Locally Private Bayesian Inference for Count Modelsas5530/ScheinWuSchofieldZhouWallach... ·...

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y 0 N N 0 ˜ y ˜ y 0 R R y μ ˜ y (±) y μ ˜ y (±) g (-) λ (+) λ (-) g (+) ˜ y (±) μ λ (+) λ (-) ˜ y (+) μ ˜ y (±) g (-) λ (+) λ (-) observed deterministic random fixed Process 2 Process 1 Process 3 Process 4 2Geo() y Pois(μ) ˜ y (±) := y + λ () Exp( 1-) g () Pois(λ () ) := g (+) - g (-) y Pois(μ) ˜ y (±) := y + λ () Exp( 1-) g (-) Pois(λ (-) ) ˜ y (+) Pois(μ+λ (+) ) ˜ y (±) := ˜ y (+) - g (-) λ () Exp( 1-) ˜ y (±) Skel ( λ (+) + μ, λ (-) ) True data y ij True μ ij Non-private ˆ μ ij Our estimate ˆ μ ij Na¨ ıve estimate ˆ μ ij Noised data ˜ y (±) ij Increasing privacy /N =1 /N =0.5 /N =0.25 True data y ij True μ ij Non-private ˆ μ ij Our estimate ˆ μ ij Na¨ ıve estimate ˆ μ ij Noised data ˜ y (±) ij Increasing privacy /N =1 /N =0.5 /N =0.25 word type v document d True topics φ k Non-private estimate ˆ φ k word type v word type v document d word type v Our estimate ˆ φ k Na¨ ıve estimate ˆ φ k word type v document d word type v Our estimate ˆ φ k Na¨ ıve estimate ˆ φ k 1 2 3 /N = ln(1/) -80.0 -75.0 -70.0 -65.0 -60.0 Coherence 1 2 3 /N = ln(1/) 0.1 0.2 0.3 0.4 Reconstruction MAE Model Type Na¨ ıve Proposed Non-private 1 2 3 /N = ln(1/) 1.0 1.5 2.0 2.5 3.0 NPMI 1 2 3 /N = ln(1/) 7.0 8.0 9.0 10.0 11.0 12.0 Held-out MAE Number of communities = 5 Model Type Na¨ ıve Proposed Non-private 1 2 3 /N = ln(1/) Number of communities = 20 1 2 3 /N = ln(1/) 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Reconstruction MAE Number of communities = 5 Model Type Na¨ ıve Proposed Non-private 1 2 3 /N = ln(1/) Number of communities = 20 Locally Private Bayesian Inference for Count Models Aaron Schein Zhiwei Steven Wu Mingyuan Zhou Hanna Wallach Alexandra Schofield UMass Amherst Univ. of Minnesota Cornell University Univ. of Texas Austin Microsoft Research -80 -60 -40 -20 0 20 40 60 80 0.00 0.10 0.20 0.30 0.40 P (| ) =0.4 =0.6 =0.8 Two-sided geometric distribution -80 -60 -40 -20 0 20 40 60 80 0.000 0.025 0.050 0.075 0.100 0.125 P (| μ 1 2 ) μ 1 =5, μ 2 =5 μ 1 =100, μ 2 =100 μ 1 =20, μ 2 =1 μ 1 =30, μ 2 =40 μ 1 =10, μ 2 =50 Skellam distribution 0 20 40 60 80 100 m 0.00 0.05 0.10 0.15 0.20 0.25 0.30 P (m | v,a) v =-0.5, a=30 v =-0.8, a=10 v =1, a=50 v =2, a=7 v =10, a=150 v =400, a=300 Bessel distribution Limited-precision local privacy Summary Auxiliary-variable MCMC Poisson factorization + geometric mechanism The geometric mechanism as Skellam noise Case study 1: Topic modeling Case study 2: Community detection Baselines Naïve method: P (R(y ) 2 S ) e P (R(y 0 ) 2 S ) , Definition: A randomized response method R(…) is an (N, ϵ)-private LPLP mechanism if, A given R(…) could be interpreted as either an (N, ϵ)-private or (N’, ϵ’)-private LPLP mechanism. ||y - y 0 || 1 N. for any two observation such that: Darker means more private (smaller ϵ). Poisson factorization y dv Pois(μ dv ) μ dv = T d φ v Model count data with Poisson factorization Privacy: Noise data using the geometric mechanism How do we do Bayesian inference given noised data? Reinterpret geometric mechanism as Skellam noise Prove link between Skellam and Bessel distribution Solution: auxiliary-variable private MCMC method Bonus: Private outperforms non-private method! Locally private Bayesian inference P M (Y,Z ) P R ( ˜ Y | Y ) 1) Build non-private model M for sensitive data Y and latent variables Z (e.g., Poisson factorization): 2) Define randomized response method R: 3) Approximate locally-private posterior: P M,R (Z | ˜ Y ) = Z dY P M (Z | Y ) P M,R (Y | ˜ Y ) non-private posterior With MCMC, we can iterate: Z (s) P M (Z | Y (s) ) Y (s) P M,R (Y | ˜ Y,Z (s - 1) ) (the challenge) (non-private update) The geometric mechanism as an LPLP mechanism We use use a generalization of “local privacy”. Many count models have the following structure: Non-private method: 2Geo() ˜ y (±) := y + 2Geo(; )= 1 - 1+ || The “geometric mechanism” [Ghosh et al. (2012)]: = exp - N Theorem 1: The geometric mechanism with parameter is an (N, ϵ)-private LPLP mechanism for any (N, ϵ) if: Skel (; μ 1 2 )= e -(μ 1 +μ 2 ) μ 1 μ 2 /2 I || (2 p μ 1 μ 2 ) Theorem 2: A two-sided geometric random variable ~ 2Geo() can be generated as an exponentially-randomized Skellam variable: λ (+) , λ (-) iid Expon 1 - Skel (λ (+) , λ (-) ) The Skellam is the difference of two Poissons g (+) Pois (λ (+) ) g (-) Pois (λ (-) ) := g (+) - g (-) Theorem 3: These two generative processes express the same joint distribution of and m: g (+) Pois(λ (+) ) g (-) Pois(λ (-) ) := g (+) - g (-) m := min{g (+) ,g (-) } Skel (λ (+) , λ (-) ) m Bessel ||, 2 p λ (+) λ (-) Four equivalent ways to generate the noised data: Solution: If we sample an auxiliary Bessel, we can “recover” the underlying (sensitive) data y: Bessel(m; v,a)= (a/2) 2m+v I |v | (a) m! Γ(m + v + 1) Synthetic data: Ours is better than the naïve method at recovering the true topics. Enron corpus data: Our method’s reconstruction error is comparable to the non- private method and its inferred topics are more coherent! Synthetic data: Ours is better than the naïve method at recovering block structure. Enron corpus data: Our method obtains lower heldout and reconstruction error than both the naïve and non-private method! Conditions on noised data Truncates any negative values to 0 Standard non-private inference P M (Z | Y ) P M (Z | ˜ Y ) Conditions on unnoised data Standard non-private inference

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observed<latexit sha1_base64="UheCCfyZS1wyN8nNAp+mhrKcVGo=">AAACC3icbZDNSgMxFIXv+FvrX9Wlm2ARXJWZbtRdwY3LCvYH2qFkMnfa0MxkSDKFWvoIbtzqW7gTtz6EL+EzmLaz0LYHAodz7oWbL0gF18Z1v52Nza3tnd3CXnH/4PDouHRy2tQyUwwbTAqp2gHVKHiCDcONwHaqkMaBwFYwvJv1rREqzWXyaMYp+jHtJzzijBobtWWgUY0w7JXKbsWdi6waLzdlyFXvlX66oWRZjIlhgmrd8dzU+BOqDGcCp8VupjGlbEj72LE2oTFqfzK/d0oubRKSSCr7EkPm6d+NCY21HseBnYypGejlbhau7RQKzZ9wXdfJTHTjT3iSZgYTtjgiygQxkszAkJArZEaMraFMcfsPwgZUUWYsvqIF5C3jWDXNasVzK95DtVy7zVEV4Bwu4Ao8uIYa3EMdGsBAwAu8wpvz7Lw7H87nYnTDyXfO4J+cr19s4JuY</latexit><latexit sha1_base64="UheCCfyZS1wyN8nNAp+mhrKcVGo=">AAACC3icbZDNSgMxFIXv+FvrX9Wlm2ARXJWZbtRdwY3LCvYH2qFkMnfa0MxkSDKFWvoIbtzqW7gTtz6EL+EzmLaz0LYHAodz7oWbL0gF18Z1v52Nza3tnd3CXnH/4PDouHRy2tQyUwwbTAqp2gHVKHiCDcONwHaqkMaBwFYwvJv1rREqzWXyaMYp+jHtJzzijBobtWWgUY0w7JXKbsWdi6waLzdlyFXvlX66oWRZjIlhgmrd8dzU+BOqDGcCp8VupjGlbEj72LE2oTFqfzK/d0oubRKSSCr7EkPm6d+NCY21HseBnYypGejlbhau7RQKzZ9wXdfJTHTjT3iSZgYTtjgiygQxkszAkJArZEaMraFMcfsPwgZUUWYsvqIF5C3jWDXNasVzK95DtVy7zVEV4Bwu4Ao8uIYa3EMdGsBAwAu8wpvz7Lw7H87nYnTDyXfO4J+cr19s4JuY</latexit><latexit sha1_base64="UheCCfyZS1wyN8nNAp+mhrKcVGo=">AAACC3icbZDNSgMxFIXv+FvrX9Wlm2ARXJWZbtRdwY3LCvYH2qFkMnfa0MxkSDKFWvoIbtzqW7gTtz6EL+EzmLaz0LYHAodz7oWbL0gF18Z1v52Nza3tnd3CXnH/4PDouHRy2tQyUwwbTAqp2gHVKHiCDcONwHaqkMaBwFYwvJv1rREqzWXyaMYp+jHtJzzijBobtWWgUY0w7JXKbsWdi6waLzdlyFXvlX66oWRZjIlhgmrd8dzU+BOqDGcCp8VupjGlbEj72LE2oTFqfzK/d0oubRKSSCr7EkPm6d+NCY21HseBnYypGejlbhau7RQKzZ9wXdfJTHTjT3iSZgYTtjgiygQxkszAkJArZEaMraFMcfsPwgZUUWYsvqIF5C3jWDXNasVzK95DtVy7zVEV4Bwu4Ao8uIYa3EMdGsBAwAu8wpvz7Lw7H87nYnTDyXfO4J+cr19s4JuY</latexit><latexit sha1_base64="UheCCfyZS1wyN8nNAp+mhrKcVGo=">AAACC3icbZDNSgMxFIXv+FvrX9Wlm2ARXJWZbtRdwY3LCvYH2qFkMnfa0MxkSDKFWvoIbtzqW7gTtz6EL+EzmLaz0LYHAodz7oWbL0gF18Z1v52Nza3tnd3CXnH/4PDouHRy2tQyUwwbTAqp2gHVKHiCDcONwHaqkMaBwFYwvJv1rREqzWXyaMYp+jHtJzzijBobtWWgUY0w7JXKbsWdi6waLzdlyFXvlX66oWRZjIlhgmrd8dzU+BOqDGcCp8VupjGlbEj72LE2oTFqfzK/d0oubRKSSCr7EkPm6d+NCY21HseBnYypGejlbhau7RQKzZ9wXdfJTHTjT3iSZgYTtjgiygQxkszAkJArZEaMraFMcfsPwgZUUWYsvqIF5C3jWDXNasVzK95DtVy7zVEV4Bwu4Ao8uIYa3EMdGsBAwAu8wpvz7Lw7H87nYnTDyXfO4J+cr19s4JuY</latexit>

