payload: {"action":"created","issue":{"url":"https://api.github.com/repos/ben18785/distribution-zoo/issues/44","repository_url":"https://api.github.com/repos/ben18785/distribution-zoo","labels_url":"https://api.github.com/repos/ben18785/distribution-zoo/issues/44/labels{/name}","comments_url":"https://api.github.com/repos/ben18785/distribution-zoo/issues/44/comments","events_url":"https://api.github.com/repos/ben18785/distribution-zoo/issues/44/events","html_url":"https://github.com/ben18785/distribution-zoo/issues/44","id":413610973,"node_id":"MDU6SXNzdWU0MTM2MTA5NzM=","number":44,"title":"Expectation of LKJ RV is the identity matrix","user":{"login":"bgoodri","id":3386282,"node_id":"MDQ6VXNlcjMzODYyODI=","avatar_url":"https://avatars2.githubusercontent.com/u/3386282?v=4","gravatar_id":"","url":"https://api.github.com/users/bgoodri","html_url":"https://github.com/bgoodri","followers_url":"https://api.github.com/users/bgoodri/followers","following_url":"https://api.github.com/users/bgoodri/following{/other_user}","gists_url":"https://api.github.com/users/bgoodri/gists{/gist_id}","starred_url":"https://api.github.com/users/bgoodri/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/bgoodri/subscriptions","organizations_url":"https://api.github.com/users/bgoodri/orgs","repos_url":"https://api.github.com/users/bgoodri/repos","events_url":"https://api.github.com/users/bgoodri/events{/privacy}","received_events_url":"https://api.github.com/users/bgoodri/received_events","type":"User","site_admin":false},"labels":[],"state":"closed","locked":false,"assignee":null,"assignees":[],"milestone":null,"comments":4,"created_at":"2019-02-22T22:54:43Z","updated_at":"2019-02-26T05:45:23Z","closed_at":"2019-02-23T15:36:43Z","author_association":"NONE","body":"The app says there is \"No simple expression\" for the expectation and variance of a correlation matrix that is distributed LKJ. The expectation is the identity matrix, which can be seen most easily from page 1997 of the LKJ paper, where they say in step 2.b.i to generate a partial correlation (on a C-vine) independently from a symmetric beta distribution on the (-1,1) interval --- which has an expectation of zero --- and then in step 2.b.ii they say to use the recursive formula in equation (2) to obtain the corresponding Pearson correlation. If you rearrange equation (2) to isolate $\\rho_{ij; L}$, then its expectation is zero if all three partial correlations have an expectation of zero.\r\n\r\nThis paper\r\nhttps://scholar.google.com/scholar?cluster=8769718558497577003&hl=en&as_sdt=0,33\r\nhas a proof in the appendix that the marginal distribution of a correlation is symmetric beta over (-1,1) with both shape parameters `eta - 1 + size / 2` from which the marginal variance could be calculated, but I don't know of anyone who has worked out what the covariances between the Pearson correlations are.\r\n"},"comment":{"url":"https://api.github.com/repos/ben18785/distribution-zoo/issues/comments/467303832","html_url":"https://github.com/ben18785/distribution-zoo/issues/44#issuecomment-467303832","issue_url":"https://api.github.com/repos/ben18785/distribution-zoo/issues/44","id":467303832,"node_id":"MDEyOklzc3VlQ29tbWVudDQ2NzMwMzgzMg==","user":{"login":"ben18785","id":4490664,"node_id":"MDQ6VXNlcjQ0OTA2NjQ=","avatar_url":"https://avatars3.githubusercontent.com/u/4490664?v=4","gravatar_id":"","url":"https://api.github.com/users/ben18785","html_url":"https://github.com/ben18785","followers_url":"https://api.github.com/users/ben18785/followers","following_url":"https://api.github.com/users/ben18785/following{/other_user}","gists_url":"https://api.github.com/users/ben18785/gists{/gist_id}","starred_url":"https://api.github.com/users/ben18785/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/ben18785/subscriptions","organizations_url":"https://api.github.com/users/ben18785/orgs","repos_url":"https://api.github.com/users/ben18785/repos","events_url":"https://api.github.com/users/ben18785/events{/privacy}","received_events_url":"https://api.github.com/users/ben18785/received_events","type":"User","site_admin":false},"created_at":"2019-02-26T05:45:23Z","updated_at":"2019-02-26T05:45:23Z","author_association":"OWNER","body":"Wow, I've made a hash of this, sorry! Been a busy few days. Changed it\nagain. Let me know if anything is still wrong...\n\nOn Mon, Feb 25, 2019 at 4:56 PM bgoodri <notifications@github.com> wrote:\n\n> Almost there. The 4 in the denominator of the marginal variance should be\n> in the numerator, since when you multiply a conventional Beta random\n> variable by 2 and shift it by -1 to put the random variable on the (-1,1)\n> interval, then the variance goes up by 2^2 .\n>\n> On Sat, Feb 23, 2019 at 5:35 PM ben18785 <notifications@github.com> wrote:\n>\n> > Thanks -- I should have twigged about the support given we're dealing\n> with\n> > correlations! I (think) I've corrected this now. Thanks again.\n> >\n> > —\n> > You are receiving this because you were mentioned.\n> > Reply to this email directly, view it on GitHub\n> > <\n> https://github.com/ben18785/distribution-zoo/issues/44#issuecomment-466705637\n> >,\n> > or mute the thread\n> > <\n> https://github.com/notifications/unsubscribe-auth/ADOrquTNvtDKeeUYxYd5K7sY9FaltYOGks5vQcIugaJpZM4bL1RA\n> >\n> > .\n> >\n>\n> —\n> You are receiving this because you modified the open/close state.\n> Reply to this email directly, view it on GitHub\n> <https://github.com/ben18785/distribution-zoo/issues/44#issuecomment-467089391>,\n> or mute the thread\n> <https://github.com/notifications/unsubscribe-auth/AESFqLgAkaJlFTZV6K3ZoRbRYwCHzw8Zks5vRBXZgaJpZM4bL1RA>\n> .\n>\n"}}
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