability-distributions logarithms convex-analysis entropy

share|cite|improve this question

edited 16 hours ago

user1936752

asked 17 hours ago

user1936752user1936752

5791513

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  What are $lambda$ and $lambda’$?
  $endgroup$
  – David M.
  16 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DavidM. sorry, my mistake with the notation. It was p and p'
  $endgroup$
  – user1936752
  16 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I'm trying to prove a result that claims the following function is concave. Let $p$ be an input probability distribution and $Q$ be a transition matrix. $q = Qp$ is a valid output probability distribution. The components of $p$ and $q$ are given as $p_i$ and $q_i$ respectively.

For any $p, p'$, define

$$J(p, p') = -D(p||p') + sum_ip_ileft(Q_ijlog(Q_ij) - sum_jq_jlog q'_jright),$$

where $q'_j = sum_k Q_jkp'_k$ and $D(p||p') = sum_i p_ilog fracp_ip'_i$, the divergence.

The function $J$ should be concave in both $p$ and $p'$. I can see this for $p$ quite easily and numerically, it seems to be true for $p'$ but how can I prove it? Somehow, I need to combine the concave term $-D(p||p')$ and the convex $-log$ term together to make a concave function but I don't know how.

probability-distributions logarithms convex-analysis entropy

share|cite|improve this question

edited 16 hours ago

user1936752

asked 17 hours ago

user1936752user1936752

5791513

$endgroup$

share|cite|improve this question

edited 16 hours ago

user1936752

asked 17 hours ago

user1936752user1936752

5791513

share|cite|improve this question

edited 16 hours ago

user1936752

asked 17 hours ago

user1936752user1936752

5791513

asked 17 hours ago

user1936752user1936752

5791513

asked 17 hours ago

user1936752user1936752

5791513