deterministic<latexit sha1_base64="QcCpJIszCaaQ0MwUZ2XhzLsUrRU=">AAACEHicbVC7TsNAEFyHVwivACWNRYREFdlpgC4SDWWQyENKrOh83iSn3J3N3RkpWPkJGlr4CzpEyx/wE3wDl8QFhIy00mhm9253woQzbTzvyymsrW9sbhW3Szu7e/sH5cOjlo5TRbFJYx6rTkg0ciaxaZjh2EkUEhFybIfj65nffkClWSzvzCTBQJChZANGibFSEKFBJZi0HzHaL1e8qjeH+5/4OalAjka//N2LYpoKlIZyonXX9xITZETZxzhOS71UY0LomAyxa6kkAnWQzZeeumdWidxBrGxJ487V3xMZEVpPRGg7BTEjvezNxJWeQq7ZI67yuqkZXAYZk0lqUNLFEoOUuyZ2Z+m4EVNIDZ9YQqhi9g6Xjogi1MakSzYgfzmO/6RVq/pe1b+tVepXeVRFOIFTOAcfLqAON9CAJlC4h2d4gVfnyXlz3p2PRWvByWeO4Q+czx+leJ3k</latexit><latexit sha1_base64="QcCpJIszCaaQ0MwUZ2XhzLsUrRU=">AAACEHicbVC7TsNAEFyHVwivACWNRYREFdlpgC4SDWWQyENKrOh83iSn3J3N3RkpWPkJGlr4CzpEyx/wE3wDl8QFhIy00mhm9253woQzbTzvyymsrW9sbhW3Szu7e/sH5cOjlo5TRbFJYx6rTkg0ciaxaZjh2EkUEhFybIfj65nffkClWSzvzCTBQJChZANGibFSEKFBJZi0HzHaL1e8qjeH+5/4OalAjka//N2LYpoKlIZyonXX9xITZETZxzhOS71UY0LomAyxa6kkAnWQzZeeumdWidxBrGxJ487V3xMZEVpPRGg7BTEjvezNxJWeQq7ZI67yuqkZXAYZk0lqUNLFEoOUuyZ2Z+m4EVNIDZ9YQqhi9g6Xjogi1MakSzYgfzmO/6RVq/pe1b+tVepXeVRFOIFTOAcfLqAON9CAJlC4h2d4gVfnyXlz3p2PRWvByWeO4Q+czx+leJ3k</latexit><latexit sha1_base64="QcCpJIszCaaQ0MwUZ2XhzLsUrRU=">AAACEHicbVC7TsNAEFyHVwivACWNRYREFdlpgC4SDWWQyENKrOh83iSn3J3N3RkpWPkJGlr4CzpEyx/wE3wDl8QFhIy00mhm9253woQzbTzvyymsrW9sbhW3Szu7e/sH5cOjlo5TRbFJYx6rTkg0ciaxaZjh2EkUEhFybIfj65nffkClWSzvzCTBQJChZANGibFSEKFBJZi0HzHaL1e8qjeH+5/4OalAjka//N2LYpoKlIZyonXX9xITZETZxzhOS71UY0LomAyxa6kkAnWQzZeeumdWidxBrGxJ487V3xMZEVpPRGg7BTEjvezNxJWeQq7ZI67yuqkZXAYZk0lqUNLFEoOUuyZ2Z+m4EVNIDZ9YQqhi9g6Xjogi1MakSzYgfzmO/6RVq/pe1b+tVepXeVRFOIFTOAcfLqAON9CAJlC4h2d4gVfnyXlz3p2PRWvByWeO4Q+czx+leJ3k</latexit><latexit sha1_base64="QcCpJIszCaaQ0MwUZ2XhzLsUrRU=">AAACEHicbVC7TsNAEFyHVwivACWNRYREFdlpgC4SDWWQyENKrOh83iSn3J3N3RkpWPkJGlr4CzpEyx/wE3wDl8QFhIy00mhm9253woQzbTzvyymsrW9sbhW3Szu7e/sH5cOjlo5TRbFJYx6rTkg0ciaxaZjh2EkUEhFybIfj65nffkClWSzvzCTBQJChZANGibFSEKFBJZi0HzHaL1e8qjeH+5/4OalAjka//N2LYpoKlIZyonXX9xITZETZxzhOS71UY0LomAyxa6kkAnWQzZeeumdWidxBrGxJ487V3xMZEVpPRGg7BTEjvezNxJWeQq7ZI67yuqkZXAYZk0lqUNLFEoOUuyZ2Z+m4EVNIDZ9YQqhi9g6Xjogi1MakSzYgfzmO/6RVq/pe1b+tVepXeVRFOIFTOAcfLqAON9CAJlC4h2d4gVfnyXlz3p2PRWvByWeO4Q+czx+leJ3k</latexit>

random<latexit sha1_base64="JMYIWor/B4JN83rBOajvBPPlkdM=">AAACCXicbZC7SgNBFIZn4y3GW9TSZjAIVmE3jdoFbCwjmAskIczOnk3GzGWZmRXikiewsdW3sBNbn8KX8BmcJFtozA8DP/9/Dpz5woQzY33/yyusrW9sbhW3Szu7e/sH5cOjllGpptCkiivdCYkBziQ0LbMcOokGIkIO7XB8PevbD6ANU/LOThLoCzKULGaUWBe1NJGREoNyxa/6c+H/JshNBeVqDMrfvUjRVIC0lBNjuoGf2H5GtGWUw7TUSw0khI7JELrOSiLA9LP5tVN85pIIx0q7Jy2ep783MiKMmYjQTQpiR2a5m4UrOw3csEdY1XVTG1/2MyaT1IKkiyPilGOr8AwLjpgGavnEGUI1c//AdEQ0odbBKzlAwTKO/6ZVqwZ+NbitVepXOaoiOkGn6BwF6ALV0Q1qoCai6B49oxf06j15b96797EYLXj5zjH6I+/zB7x8mqs=</latexit><latexit sha1_base64="JMYIWor/B4JN83rBOajvBPPlkdM=">AAACCXicbZC7SgNBFIZn4y3GW9TSZjAIVmE3jdoFbCwjmAskIczOnk3GzGWZmRXikiewsdW3sBNbn8KX8BmcJFtozA8DP/9/Dpz5woQzY33/yyusrW9sbhW3Szu7e/sH5cOjllGpptCkiivdCYkBziQ0LbMcOokGIkIO7XB8PevbD6ANU/LOThLoCzKULGaUWBe1NJGREoNyxa/6c+H/JshNBeVqDMrfvUjRVIC0lBNjuoGf2H5GtGWUw7TUSw0khI7JELrOSiLA9LP5tVN85pIIx0q7Jy2ep783MiKMmYjQTQpiR2a5m4UrOw3csEdY1XVTG1/2MyaT1IKkiyPilGOr8AwLjpgGavnEGUI1c//AdEQ0odbBKzlAwTKO/6ZVqwZ+NbitVepXOaoiOkGn6BwF6ALV0Q1qoCai6B49oxf06j15b96797EYLXj5zjH6I+/zB7x8mqs=</latexit><latexit sha1_base64="JMYIWor/B4JN83rBOajvBPPlkdM=">AAACCXicbZC7SgNBFIZn4y3GW9TSZjAIVmE3jdoFbCwjmAskIczOnk3GzGWZmRXikiewsdW3sBNbn8KX8BmcJFtozA8DP/9/Dpz5woQzY33/yyusrW9sbhW3Szu7e/sH5cOjllGpptCkiivdCYkBziQ0LbMcOokGIkIO7XB8PevbD6ANU/LOThLoCzKULGaUWBe1NJGREoNyxa/6c+H/JshNBeVqDMrfvUjRVIC0lBNjuoGf2H5GtGWUw7TUSw0khI7JELrOSiLA9LP5tVN85pIIx0q7Jy2ep783MiKMmYjQTQpiR2a5m4UrOw3csEdY1XVTG1/2MyaT1IKkiyPilGOr8AwLjpgGavnEGUI1c//AdEQ0odbBKzlAwTKO/6ZVqwZ+NbitVepXOaoiOkGn6BwF6ALV0Q1qoCai6B49oxf06j15b96797EYLXj5zjH6I+/zB7x8mqs=</latexit><latexit sha1_base64="JMYIWor/B4JN83rBOajvBPPlkdM=">AAACCXicbZC7SgNBFIZn4y3GW9TSZjAIVmE3jdoFbCwjmAskIczOnk3GzGWZmRXikiewsdW3sBNbn8KX8BmcJFtozA8DP/9/Dpz5woQzY33/yyusrW9sbhW3Szu7e/sH5cOjllGpptCkiivdCYkBziQ0LbMcOokGIkIO7XB8PevbD6ANU/LOThLoCzKULGaUWBe1NJGREoNyxa/6c+H/JshNBeVqDMrfvUjRVIC0lBNjuoGf2H5GtGWUw7TUSw0khI7JELrOSiLA9LP5tVN85pIIx0q7Jy2ep783MiKMmYjQTQpiR2a5m4UrOw3csEdY1XVTG1/2MyaT1IKkiyPilGOr8AwLjpgGavnEGUI1c//AdEQ0odbBKzlAwTKO/6ZVqwZ+NbitVepXOaoiOkGn6BwF6ALV0Q1qoCai6B49oxf06j15b96797EYLXj5zjH6I+/zB7x8mqs=</latexit>fixed

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Process 2<latexit sha1_base64="vbW0jF95gvVCOtsd+ZsH81+1tqs=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbuyy4cVnBPqAJZTK9aYdOJmFmItbQlb/ixq3+hTtxJfgNfoPTNgtte+DC4Zx7L/eeMOVMadf9stbWNza3tks79u7e/sGhc3TcUkkmKTRpwhPZCYkCzgQ0NdMcOqkEEocc2uHoeuq370Eqlog7PU4hiMlAsIhRoo3Uc858CkKDZGJg+xoedBjlDZlQUApXJz2n7FbcGfAy8QpSRgUaPefH7yc0i81OyolSXc9NdZATqRnlMLH9TEFK6IgMoGuoIDGoIJ+9McEXRunjKJGmhMYz9e9ETmKlxnFoOmOih2rRm4orPQlcsUdY5XUzHdWCnIk00yDo/Igo41gneJoX7jMJVPOxIYRKZv7AdEgkoSY0ZZuAvMU4lkmrWvHcindbLddrRVQldIrO0SXy0BWqoxvUQE1E0RN6Qa/ozXq23q0P63PeumYVMyfoH6zvX1EUpHc=</latexit><latexit sha1_base64="vbW0jF95gvVCOtsd+ZsH81+1tqs=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbuyy4cVnBPqAJZTK9aYdOJmFmItbQlb/ixq3+hTtxJfgNfoPTNgtte+DC4Zx7L/eeMOVMadf9stbWNza3tks79u7e/sGhc3TcUkkmKTRpwhPZCYkCzgQ0NdMcOqkEEocc2uHoeuq370Eqlog7PU4hiMlAsIhRoo3Uc858CkKDZGJg+xoedBjlDZlQUApXJz2n7FbcGfAy8QpSRgUaPefH7yc0i81OyolSXc9NdZATqRnlMLH9TEFK6IgMoGuoIDGoIJ+9McEXRunjKJGmhMYz9e9ETmKlxnFoOmOih2rRm4orPQlcsUdY5XUzHdWCnIk00yDo/Igo41gneJoX7jMJVPOxIYRKZv7AdEgkoSY0ZZuAvMU4lkmrWvHcindbLddrRVQldIrO0SXy0BWqoxvUQE1E0RN6Qa/ozXq23q0P63PeumYVMyfoH6zvX1EUpHc=</latexit><latexit sha1_base64="vbW0jF95gvVCOtsd+ZsH81+1tqs=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbuyy4cVnBPqAJZTK9aYdOJmFmItbQlb/ixq3+hTtxJfgNfoPTNgtte+DC4Zx7L/eeMOVMadf9stbWNza3tks79u7e/sGhc3TcUkkmKTRpwhPZCYkCzgQ0NdMcOqkEEocc2uHoeuq370Eqlog7PU4hiMlAsIhRoo3Uc858CkKDZGJg+xoedBjlDZlQUApXJz2n7FbcGfAy8QpSRgUaPefH7yc0i81OyolSXc9NdZATqRnlMLH9TEFK6IgMoGuoIDGoIJ+9McEXRunjKJGmhMYz9e9ETmKlxnFoOmOih2rRm4orPQlcsUdY5XUzHdWCnIk00yDo/Igo41gneJoX7jMJVPOxIYRKZv7AdEgkoSY0ZZuAvMU4lkmrWvHcindbLddrRVQldIrO0SXy0BWqoxvUQE1E0RN6Qa/ozXq23q0P63PeumYVMyfoH6zvX1EUpHc=</latexit><latexit sha1_base64="vbW0jF95gvVCOtsd+ZsH81+1tqs=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbuyy4cVnBPqAJZTK9aYdOJmFmItbQlb/ixq3+hTtxJfgNfoPTNgtte+DC4Zx7L/eeMOVMadf9stbWNza3tks79u7e/sGhc3TcUkkmKTRpwhPZCYkCzgQ0NdMcOqkEEocc2uHoeuq370Eqlog7PU4hiMlAsIhRoo3Uc858CkKDZGJg+xoedBjlDZlQUApXJz2n7FbcGfAy8QpSRgUaPefH7yc0i81OyolSXc9NdZATqRnlMLH9TEFK6IgMoGuoIDGoIJ+9McEXRunjKJGmhMYz9e9ETmKlxnFoOmOih2rRm4orPQlcsUdY5XUzHdWCnIk00yDo/Igo41gneJoX7jMJVPOxIYRKZv7AdEgkoSY0ZZuAvMU4lkmrWvHcindbLddrRVQldIrO0SXy0BWqoxvUQE1E0RN6Qa/ozXq23q0P63PeumYVMyfoH6zvX1EUpHc=</latexit>Process 1

<latexit sha1_base64="MYcfOc2RAVFvCMSZM84sQYqsFPc=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbdVdw47KCfUATymR60w6dTMLMRKyhK3/FjVv9C3fiSvAb/AanbRba9sCFwzn3Xu49YcqZ0q77Za2srq1vbJa27O2d3b195+CwqZJMUmjQhCeyHRIFnAloaKY5tFMJJA45tMLh9cRv3YNULBF3epRCEJO+YBGjRBup65z4FIQGyUTf9jU86DDK6zKhoBT2xl2n7FbcKfAi8QpSRgXqXefH7yU0i81OyolSHc9NdZATqRnlMLb9TEFK6JD0oWOoIDGoIJ++McZnRunhKJGmhMZT9e9ETmKlRnFoOmOiB2rem4hLPQlcsUdY5nUyHV0GORNppkHQ2RFRxrFO8CQv3GMSqOYjQwiVzPyB6YBIQk1oyjYBefNxLJJmteK5Fe+2Wq5dFVGV0DE6RefIQxeohm5QHTUQRU/oBb2iN+vZerc+rM9Z64pVzByhf7C+fwFPwKR3</latexit><latexit sha1_base64="MYcfOc2RAVFvCMSZM84sQYqsFPc=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbdVdw47KCfUATymR60w6dTMLMRKyhK3/FjVv9C3fiSvAb/AanbRba9sCFwzn3Xu49YcqZ0q77Za2srq1vbJa27O2d3b195+CwqZJMUmjQhCeyHRIFnAloaKY5tFMJJA45tMLh9cRv3YNULBF3epRCEJO+YBGjRBup65z4FIQGyUTf9jU86DDK6zKhoBT2xl2n7FbcKfAi8QpSRgXqXefH7yU0i81OyolSHc9NdZATqRnlMLb9TEFK6JD0oWOoIDGoIJ++McZnRunhKJGmhMZT9e9ETmKlRnFoOmOiB2rem4hLPQlcsUdY5nUyHV0GORNppkHQ2RFRxrFO8CQv3GMSqOYjQwiVzPyB6YBIQk1oyjYBefNxLJJmteK5Fe+2Wq5dFVGV0DE6RefIQxeohm5QHTUQRU/oBb2iN+vZerc+rM9Z64pVzByhf7C+fwFPwKR3</latexit><latexit sha1_base64="MYcfOc2RAVFvCMSZM84sQYqsFPc=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbdVdw47KCfUATymR60w6dTMLMRKyhK3/FjVv9C3fiSvAb/AanbRba9sCFwzn3Xu49YcqZ0q77Za2srq1vbJa27O2d3b195+CwqZJMUmjQhCeyHRIFnAloaKY5tFMJJA45tMLh9cRv3YNULBF3epRCEJO+YBGjRBup65z4FIQGyUTf9jU86DDK6zKhoBT2xl2n7FbcKfAi8QpSRgXqXefH7yU0i81OyolSHc9NdZATqRnlMLb9TEFK6JD0oWOoIDGoIJ++McZnRunhKJGmhMZT9e9ETmKlRnFoOmOiB2rem4hLPQlcsUdY5nUyHV0GORNppkHQ2RFRxrFO8CQv3GMSqOYjQwiVzPyB6YBIQk1oyjYBefNxLJJmteK5Fe+2Wq5dFVGV0DE6RefIQxeohm5QHTUQRU/oBb2iN+vZerc+rM9Z64pVzByhf7C+fwFPwKR3</latexit><latexit sha1_base64="MYcfOc2RAVFvCMSZM84sQYqsFPc=">AAACInicbVDLSsNAFJ34rPEVdSnCYBFclaQbdVdw47KCfUATymR60w6dTMLMRKyhK3/FjVv9C3fiSvAb/AanbRba9sCFwzn3Xu49YcqZ0q77Za2srq1vbJa27O2d3b195+CwqZJMUmjQhCeyHRIFnAloaKY5tFMJJA45tMLh9cRv3YNULBF3epRCEJO+YBGjRBup65z4FIQGyUTf9jU86DDK6zKhoBT2xl2n7FbcKfAi8QpSRgXqXefH7yU0i81OyolSHc9NdZATqRnlMLb9TEFK6JD0oWOoIDGoIJ++McZnRunhKJGmhMZT9e9ETmKlRnFoOmOiB2rem4hLPQlcsUdY5nUyHV0GORNppkHQ2RFRxrFO8CQv3GMSqOYjQwiVzPyB6YBIQk1oyjYBefNxLJJmteK5Fe+2Wq5dFVGV0DE6RefIQxeohm5QHTUQRU/oBb2iN+vZerc+rM9Z64pVzByhf7C+fwFPwKR3</latexit>

Process 3<latexit sha1_base64="M+DFRKJgeIxw8z9uzqdWjNQfvwE=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiR1oe4KblxWsA9oSplMb9qhk0mYmYg1dOWvuHGrf+FOXAl+g9/gtM1C2x64cDjn3su9J0g4U9p1v6zCyura+kZx097a3tndc/YPGipOJYU6jXksWwFRwJmAumaaQyuRQKKAQzMYXk/85j1IxWJxp0cJdCLSFyxklGgjdZ1jn4LQIJno276GBx2EWU3GFJTC5+OuU3LL7hR4kXg5KaEcta7z4/dimkZmJ+VEqbbnJrqTEakZ5TC2/VRBQuiQ9KFtqCARqE42fWOMT43Sw2EsTQmNp+rfiYxESo2iwHRGRA/UvDcRl3oSuGKPsMxrpzq87GRMJKkGQWdHhCnHOsaTvHCPSaCajwwhVDLzB6YDIgk1oSnbBOTNx7FIGpWy55a920qpepVHVURH6ASdIQ9doCq6QTVURxQ9oRf0it6sZ+vd+rA+Z60FK585RP9gff8CUwKkeQ==</latexit><latexit sha1_base64="M+DFRKJgeIxw8z9uzqdWjNQfvwE=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiR1oe4KblxWsA9oSplMb9qhk0mYmYg1dOWvuHGrf+FOXAl+g9/gtM1C2x64cDjn3su9J0g4U9p1v6zCyura+kZx097a3tndc/YPGipOJYU6jXksWwFRwJmAumaaQyuRQKKAQzMYXk/85j1IxWJxp0cJdCLSFyxklGgjdZ1jn4LQIJno276GBx2EWU3GFJTC5+OuU3LL7hR4kXg5KaEcta7z4/dimkZmJ+VEqbbnJrqTEakZ5TC2/VRBQuiQ9KFtqCARqE42fWOMT43Sw2EsTQmNp+rfiYxESo2iwHRGRA/UvDcRl3oSuGKPsMxrpzq87GRMJKkGQWdHhCnHOsaTvHCPSaCajwwhVDLzB6YDIgk1oSnbBOTNx7FIGpWy55a920qpepVHVURH6ASdIQ9doCq6QTVURxQ9oRf0it6sZ+vd+rA+Z60FK585RP9gff8CUwKkeQ==</latexit><latexit sha1_base64="M+DFRKJgeIxw8z9uzqdWjNQfvwE=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiR1oe4KblxWsA9oSplMb9qhk0mYmYg1dOWvuHGrf+FOXAl+g9/gtM1C2x64cDjn3su9J0g4U9p1v6zCyura+kZx097a3tndc/YPGipOJYU6jXksWwFRwJmAumaaQyuRQKKAQzMYXk/85j1IxWJxp0cJdCLSFyxklGgjdZ1jn4LQIJno276GBx2EWU3GFJTC5+OuU3LL7hR4kXg5KaEcta7z4/dimkZmJ+VEqbbnJrqTEakZ5TC2/VRBQuiQ9KFtqCARqE42fWOMT43Sw2EsTQmNp+rfiYxESo2iwHRGRA/UvDcRl3oSuGKPsMxrpzq87GRMJKkGQWdHhCnHOsaTvHCPSaCajwwhVDLzB6YDIgk1oSnbBOTNx7FIGpWy55a920qpepVHVURH6ASdIQ9doCq6QTVURxQ9oRf0it6sZ+vd+rA+Z60FK585RP9gff8CUwKkeQ==</latexit><latexit sha1_base64="M+DFRKJgeIxw8z9uzqdWjNQfvwE=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiR1oe4KblxWsA9oSplMb9qhk0mYmYg1dOWvuHGrf+FOXAl+g9/gtM1C2x64cDjn3su9J0g4U9p1v6zCyura+kZx097a3tndc/YPGipOJYU6jXksWwFRwJmAumaaQyuRQKKAQzMYXk/85j1IxWJxp0cJdCLSFyxklGgjdZ1jn4LQIJno276GBx2EWU3GFJTC5+OuU3LL7hR4kXg5KaEcta7z4/dimkZmJ+VEqbbnJrqTEakZ5TC2/VRBQuiQ9KFtqCARqE42fWOMT43Sw2EsTQmNp+rfiYxESo2iwHRGRA/UvDcRl3oSuGKPsMxrpzq87GRMJKkGQWdHhCnHOsaTvHCPSaCajwwhVDLzB6YDIgk1oSnbBOTNx7FIGpWy55a920qpepVHVURH6ASdIQ9doCq6QTVURxQ9oRf0it6sZ+vd+rA+Z60FK585RP9gff8CUwKkeQ==</latexit> Process 4

<latexit sha1_base64="qCiG5N8/wAn0jCb70h1ejyHuKuY=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiRFUHcFNy4r2Ac0pUymN+3QySTMTMQauvJX3LjVv3AnrgS/wW9w2mahbQ9cOJxz7+XeEyScKe26X1ZhZXVtfaO4aW9t7+zuOfsHDRWnkkKdxjyWrYAo4ExAXTPNoZVIIFHAoRkMryd+8x6kYrG406MEOhHpCxYySrSRus6xT0FokEz0bV/Dgw7CrCZjCkrh83HXKblldwq8SLyclFCOWtf58XsxTSOzk3KiVNtzE93JiNSMchjbfqogIXRI+tA2VJAIVCebvjHGp0bp4TCWpoTGU/XvREYipUZRYDojogdq3puISz0JXLFHWOa1Ux1edjImklSDoLMjwpRjHeNJXrjHJFDNR4YQKpn5A9MBkYSa0JRtAvLm41gkjUrZc8vebaVUvcqjKqIjdILOkIcuUBXdoBqqI4qe0At6RW/Ws/VufVifs9aClc8con+wvn8BVKOkeg==</latexit><latexit sha1_base64="qCiG5N8/wAn0jCb70h1ejyHuKuY=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiRFUHcFNy4r2Ac0pUymN+3QySTMTMQauvJX3LjVv3AnrgS/wW9w2mahbQ9cOJxz7+XeEyScKe26X1ZhZXVtfaO4aW9t7+zuOfsHDRWnkkKdxjyWrYAo4ExAXTPNoZVIIFHAoRkMryd+8x6kYrG406MEOhHpCxYySrSRus6xT0FokEz0bV/Dgw7CrCZjCkrh83HXKblldwq8SLyclFCOWtf58XsxTSOzk3KiVNtzE93JiNSMchjbfqogIXRI+tA2VJAIVCebvjHGp0bp4TCWpoTGU/XvREYipUZRYDojogdq3puISz0JXLFHWOa1Ux1edjImklSDoLMjwpRjHeNJXrjHJFDNR4YQKpn5A9MBkYSa0JRtAvLm41gkjUrZc8vebaVUvcqjKqIjdILOkIcuUBXdoBqqI4qe0At6RW/Ws/VufVifs9aClc8con+wvn8BVKOkeg==</latexit><latexit sha1_base64="qCiG5N8/wAn0jCb70h1ejyHuKuY=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiRFUHcFNy4r2Ac0pUymN+3QySTMTMQauvJX3LjVv3AnrgS/wW9w2mahbQ9cOJxz7+XeEyScKe26X1ZhZXVtfaO4aW9t7+zuOfsHDRWnkkKdxjyWrYAo4ExAXTPNoZVIIFHAoRkMryd+8x6kYrG406MEOhHpCxYySrSRus6xT0FokEz0bV/Dgw7CrCZjCkrh83HXKblldwq8SLyclFCOWtf58XsxTSOzk3KiVNtzE93JiNSMchjbfqogIXRI+tA2VJAIVCebvjHGp0bp4TCWpoTGU/XvREYipUZRYDojogdq3puISz0JXLFHWOa1Ux1edjImklSDoLMjwpRjHeNJXrjHJFDNR4YQKpn5A9MBkYSa0JRtAvLm41gkjUrZc8vebaVUvcqjKqIjdILOkIcuUBXdoBqqI4qe0At6RW/Ws/VufVifs9aClc8con+wvn8BVKOkeg==</latexit><latexit sha1_base64="qCiG5N8/wAn0jCb70h1ejyHuKuY=">AAACInicbVDLSsNAFJ3UV42vqEsRBovgqiRFUHcFNy4r2Ac0pUymN+3QySTMTMQauvJX3LjVv3AnrgS/wW9w2mahbQ9cOJxz7+XeEyScKe26X1ZhZXVtfaO4aW9t7+zuOfsHDRWnkkKdxjyWrYAo4ExAXTPNoZVIIFHAoRkMryd+8x6kYrG406MEOhHpCxYySrSRus6xT0FokEz0bV/Dgw7CrCZjCkrh83HXKblldwq8SLyclFCOWtf58XsxTSOzk3KiVNtzE93JiNSMchjbfqogIXRI+tA2VJAIVCebvjHGp0bp4TCWpoTGU/XvREYipUZRYDojogdq3puISz0JXLFHWOa1Ux1edjImklSDoLMjwpRjHeNJXrjHJFDNR4YQKpn5A9MBkYSa0JRtAvLm41gkjUrZc8vebaVUvcqjKqIjdILOkIcuUBXdoBqqI4qe0At6RW/Ws/VufVifs9aClc8con+wvn8BVKOkeg==</latexit>

⌧ ⇠ 2Geo(↵)

y ⇠ Pois(µ)

y(±) := y + ⌧<latexit sha1_base64="pBV9WgJP/JfCuWpdQD4sOWUGEZQ=">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</latexit><latexit sha1_base64="pBV9WgJP/JfCuWpdQD4sOWUGEZQ=">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</latexit><latexit sha1_base64="pBV9WgJP/JfCuWpdQD4sOWUGEZQ=">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</latexit><latexit sha1_base64="pBV9WgJP/JfCuWpdQD4sOWUGEZQ=">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</latexit>

�(⇤) ⇠ Exp( ↵1�↵ )

g(⇤) ⇠ Pois(�(⇤))

⌧ := g(+) � g(�)

y ⇠ Pois(µ)

y(±) := y + ⌧<latexit sha1_base64="Eg59HZOJ+lbkOiY3p4FrRZnuJlA=">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</latexit><latexit sha1_base64="Eg59HZOJ+lbkOiY3p4FrRZnuJlA=">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</latexit><latexit sha1_base64="Eg59HZOJ+lbkOiY3p4FrRZnuJlA=">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</latexit><latexit sha1_base64="Eg59HZOJ+lbkOiY3p4FrRZnuJlA=">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</latexit>

�(⇤) ⇠ Exp( ↵1�↵ )

g(�) ⇠ Pois(�(�))

y(+) ⇠ Pois(µ+�(+))

y(±) := y(+) � g(�)<latexit sha1_base64="nmumql7nYhfXtkXrJMCFqkA+jpc=">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</latexit><latexit sha1_base64="nmumql7nYhfXtkXrJMCFqkA+jpc=">AAADTnichVJLa9tAEF4pfbjuy0mPvSw1LU6NjVQCDYFAoBRydKFOAl7XrFYje8muJHZXIc6i39S/0ksPvbT/oMfeSrtSdGhtpx1Y+Jj55pvHTpQLrk0QfPH8rVu379xt3Wvff/Dw0ePO9s6JzgrFYMwykamziGoQPIWx4UbAWa6AykjAaXT+poqfXoDSPEvfm2UOU0nnKU84o8a5ZtveMRGOHdMPlkhqFlpSIUDZ3svdssQviOYSEwOXRkn79jIve8QkijJLqMgXtLQhHuAG7xLSnq/KDNZlRhnXTmdz2YpfCRHDRQx2Wa4S+jcKygLbfolvEO7/T5jkspY+OMT/Lj7AG6ecdbrBMKgNr4OwAV3U2GjW+U7ijBUSUsME1XoSBrmZWqoMZwLKNik05JSd0zlMHEypBD219ZeX+LnzxDjJlHupwbX3zwxLpdZLGTlm3elqrHJujCkQml/BptikMMn+1PI0Lwyk7LqJpBDYZLi6LRxzBcyIpQOUKe7mwGxB3bkYd4Ftt6BwdR3r4OTVMAyG4bu97tFes6oWeoqeoR4K0Wt0hI7RCI0R8z56n72v3jf/k//D/+n/uqb6XpPzBP1lW63fHPUVEw==</latexit><latexit sha1_base64="nmumql7nYhfXtkXrJMCFqkA+jpc=">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</latexit><latexit sha1_base64="nmumql7nYhfXtkXrJMCFqkA+jpc=">AAADTnichVJLa9tAEF4pfbjuy0mPvSw1LU6NjVQCDYFAoBRydKFOAl7XrFYje8muJHZXIc6i39S/0ksPvbT/oMfeSrtSdGhtpx1Y+Jj55pvHTpQLrk0QfPH8rVu379xt3Wvff/Dw0ePO9s6JzgrFYMwykamziGoQPIWx4UbAWa6AykjAaXT+poqfXoDSPEvfm2UOU0nnKU84o8a5ZtveMRGOHdMPlkhqFlpSIUDZ3svdssQviOYSEwOXRkn79jIve8QkijJLqMgXtLQhHuAG7xLSnq/KDNZlRhnXTmdz2YpfCRHDRQx2Wa4S+jcKygLbfolvEO7/T5jkspY+OMT/Lj7AG6ecdbrBMKgNr4OwAV3U2GjW+U7ijBUSUsME1XoSBrmZWqoMZwLKNik05JSd0zlMHEypBD219ZeX+LnzxDjJlHupwbX3zwxLpdZLGTlm3elqrHJujCkQml/BptikMMn+1PI0Lwyk7LqJpBDYZLi6LRxzBcyIpQOUKe7mwGxB3bkYd4Ftt6BwdR3r4OTVMAyG4bu97tFes6oWeoqeoR4K0Wt0hI7RCI0R8z56n72v3jf/k//D/+n/uqb6XpPzBP1lW63fHPUVEw==</latexit>

�(⇤) ⇠ Exp( ↵1�↵ )

y(±) ⇠ Skel��(+)+µ, �(�)

�<latexit sha1_base64="idxw1tNbXhiZuiNvUTkk3kTnkCc=">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</latexit><latexit sha1_base64="idxw1tNbXhiZuiNvUTkk3kTnkCc=">AAAC03icdVJNbxMxEPUuXyV8NIUjF5cIlNA22kWR4FgJIXEsgrQVcYi83tnEir27smerBmsviCt/jTt/gt+AN90DJO1Ilp7eezMaPzsplbQYRb+D8NbtO3fv7dzvPHj46PFud+/JqS0qI2AsClWY84RbUDKHMUpUcF4a4DpRcJYs3zX62QUYK4v8M65KmGo+z2UmBUdPzbq/mPLmlH91THNcWM2VAuP6rwZ1TV8yKzVlCJdotHt/WdZ9hpnhwjGuygWvXUyPaIsHjHUYSpWCW9Wb41iptwd+WoKqWSLn/RuWOGh62P4B26dMV4fskN5gPPLGZtBg1u1Fw2hddBvELeiRtk5m3T8sLUSlIUehuLWTOCpx6rhBKRTUHVZZKLlY8jlMPMy5Bjt169xr+sIzKc0K40+OdM3+2+G4tnalE+9cr7upNeS1mgFl5Te4TptUmL2dOpmXFUIurpbIKkWxoM0D01QaEKhWHnBhpL8HFQvuHw39N+j4gOLNOLbB6ethHA3jj6Pe8aiNaoc8I89Jn8TkDTkmH8gJGRMRjIIvgQjScBy68Hv448oaBm3PU/JfhT//AiJ7480=</latexit><latexit sha1_base64="idxw1tNbXhiZuiNvUTkk3kTnkCc=">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</latexit><latexit sha1_base64="idxw1tNbXhiZuiNvUTkk3kTnkCc=">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</latexit>

True data yij

True µ⇤ij

Non-private µij Our estimate µij

Naıve estimate µij

Noised data y(±)ij

Increasing privacy ✏/N=1<latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit><latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit><latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit><latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit>

✏/N=0.5<latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit><latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit><latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit><latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit>

✏/N=0.25<latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit><latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit><latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit><latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit>

True data yij

True µ⇤ij

Non-private µij Our estimate µij

Naıve estimate µij

Noised data y(±)ij

Increasing privacy ✏/N=1<latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit><latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit><latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit><latexit sha1_base64="j3BuK7EHK7lxhCJ2ICD/7fEl4Ms=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM7UIXdVEBN0IBTeupIJ9QBPKZDppByeTMDMRaij+ihsXirj1P9z5N07bLLT1wIXDOfdy7z1BwpnSjvNtFZaWV1bXiuuljc2t7R17d6+l4lQS2iQxj2UnwIpyJmhTM81pJ5EURwGn7eD+auK3H6hULBZ3epRQP8IDwUJGsDZSzz7waKIYjwU6QTfIK196ZeT27IpTc6ZAi8TNSQVyNHr2l9ePSRpRoQnHSnVdJ9F+hqVmhNNxyUsVTTC5xwPaNVTgiCo/m14/RsdG6aMwlqaERlP190SGI6VGUWA6I6yHat6biP953VSHF37GRJJqKshsUZhypGM0iQL1maRE85EhmEhmbkVkiCUm2gRWMiG48y8vktZpzXVq7u1ZpV7N4yjCIRxBFVw4hzpcQwOaQOARnuEV3qwn68V6tz5mrQUrn9mHP7A+fwBPfpMU</latexit>

✏/N=0.5<latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit><latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit><latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit><latexit sha1_base64="Ng0gtFtWYod03th7XEOH2TNuvkg=">AAAB/XicbVDLSsNAFJ3UV62v+Ni5mVqErmIiim6EghtXUsE+oAllMp20QyczYWYi1FD8FTcuFHHrf7jzb5y2WWjrgQuHc+7l3nvChFGlXffbKiwtr6yuFddLG5tb2zv27l5TiVRi0sCCCdkOkSKMctLQVDPSTiRBcchIKxxeT/zWA5GKCn6vRwkJYtTnNKIYaSN17QOfJIoywU9uoV++8svQdc67dsV13CngIvFyUgE56l37y+8JnMaEa8yQUh3PTXSQIakpZmRc8lNFEoSHqE86hnIUExVk0+vH8NgoPRgJaYprOFV/T2QoVmoUh6YzRnqg5r2J+J/XSXV0GWSUJ6kmHM8WRSmDWsBJFLBHJcGajQxBWFJzK8QDJBHWJrCSCcGbf3mRNE8dz3W8u7NKrZrHUQSH4AhUgQcuQA3cgDpoAAwewTN4BW/Wk/VivVsfs9aClc/sgz+wPn8AhM2TNg==</latexit>

✏/N=0.25<latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit><latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit><latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit><latexit sha1_base64="P8JoUQIFhMtPZEZq/fLaV9Xkt6E=">AAAB/nicbVDLSsNAFJ3UV62vqLhyM7UIXcWkKLoRCm5cSQX7gCaUyXTSDp3MhJmJUELBX3HjQhG3foc7/8Zpm4W2HrhwOOde7r0nTBhV2nW/rcLK6tr6RnGztLW9s7tn7x+0lEglJk0smJCdECnCKCdNTTUjnUQSFIeMtMPRzdRvPxKpqOAPepyQIEYDTiOKkTZSzz7ySaIoE/zsDvrla78MXad20bMrruPOAJeJl5MKyNHo2V9+X+A0JlxjhpTqem6igwxJTTEjk5KfKpIgPEID0jWUo5ioIJudP4GnRunDSEhTXMOZ+nsiQ7FS4zg0nTHSQ7XoTcX/vG6qo6sgozxJNeF4vihKGdQCTrOAfSoJ1mxsCMKSmlshHiKJsDaJlUwI3uLLy6RVczzX8e7PK/VqHkcRHIMTUAUeuAR1cAsaoAkwyMAzeAVv1pP1Yr1bH/PWgpXPHII/sD5/APuYk3I=</latexit>

word type v

documentd

True topics �⇤k

Non-private estimate �k

word type v word type v

documentd

word type v

Our estimate �k

Naıve estimate �k

word type v

documentd

word type v

Naıve estimate �k

Our estimate �k

Naıve estimate �k

123✏/N = ln(1/↵)

-80.0

-75.0

-70.0

-65.0

-60.0

Coh

eren

ce

123✏/N = ln(1/↵)

0.1

0.2

0.3

0.4

Rec

onst

ruct

ion

MAE

Model Type

Naıve

Proposed

Non-private

123✏/N = ln(1/↵)

1.0

1.5

2.0

2.5

3.0

NPM

I

123✏/N = ln(1/↵)

7.0

8.0

9.0

10.0

11.0

12.0H

eld-

out

MAE

Number of communities = 5

Model Type

Naıve

Proposed

Non-private

123✏/N = ln(1/↵)

Number of communities = 20

123✏/N = ln(1/↵)

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

Rec

onst

ruct

ion

MAE

Number of communities = 5Model Type

Naıve

Proposed

Non-private

123✏/N = ln(1/↵)

Number of communities = 20

Locally Private Bayesian Inference for Count ModelsAaron Schein Zhiwei Steven Wu Mingyuan Zhou Hanna WallachAlexandra SchofieldUMass Amherst Univ. of Minnesota Cornell University Univ. of Texas Austin Microsoft Research

�80 �60 �40 �20 0 20 40 60 80⌧

0.00

0.10

0.20

0.30

0.40

P(⌧

|↵)

↵=0.4

↵=0.6

↵=0.8

Two-sided geometric distribution

�80 �60 �40 �20 0 20 40 60 80⌧

0.000

0.025

0.050

0.075

0.100

0.125

P(⌧

|µ1,

µ2)

µ1=5, µ2=5

µ1=100, µ2=100

µ1=20, µ2=1

µ1=30, µ2=40

µ1=10, µ2=50

Skellam distribution

0 20 40 60 80 100m

0.00

0.05

0.10

0.15

0.20

0.25

0.30

P(m

|v,a

)

v=-0.5, a=30

v=-0.8, a=10

v=1, a=50

v=2, a=7

v=10, a=150

v=400, a=300

Bessel distribution

Limited-precision local privacy

Summary

Auxiliary-variable MCMC

Poisson factorization + geometric mechanism

The geometric mechanism as Skellam noise

Case study 1: Topic modeling

Case study 2: Community detection

BaselinesNaïve method:

P (R(y) 2 S) e✏P (R(y0) 2 S) ,<latexit sha1_base64="Ev7ZWS9HQph5wjTgegxF40GH/YY=">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</latexit><latexit sha1_base64="Ev7ZWS9HQph5wjTgegxF40GH/YY=">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</latexit><latexit sha1_base64="Ev7ZWS9HQph5wjTgegxF40GH/YY=">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</latexit><latexit sha1_base64="Ev7ZWS9HQph5wjTgegxF40GH/YY=">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</latexit>

Definition: A randomized response method R(…) is

an (N, ϵ)-private LPLP mechanism if,

A given R(…) could be interpreted as either an (N, ϵ)-private or (N’, ϵ’)-private LPLP mechanism.

||y � y0||1 N.<latexit sha1_base64="BLl7HEIkH8giReC89e1ICwOdhuE=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UUwZCIoMuiG1dSwT6gCWEynbRDJ5M4MxFC6D+48VfcuFDErRt3/o3TNAttPTCXwzn3cueeIGFUKtv+NioLi0vLK9XV2tr6xuaWub3TlnEqMGnhmMWiGyBJGOWkpahipJsIgqKAkU4wupr4nQciJI35ncoS4kVowGlIMVJa8s1jl2lbTSvM4AnMDqErCq2ovgNdRu7hjeWbdduyC8B54pSkDko0ffPL7cc4jQhXmCEpe46dKC9HQlHMyLjmppIkCI/QgPQ05Sgi0suLm8bwQCt9GMZCP65gof6eyFEkZRYFujNCaihnvYn4n9dLVXjh5ZQnqSIcTxeFKYMqhpOAYJ8KghXLNEFYUP1XiIdIIKx0jDUdgjN78jxpn1qObTm3Z/XGZRlHFeyBfXAEHHAOGuAaNEELYPAInsEreDOejBfj3fiYtlaMcmYX/IHx+QNk8J0v</latexit><latexit sha1_base64="BLl7HEIkH8giReC89e1ICwOdhuE=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UUwZCIoMuiG1dSwT6gCWEynbRDJ5M4MxFC6D+48VfcuFDErRt3/o3TNAttPTCXwzn3cueeIGFUKtv+NioLi0vLK9XV2tr6xuaWub3TlnEqMGnhmMWiGyBJGOWkpahipJsIgqKAkU4wupr4nQciJI35ncoS4kVowGlIMVJa8s1jl2lbTSvM4AnMDqErCq2ovgNdRu7hjeWbdduyC8B54pSkDko0ffPL7cc4jQhXmCEpe46dKC9HQlHMyLjmppIkCI/QgPQ05Sgi0suLm8bwQCt9GMZCP65gof6eyFEkZRYFujNCaihnvYn4n9dLVXjh5ZQnqSIcTxeFKYMqhpOAYJ8KghXLNEFYUP1XiIdIIKx0jDUdgjN78jxpn1qObTm3Z/XGZRlHFeyBfXAEHHAOGuAaNEELYPAInsEreDOejBfj3fiYtlaMcmYX/IHx+QNk8J0v</latexit><latexit sha1_base64="BLl7HEIkH8giReC89e1ICwOdhuE=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UUwZCIoMuiG1dSwT6gCWEynbRDJ5M4MxFC6D+48VfcuFDErRt3/o3TNAttPTCXwzn3cueeIGFUKtv+NioLi0vLK9XV2tr6xuaWub3TlnEqMGnhmMWiGyBJGOWkpahipJsIgqKAkU4wupr4nQciJI35ncoS4kVowGlIMVJa8s1jl2lbTSvM4AnMDqErCq2ovgNdRu7hjeWbdduyC8B54pSkDko0ffPL7cc4jQhXmCEpe46dKC9HQlHMyLjmppIkCI/QgPQ05Sgi0suLm8bwQCt9GMZCP65gof6eyFEkZRYFujNCaihnvYn4n9dLVXjh5ZQnqSIcTxeFKYMqhpOAYJ8KghXLNEFYUP1XiIdIIKx0jDUdgjN78jxpn1qObTm3Z/XGZRlHFeyBfXAEHHAOGuAaNEELYPAInsEreDOejBfj3fiYtlaMcmYX/IHx+QNk8J0v</latexit><latexit sha1_base64="BLl7HEIkH8giReC89e1ICwOdhuE=">AAACE3icbVDLSsNAFJ3UV62vqEs3g0UUwZCIoMuiG1dSwT6gCWEynbRDJ5M4MxFC6D+48VfcuFDErRt3/o3TNAttPTCXwzn3cueeIGFUKtv+NioLi0vLK9XV2tr6xuaWub3TlnEqMGnhmMWiGyBJGOWkpahipJsIgqKAkU4wupr4nQciJI35ncoS4kVowGlIMVJa8s1jl2lbTSvM4AnMDqErCq2ovgNdRu7hjeWbdduyC8B54pSkDko0ffPL7cc4jQhXmCEpe46dKC9HQlHMyLjmppIkCI/QgPQ05Sgi0suLm8bwQCt9GMZCP65gof6eyFEkZRYFujNCaihnvYn4n9dLVXjh5ZQnqSIcTxeFKYMqhpOAYJ8KghXLNEFYUP1XiIdIIKx0jDUdgjN78jxpn1qObTm3Z/XGZRlHFeyBfXAEHHAOGuAaNEELYPAInsEreDOejBfj3fiYtlaMcmYX/IHx+QNk8J0v</latexit>

for any two observation such that:

Darker means more private

(smaller ϵ).

Poisson factorization

ydv ⇠ Pois(µdv) µdv = ✓Td �v

▪ Model count data with Poisson factorization▪ Privacy: Noise data using the geometric mechanism▪ How do we do Bayesian inference given noised data?▪ Reinterpret geometric mechanism as Skellam noise▪ Prove link between Skellam and Bessel distribution▪ Solution: auxiliary-variable private MCMC method▪ Bonus: Private outperforms non-private method!

Locally private Bayesian inference

PM(Y, Z)<latexit sha1_base64="aynb1y7E4ZsK6bAJY4vD2Lz1l9I=">AAAB/nicdVDLSgMxFM34rPVVFVdugkWoIEOmz3FXdONGqGAf2paSSTNtaOZBkhHKMOCvuHGhiFu/w51/Y/oQVPRA4HDOvdyT44ScSYXQh7GwuLS8sppaS69vbG5tZ3Z2GzKIBKF1EvBAtBwsKWc+rSumOG2FgmLP4bTpjM4nfvOOCskC/1qNQ9r18MBnLiNYaamX2a/14o6H1ZBgHl8mSe7mBN4e9zJZZFZsO18+hcgslcuFAtKkYJeKCEHLRFNkwRy1Xua90w9I5FFfEY6lbFsoVN0YC8UIp0m6E0kaYjLCA9rW1Mceld14Gj+BR1rpQzcQ+vkKTtXvGzH2pBx7jp6cJJW/vYn4l9eOlGt3Y+aHkaI+mR1yIw5VACddwD4TlCg+1gQTwXRWSIZYYKJ0Y2ldwtdP4f+kkTctZFpXxWz1bF5HChyAQ5ADFqiAKrgANVAHBMTgATyBZ+PeeDRejNfZ6IIx39kDP2C8fQKgS5VC</latexit><latexit sha1_base64="aynb1y7E4ZsK6bAJY4vD2Lz1l9I=">AAAB/nicdVDLSgMxFM34rPVVFVdugkWoIEOmz3FXdONGqGAf2paSSTNtaOZBkhHKMOCvuHGhiFu/w51/Y/oQVPRA4HDOvdyT44ScSYXQh7GwuLS8sppaS69vbG5tZ3Z2GzKIBKF1EvBAtBwsKWc+rSumOG2FgmLP4bTpjM4nfvOOCskC/1qNQ9r18MBnLiNYaamX2a/14o6H1ZBgHl8mSe7mBN4e9zJZZFZsO18+hcgslcuFAtKkYJeKCEHLRFNkwRy1Xua90w9I5FFfEY6lbFsoVN0YC8UIp0m6E0kaYjLCA9rW1Mceld14Gj+BR1rpQzcQ+vkKTtXvGzH2pBx7jp6cJJW/vYn4l9eOlGt3Y+aHkaI+mR1yIw5VACddwD4TlCg+1gQTwXRWSIZYYKJ0Y2ldwtdP4f+kkTctZFpXxWz1bF5HChyAQ5ADFqiAKrgANVAHBMTgATyBZ+PeeDRejNfZ6IIx39kDP2C8fQKgS5VC</latexit><latexit sha1_base64="aynb1y7E4ZsK6bAJY4vD2Lz1l9I=">AAAB/nicdVDLSgMxFM34rPVVFVdugkWoIEOmz3FXdONGqGAf2paSSTNtaOZBkhHKMOCvuHGhiFu/w51/Y/oQVPRA4HDOvdyT44ScSYXQh7GwuLS8sppaS69vbG5tZ3Z2GzKIBKF1EvBAtBwsKWc+rSumOG2FgmLP4bTpjM4nfvOOCskC/1qNQ9r18MBnLiNYaamX2a/14o6H1ZBgHl8mSe7mBN4e9zJZZFZsO18+hcgslcuFAtKkYJeKCEHLRFNkwRy1Xua90w9I5FFfEY6lbFsoVN0YC8UIp0m6E0kaYjLCA9rW1Mceld14Gj+BR1rpQzcQ+vkKTtXvGzH2pBx7jp6cJJW/vYn4l9eOlGt3Y+aHkaI+mR1yIw5VACddwD4TlCg+1gQTwXRWSIZYYKJ0Y2ldwtdP4f+kkTctZFpXxWz1bF5HChyAQ5ADFqiAKrgANVAHBMTgATyBZ+PeeDRejNfZ6IIx39kDP2C8fQKgS5VC</latexit><latexit sha1_base64="aynb1y7E4ZsK6bAJY4vD2Lz1l9I=">AAAB/nicdVDLSgMxFM34rPVVFVdugkWoIEOmz3FXdONGqGAf2paSSTNtaOZBkhHKMOCvuHGhiFu/w51/Y/oQVPRA4HDOvdyT44ScSYXQh7GwuLS8sppaS69vbG5tZ3Z2GzKIBKF1EvBAtBwsKWc+rSumOG2FgmLP4bTpjM4nfvOOCskC/1qNQ9r18MBnLiNYaamX2a/14o6H1ZBgHl8mSe7mBN4e9zJZZFZsO18+hcgslcuFAtKkYJeKCEHLRFNkwRy1Xua90w9I5FFfEY6lbFsoVN0YC8UIp0m6E0kaYjLCA9rW1Mceld14Gj+BR1rpQzcQ+vkKTtXvGzH2pBx7jp6cJJW/vYn4l9eOlGt3Y+aHkaI+mR1yIw5VACddwD4TlCg+1gQTwXRWSIZYYKJ0Y2ldwtdP4f+kkTctZFpXxWz1bF5HChyAQ5ADFqiAKrgANVAHBMTgATyBZ+PeeDRejNfZ6IIx39kDP2C8fQKgS5VC</latexit>

PR(Y |Y )<latexit sha1_base64="Oq5/Q6maoK+gz7tlSn/37oRy+SY=">AAACCXicdVDLSsNAFJ3UV62vqks3g0WoUEJS+loW3bisYl80pUwmk3bo5MHMRCgxWzf+ihsXirj1D9z5N07bCCp64MLhnHu59x47ZFRIw/jQMiura+sb2c3c1vbO7l5+/6Ajgohj0sYBC3jPRoIw6pO2pJKRXsgJ8mxGuvb0fO53bwgXNPCv5SwkQw+NfepSjKSSRnnYGsWWh+QEIxZfJUnRkpQ5JO4nVunWKvVPR/mCoRsLQEMvm9Vao6ZIpVauV6rQTK0CSNEa5d8tJ8CRR3yJGRJiYBqhHMaIS4oZSXJWJEiI8BSNyUBRH3lEDOPFJwk8UYoD3YCr8iVcqN8nYuQJMfNs1Tk/Wvz25uJf3iCSbmMYUz+MJPHxcpEbMSgDOI8FOpQTLNlMEYQ5VbdCPEEcYanCy6kQvj6F/5NOWTcN3bysFJpnaRxZcASOQRGYoA6a4AK0QBtgcAcewBN41u61R+1Fe122ZrR05hD8gPb2CZVomj8=</latexit><latexit sha1_base64="Oq5/Q6maoK+gz7tlSn/37oRy+SY=">AAACCXicdVDLSsNAFJ3UV62vqks3g0WoUEJS+loW3bisYl80pUwmk3bo5MHMRCgxWzf+ihsXirj1D9z5N07bCCp64MLhnHu59x47ZFRIw/jQMiura+sb2c3c1vbO7l5+/6Ajgohj0sYBC3jPRoIw6pO2pJKRXsgJ8mxGuvb0fO53bwgXNPCv5SwkQw+NfepSjKSSRnnYGsWWh+QEIxZfJUnRkpQ5JO4nVunWKvVPR/mCoRsLQEMvm9Vao6ZIpVauV6rQTK0CSNEa5d8tJ8CRR3yJGRJiYBqhHMaIS4oZSXJWJEiI8BSNyUBRH3lEDOPFJwk8UYoD3YCr8iVcqN8nYuQJMfNs1Tk/Wvz25uJf3iCSbmMYUz+MJPHxcpEbMSgDOI8FOpQTLNlMEYQ5VbdCPEEcYanCy6kQvj6F/5NOWTcN3bysFJpnaRxZcASOQRGYoA6a4AK0QBtgcAcewBN41u61R+1Fe122ZrR05hD8gPb2CZVomj8=</latexit><latexit sha1_base64="Oq5/Q6maoK+gz7tlSn/37oRy+SY=">AAACCXicdVDLSsNAFJ3UV62vqks3g0WoUEJS+loW3bisYl80pUwmk3bo5MHMRCgxWzf+ihsXirj1D9z5N07bCCp64MLhnHu59x47ZFRIw/jQMiura+sb2c3c1vbO7l5+/6Ajgohj0sYBC3jPRoIw6pO2pJKRXsgJ8mxGuvb0fO53bwgXNPCv5SwkQw+NfepSjKSSRnnYGsWWh+QEIxZfJUnRkpQ5JO4nVunWKvVPR/mCoRsLQEMvm9Vao6ZIpVauV6rQTK0CSNEa5d8tJ8CRR3yJGRJiYBqhHMaIS4oZSXJWJEiI8BSNyUBRH3lEDOPFJwk8UYoD3YCr8iVcqN8nYuQJMfNs1Tk/Wvz25uJf3iCSbmMYUz+MJPHxcpEbMSgDOI8FOpQTLNlMEYQ5VbdCPEEcYanCy6kQvj6F/5NOWTcN3bysFJpnaRxZcASOQRGYoA6a4AK0QBtgcAcewBN41u61R+1Fe122ZrR05hD8gPb2CZVomj8=</latexit><latexit sha1_base64="Oq5/Q6maoK+gz7tlSn/37oRy+SY=">AAACCXicdVDLSsNAFJ3UV62vqks3g0WoUEJS+loW3bisYl80pUwmk3bo5MHMRCgxWzf+ihsXirj1D9z5N07bCCp64MLhnHu59x47ZFRIw/jQMiura+sb2c3c1vbO7l5+/6Ajgohj0sYBC3jPRoIw6pO2pJKRXsgJ8mxGuvb0fO53bwgXNPCv5SwkQw+NfepSjKSSRnnYGsWWh+QEIxZfJUnRkpQ5JO4nVunWKvVPR/mCoRsLQEMvm9Vao6ZIpVauV6rQTK0CSNEa5d8tJ8CRR3yJGRJiYBqhHMaIS4oZSXJWJEiI8BSNyUBRH3lEDOPFJwk8UYoD3YCr8iVcqN8nYuQJMfNs1Tk/Wvz25uJf3iCSbmMYUz+MJPHxcpEbMSgDOI8FOpQTLNlMEYQ5VbdCPEEcYanCy6kQvj6F/5NOWTcN3bysFJpnaRxZcASOQRGYoA6a4AK0QBtgcAcewBN41u61R+1Fe122ZrR05hD8gPb2CZVomj8=</latexit>

1) Build non-private model M for sensitive data Y and latent variables Z (e.g., Poisson factorization):

2) Define randomized response method R:

3) Approximate locally-private posterior:

PM,R(Z | Y )<latexit sha1_base64="SA9vJcmR80YZHj1Bg0DeM12k/LM=">AAACFXicdVDLSgMxFM3UV62vqks3wSJUKCVTh9ruim7cCFXsQzulZNK0Dc08SDJCGecn3Pgrblwo4lZw59+YPhQVPXDhcM693HuPE3AmFULvRmJufmFxKbmcWlldW99Ib27VpR8KQmvE575oOlhSzjxaU0xx2gwExa7DacMZHo/9xjUVkvnehRoFtO3ivsd6jGClpU46V+1EtovVgGAenca5L34ex9krO3dj52zFeJdGl/F+J51BebNolVABorxloXIRaVI4MMslC5p5NEEGzFDtpN/srk9Cl3qKcCxly0SBakdYKEY4jVN2KGmAyRD3aUtTD7tUtqPJVzHc00oX9nyhy1Nwon6fiLAr5ch1dOf4aPnbG4t/ea1Q9UrtiHlBqKhHpot6IYfKh+OIYJcJShQfaYKJYPpWSAZYYKJ0kCkdwuen8H9SL+RNndWZlakczeJIgh2wC7LABIegAk5AFdQAAbfgHjyCJ+POeDCejZdpa8KYzWyDHzBePwAk859u</latexit><latexit sha1_base64="SA9vJcmR80YZHj1Bg0DeM12k/LM=">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</latexit><latexit sha1_base64="SA9vJcmR80YZHj1Bg0DeM12k/LM=">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</latexit><latexit sha1_base64="SA9vJcmR80YZHj1Bg0DeM12k/LM=">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</latexit>

=

ZdY

<latexit sha1_base64="8rryensyEOUrexLjsfPniWrXodE=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyURQTdC0Y3LCvYhTSiT6aQdOpmEmYlQQ/FX3LhQxK3/4c6/cdJmoa0HBg7n3Ms9c4KEM6Ud59sqLS2vrK6V1ysbm1vbO/buXkvFqSS0SWIey06AFeVM0KZmmtNOIimOAk7bweg699sPVCoWizs9Tqgf4YFgISNYG6lnH6BL5DGhkRdhPQzCrD+5Rz276tScKdAicQtShQKNnv3l9WOSRlRowrFSXddJtJ9hqRnhdFLxUkUTTEZ4QLuGChxR5WfT9BN0bJQ+CmNpngkyVX9vZDhSahwFZjLPqOa9XPzP66Y6vPAzJpJUU0Fmh8KUIx2jvArUZ5ISzceGYCKZyYrIEEtMtCmsYkpw57+8SFqnNdepubdn1fpVUUcZDuEITsCFc6jDDTSgCQQe4Rle4c16sl6sd+tjNlqyip19+APr8wcavpRa</latexit><latexit sha1_base64="8rryensyEOUrexLjsfPniWrXodE=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyURQTdC0Y3LCvYhTSiT6aQdOpmEmYlQQ/FX3LhQxK3/4c6/cdJmoa0HBg7n3Ms9c4KEM6Ud59sqLS2vrK6V1ysbm1vbO/buXkvFqSS0SWIey06AFeVM0KZmmtNOIimOAk7bweg699sPVCoWizs9Tqgf4YFgISNYG6lnH6BL5DGhkRdhPQzCrD+5Rz276tScKdAicQtShQKNnv3l9WOSRlRowrFSXddJtJ9hqRnhdFLxUkUTTEZ4QLuGChxR5WfT9BN0bJQ+CmNpngkyVX9vZDhSahwFZjLPqOa9XPzP66Y6vPAzJpJUU0Fmh8KUIx2jvArUZ5ISzceGYCKZyYrIEEtMtCmsYkpw57+8SFqnNdepubdn1fpVUUcZDuEITsCFc6jDDTSgCQQe4Rle4c16sl6sd+tjNlqyip19+APr8wcavpRa</latexit><latexit sha1_base64="8rryensyEOUrexLjsfPniWrXodE=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyURQTdC0Y3LCvYhTSiT6aQdOpmEmYlQQ/FX3LhQxK3/4c6/cdJmoa0HBg7n3Ms9c4KEM6Ud59sqLS2vrK6V1ysbm1vbO/buXkvFqSS0SWIey06AFeVM0KZmmtNOIimOAk7bweg699sPVCoWizs9Tqgf4YFgISNYG6lnH6BL5DGhkRdhPQzCrD+5Rz276tScKdAicQtShQKNnv3l9WOSRlRowrFSXddJtJ9hqRnhdFLxUkUTTEZ4QLuGChxR5WfT9BN0bJQ+CmNpngkyVX9vZDhSahwFZjLPqOa9XPzP66Y6vPAzJpJUU0Fmh8KUIx2jvArUZ5ISzceGYCKZyYrIEEtMtCmsYkpw57+8SFqnNdepubdn1fpVUUcZDuEITsCFc6jDDTSgCQQe4Rle4c16sl6sd+tjNlqyip19+APr8wcavpRa</latexit><latexit sha1_base64="8rryensyEOUrexLjsfPniWrXodE=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyURQTdC0Y3LCvYhTSiT6aQdOpmEmYlQQ/FX3LhQxK3/4c6/cdJmoa0HBg7n3Ms9c4KEM6Ud59sqLS2vrK6V1ysbm1vbO/buXkvFqSS0SWIey06AFeVM0KZmmtNOIimOAk7bweg699sPVCoWizs9Tqgf4YFgISNYG6lnH6BL5DGhkRdhPQzCrD+5Rz276tScKdAicQtShQKNnv3l9WOSRlRowrFSXddJtJ9hqRnhdFLxUkUTTEZ4QLuGChxR5WfT9BN0bJQ+CmNpngkyVX9vZDhSahwFZjLPqOa9XPzP66Y6vPAzJpJUU0Fmh8KUIx2jvArUZ5ISzceGYCKZyYrIEEtMtCmsYkpw57+8SFqnNdepubdn1fpVUUcZDuEITsCFc6jDDTSgCQQe4Rle4c16sl6sd+tjNlqyip19+APr8wcavpRa</latexit>

PM(Z |Y )<latexit sha1_base64="wMU/ChERHazBxANVtWvD+BtSdIQ=">AAACAXicdVDJSgNBEO2JW4xb1IvgpTEIEcLQk91b0IsXIYJZNBNCT6eTNOlZ6O4Rwhgv/ooXD4p49S+8+Td2FkFFHxQ83quiqp4TcCYVQh9GbGFxaXklvppYW9/Y3Epu79SlHwpCa8Tnvmg6WFLOPFpTTHHaDATFrsNpwxmeTvzGDRWS+d6lGgW07eK+x3qMYKWlTnKv2olsF6sBwTw6H4/T13bm1s5cHXWSKWSWyuVs8Rgis1As5nJIk1y5kEcIWiaaIgXmqHaS73bXJ6FLPUU4lrJloUC1IywUI5yOE3YoaYDJEPdpS1MPu1S2o+kHY3iolS7s+UKXp+BU/T4RYVfKkevozsmx8rc3Ef/yWqHqldsR84JQUY/MFvVCDpUPJ3HALhOUKD7SBBPB9K2QDLDAROnQEjqEr0/h/6SeNS1kWhf5VOVkHkcc7IMDkAYWKIEKOANVUAME3IEH8ASejXvj0XgxXmetMWM+swt+wHj7BA6KlqA=</latexit><latexit sha1_base64="wMU/ChERHazBxANVtWvD+BtSdIQ=">AAACAXicdVDJSgNBEO2JW4xb1IvgpTEIEcLQk91b0IsXIYJZNBNCT6eTNOlZ6O4Rwhgv/ooXD4p49S+8+Td2FkFFHxQ83quiqp4TcCYVQh9GbGFxaXklvppYW9/Y3Epu79SlHwpCa8Tnvmg6WFLOPFpTTHHaDATFrsNpwxmeTvzGDRWS+d6lGgW07eK+x3qMYKWlTnKv2olsF6sBwTw6H4/T13bm1s5cHXWSKWSWyuVs8Rgis1As5nJIk1y5kEcIWiaaIgXmqHaS73bXJ6FLPUU4lrJloUC1IywUI5yOE3YoaYDJEPdpS1MPu1S2o+kHY3iolS7s+UKXp+BU/T4RYVfKkevozsmx8rc3Ef/yWqHqldsR84JQUY/MFvVCDpUPJ3HALhOUKD7SBBPB9K2QDLDAROnQEjqEr0/h/6SeNS1kWhf5VOVkHkcc7IMDkAYWKIEKOANVUAME3IEH8ASejXvj0XgxXmetMWM+swt+wHj7BA6KlqA=</latexit><latexit sha1_base64="wMU/ChERHazBxANVtWvD+BtSdIQ=">AAACAXicdVDJSgNBEO2JW4xb1IvgpTEIEcLQk91b0IsXIYJZNBNCT6eTNOlZ6O4Rwhgv/ooXD4p49S+8+Td2FkFFHxQ83quiqp4TcCYVQh9GbGFxaXklvppYW9/Y3Epu79SlHwpCa8Tnvmg6WFLOPFpTTHHaDATFrsNpwxmeTvzGDRWS+d6lGgW07eK+x3qMYKWlTnKv2olsF6sBwTw6H4/T13bm1s5cHXWSKWSWyuVs8Rgis1As5nJIk1y5kEcIWiaaIgXmqHaS73bXJ6FLPUU4lrJloUC1IywUI5yOE3YoaYDJEPdpS1MPu1S2o+kHY3iolS7s+UKXp+BU/T4RYVfKkevozsmx8rc3Ef/yWqHqldsR84JQUY/MFvVCDpUPJ3HALhOUKD7SBBPB9K2QDLDAROnQEjqEr0/h/6SeNS1kWhf5VOVkHkcc7IMDkAYWKIEKOANVUAME3IEH8ASejXvj0XgxXmetMWM+swt+wHj7BA6KlqA=</latexit><latexit sha1_base64="wMU/ChERHazBxANVtWvD+BtSdIQ=">AAACAXicdVDJSgNBEO2JW4xb1IvgpTEIEcLQk91b0IsXIYJZNBNCT6eTNOlZ6O4Rwhgv/ooXD4p49S+8+Td2FkFFHxQ83quiqp4TcCYVQh9GbGFxaXklvppYW9/Y3Epu79SlHwpCa8Tnvmg6WFLOPFpTTHHaDATFrsNpwxmeTvzGDRWS+d6lGgW07eK+x3qMYKWlTnKv2olsF6sBwTw6H4/T13bm1s5cHXWSKWSWyuVs8Rgis1As5nJIk1y5kEcIWiaaIgXmqHaS73bXJ6FLPUU4lrJloUC1IywUI5yOE3YoaYDJEPdpS1MPu1S2o+kHY3iolS7s+UKXp+BU/T4RYVfKkevozsmx8rc3Ef/yWqHqldsR84JQUY/MFvVCDpUPJ3HALhOUKD7SBBPB9K2QDLDAROnQEjqEr0/h/6SeNS1kWhf5VOVkHkcc7IMDkAYWKIEKOANVUAME3IEH8ASejXvj0XgxXmetMWM+swt+wHj7BA6KlqA=</latexit>

PM,R(Y | Y )<latexit sha1_base64="BskvjlxyY01qFclOVQTZjK8jCko=">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</latexit><latexit sha1_base64="BskvjlxyY01qFclOVQTZjK8jCko=">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</latexit><latexit sha1_base64="BskvjlxyY01qFclOVQTZjK8jCko=">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</latexit><latexit sha1_base64="BskvjlxyY01qFclOVQTZjK8jCko=">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</latexit>

non-privateposterior

With MCMC, we can iterate:

Z(s) ⇠ PM(Z |Y (s))<latexit sha1_base64="7F1azSLCiXFU/GmkaLPBTH556AE=">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</latexit><latexit sha1_base64="7F1azSLCiXFU/GmkaLPBTH556AE=">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</latexit><latexit sha1_base64="7F1azSLCiXFU/GmkaLPBTH556AE=">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</latexit><latexit sha1_base64="7F1azSLCiXFU/GmkaLPBTH556AE=">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</latexit>

Y (s) ⇠ PM,R(Y | Y , Z(s�1))<latexit sha1_base64="KMJE/6Kzy19dVFTqOtwq/0HIAPQ=">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</latexit><latexit sha1_base64="KMJE/6Kzy19dVFTqOtwq/0HIAPQ=">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</latexit><latexit sha1_base64="KMJE/6Kzy19dVFTqOtwq/0HIAPQ=">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</latexit><latexit sha1_base64="KMJE/6Kzy19dVFTqOtwq/0HIAPQ=">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</latexit>

(the challenge)

(non-private update)

The geometric mechanism as an LPLP mechanism

We use use a generalization of “local privacy”.

Many count models have the following structure:

Non-private method:

⌧ ⇠ 2Geo(↵)

y(±) := y + ⌧<latexit sha1_base64="Z2sO5ZFXvVo6y4cQDCqMtyuYbAE=">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</latexit><latexit sha1_base64="Z2sO5ZFXvVo6y4cQDCqMtyuYbAE=">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</latexit><latexit sha1_base64="Z2sO5ZFXvVo6y4cQDCqMtyuYbAE=">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</latexit><latexit sha1_base64="Z2sO5ZFXvVo6y4cQDCqMtyuYbAE=">AAACP3icbVDBShxBEO3RmJjVxI0ec2myRFYCy4wIhoCg8RCPCq4K9rrU9Na6jd0zQ3eNOA7zPX5HPsCr4hd4C1692bvuQdc8KHi8V4+iXpxp5SgM74Kp6Xcz7z/MfqzNzX/6vFD/snjg0txKbMtUp/YoBodaJdgmRRqPMotgYo2H8dn20D88R+tUmuxTkWHHwGmi+koCealb3xIEOV8WThkuCC/ImnL1D6ZVU4DOBrAiRE2Q0j0si+qkbIrMrFR8+dcGL/gPPgx3642wFY7A35JoTBpsjN1u/V70UpkbTEhqcO44CjPqlGBJSY1VTeQOM5BncIrHniZg0HXK0asV/+6VHu+n1k9CfKS+TJRgnCtM7DcN0MBNekPxv55F7dTl5Hnq/+yUKslywkQ+X+/nmlPKh2XynrIoSReegLTKP8DlACxI8pXXfDPRZA9vycFqKwpb0d5aY/P3uKNZ9pV9Y00WsXW2yXbYLmszya7YNbtht8Hf4D74Fzw8r04F48wSe4Xg8QnvV665</latexit>

2Geo(⌧ ; ↵) =1� ↵

1 + ↵↵|⌧ |

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The “geometric mechanism” [Ghosh et al. (2012)]:

↵ = exp⇣� ✏

N

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Theorem 1: The geometric mechanism with parameter ! is an (N, ϵ)-private LPLP mechanism for any (N, ϵ) if:

Skel (⌧ ;µ1, µ2) = e�(µ1+µ2)

✓µ1

µ2

◆⌧/2

I|⌧ |(2pµ1µ2)

<latexit sha1_base64="MXTWitwGl/zInCmegd51xeKYpJU=">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</latexit><latexit sha1_base64="MXTWitwGl/zInCmegd51xeKYpJU=">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</latexit><latexit sha1_base64="MXTWitwGl/zInCmegd51xeKYpJU=">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</latexit><latexit sha1_base64="MXTWitwGl/zInCmegd51xeKYpJU=">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</latexit>

Theorem 2: A two-sided geometric random variable " ~ 2Geo(!) can be generated as an exponentially-randomized Skellam variable:

�(+), �(�) iid⇠ Expon⇣ ↵

1� ↵

⌧ ⇠ Skel (�(+), �(�))<latexit sha1_base64="UNiu86Bj1w6tUjnElqg9yWt0NS0=">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</latexit><latexit sha1_base64="UNiu86Bj1w6tUjnElqg9yWt0NS0=">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</latexit><latexit sha1_base64="UNiu86Bj1w6tUjnElqg9yWt0NS0=">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</latexit><latexit sha1_base64="UNiu86Bj1w6tUjnElqg9yWt0NS0=">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</latexit>

The Skellam is the difference of two Poissonsg(+) ⇠ Pois (�(+))

<latexit sha1_base64="lw7CBecIb3hUHbdDTMemXaJqsWE=">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</latexit><latexit sha1_base64="lw7CBecIb3hUHbdDTMemXaJqsWE=">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</latexit><latexit sha1_base64="lw7CBecIb3hUHbdDTMemXaJqsWE=">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</latexit><latexit sha1_base64="lw7CBecIb3hUHbdDTMemXaJqsWE=">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</latexit>

g(�) ⇠ Pois (�(�))<latexit sha1_base64="AvR5Q+eK2WnaO0e+NqqCvkmdd+I=">AAACVnicbVHLTsMwEHTDo1BeAY5cLCpQOVAlCAmOCC4cW4kCUlMqx920FnYS2RtEifJd/AocuMKNT0C4jwOPjmRpNLO72h2HqRQGPe+15MzNLyyWl5YrK6tr6xvu5ta1STLNocUTmejbkBmQIoYWCpRwm2pgKpRwE95fjPybB9BGJPEVDlPoKNaPRSQ4Qyt13Wb/Lg8Uw4FRTErQee3woCjofmCEogHCI2qVNxJhikBChLVA2tk9Nqsp0KI/wIOuW/Xq3hj0P/GnpEqmaHTdz6CX8ExBjFwyY9q+l2InZxoFl1BUgsxAyvg960Pb0pgpMJ18fHpB96zSo1Gi7YuRjtWfHTlTxgxVaCvHC//1RuJMT4M04glmee0Mo9NOLuI0Q4j5ZIkokxQTOsqY9oQGjnJoCeNa2DsoHzDNONqfqNiA/L9x/CfXR3Xfq/vN4+rZ+TSqJbJDdkmN+OSEnJFL0iAtwskzeSPv5KP0UvpyFpzypNQpTXu2yS847jc+grdv</latexit><latexit sha1_base64="AvR5Q+eK2WnaO0e+NqqCvkmdd+I=">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</latexit><latexit sha1_base64="AvR5Q+eK2WnaO0e+NqqCvkmdd+I=">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</latexit><latexit sha1_base64="AvR5Q+eK2WnaO0e+NqqCvkmdd+I=">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</latexit>

⌧ := g(+) � g(�)<latexit sha1_base64="ciPj0IB9Vopf2JEx9EAWfDuFnjA=">AAACOnicbZDLSgNBEEV74ivG16hLN41BiEjCjCiKIATduIxgHpCJoadTSZr0POjuEeIwX+KvuHGrX+DWnQiu/AA7kyw0SUHB5dwqqLpuyJlUlvVuZBYWl5ZXsqu5tfWNzS1ze6cmg0hQqNKAB6LhEgmc+VBVTHFohAKI53Kou4PrkV9/ACFZ4N+pYQgtj/R81mWUKI3a5qmjSIQvLnHvPnY8ovrSI5yDiAtHh0mCi7O8qHnbzFslKy08K+yJyKNJVdrmt9MJaOSBrygnUjZtK1StmAjFKIck50QSQkIHpAdNLX3igWzF6XsJPtCkg7uB0O0rnNK/GzHxpBx6rp5ML532RnCuJ4BL9gjzvGakuuetmPlhpMCn4yO6EccqwKMccYcJoIoPtSBUMP0Hpn0iCFU67ZwOyJ6OY1bUjku2VbJvT/Llq0lUWbSH9lEB2egMldENqqAqougJvaBX9GY8Gx/Gp/E1Hs0Yk51d9K+Mn1/0IK1W</latexit><latexit sha1_base64="ciPj0IB9Vopf2JEx9EAWfDuFnjA=">AAACOnicbZDLSgNBEEV74ivG16hLN41BiEjCjCiKIATduIxgHpCJoadTSZr0POjuEeIwX+KvuHGrX+DWnQiu/AA7kyw0SUHB5dwqqLpuyJlUlvVuZBYWl5ZXsqu5tfWNzS1ze6cmg0hQqNKAB6LhEgmc+VBVTHFohAKI53Kou4PrkV9/ACFZ4N+pYQgtj/R81mWUKI3a5qmjSIQvLnHvPnY8ovrSI5yDiAtHh0mCi7O8qHnbzFslKy08K+yJyKNJVdrmt9MJaOSBrygnUjZtK1StmAjFKIck50QSQkIHpAdNLX3igWzF6XsJPtCkg7uB0O0rnNK/GzHxpBx6rp5ML532RnCuJ4BL9gjzvGakuuetmPlhpMCn4yO6EccqwKMccYcJoIoPtSBUMP0Hpn0iCFU67ZwOyJ6OY1bUjku2VbJvT/Llq0lUWbSH9lEB2egMldENqqAqougJvaBX9GY8Gx/Gp/E1Hs0Yk51d9K+Mn1/0IK1W</latexit><latexit sha1_base64="ciPj0IB9Vopf2JEx9EAWfDuFnjA=">AAACOnicbZDLSgNBEEV74ivG16hLN41BiEjCjCiKIATduIxgHpCJoadTSZr0POjuEeIwX+KvuHGrX+DWnQiu/AA7kyw0SUHB5dwqqLpuyJlUlvVuZBYWl5ZXsqu5tfWNzS1ze6cmg0hQqNKAB6LhEgmc+VBVTHFohAKI53Kou4PrkV9/ACFZ4N+pYQgtj/R81mWUKI3a5qmjSIQvLnHvPnY8ovrSI5yDiAtHh0mCi7O8qHnbzFslKy08K+yJyKNJVdrmt9MJaOSBrygnUjZtK1StmAjFKIck50QSQkIHpAdNLX3igWzF6XsJPtCkg7uB0O0rnNK/GzHxpBx6rp5ML532RnCuJ4BL9gjzvGakuuetmPlhpMCn4yO6EccqwKMccYcJoIoPtSBUMP0Hpn0iCFU67ZwOyJ6OY1bUjku2VbJvT/Llq0lUWbSH9lEB2egMldENqqAqougJvaBX9GY8Gx/Gp/E1Hs0Yk51d9K+Mn1/0IK1W</latexit><latexit sha1_base64="ciPj0IB9Vopf2JEx9EAWfDuFnjA=">AAACOnicbZDLSgNBEEV74ivG16hLN41BiEjCjCiKIATduIxgHpCJoadTSZr0POjuEeIwX+KvuHGrX+DWnQiu/AA7kyw0SUHB5dwqqLpuyJlUlvVuZBYWl5ZXsqu5tfWNzS1ze6cmg0hQqNKAB6LhEgmc+VBVTHFohAKI53Kou4PrkV9/ACFZ4N+pYQgtj/R81mWUKI3a5qmjSIQvLnHvPnY8ovrSI5yDiAtHh0mCi7O8qHnbzFslKy08K+yJyKNJVdrmt9MJaOSBrygnUjZtK1StmAjFKIck50QSQkIHpAdNLX3igWzF6XsJPtCkg7uB0O0rnNK/GzHxpBx6rp5ML532RnCuJ4BL9gjzvGakuuetmPlhpMCn4yO6EccqwKMccYcJoIoPtSBUMP0Hpn0iCFU67ZwOyJ6OY1bUjku2VbJvT/Llq0lUWbSH9lEB2egMldENqqAqougJvaBX9GY8Gx/Gp/E1Hs0Yk51d9K+Mn1/0IK1W</latexit>

Theorem 3: These two generative processes express the same joint distribution of # and m:

g(+) ⇠ Pois(�(+))

g(�) ⇠ Pois(�(�))

⌧ := g(+) � g(�)

m := min{g(+), g(�)}<latexit sha1_base64="yd28PvMce7/InJTkPS1TCOMYfKo=">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</latexit><latexit sha1_base64="yd28PvMce7/InJTkPS1TCOMYfKo=">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</latexit><latexit sha1_base64="yd28PvMce7/InJTkPS1TCOMYfKo=">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</latexit><latexit sha1_base64="yd28PvMce7/InJTkPS1TCOMYfKo=">AAADDHiclVLLbhMxFPVMoZTwaFqWbCwiqlSQ0RilSoMEqsSGZZBIWykOkce5Sa3anpHtQYTR/AIbtvAX7BBb/oGf4BvwTIMEdLrgSpaOzrnn6PraSSaFdXH8Iwg3rl3fvLF1s3Xr9p272+2d3WOb5obDmKcyNacJsyCFhrETTsJpZoCpRMJJcv6i0k/egrEi1a/dKoOpYkstFoIz56nZTrC5fFNQxdyZVUxKMEX30X5Z4j1qhcLUwTtnVDFKhS27VPrcOWvq36e0dSmo959BvXUQdSzHe0+f4ebRepf5yumNqnZRJTSmRaP58RXectbuxNFBv0+GQxxHMRkMyWEFhgeEDDCJ4ro6aF2jWfsnnac8V6Adl8zaCYkzNy2YcYJLKFs0t5Axfs6WMPFQMwV2WtQPVuKHnpnjRWr80Q7X7J+OgilrVyrxnfWs/2oV2agZkFa8hyZtkrvF4bQQOssdaH4xxCKX2KW4+hl4LgxwJ1ceMG6EvwfmZ8ww7vz/afkF/d4CvhocP4lIHJFX/c7R8/WqttB99AB1EUEDdIReohEaIx6o4GPwKfgcfgi/hF/DbxetYbD23EN/Vfj9F64t+M0=</latexit>

⌧ ⇠ Skel (�(+),�(�))

m ⇠ Bessel⇣|⌧ |, 2

p�(+)�(�)

<latexit sha1_base64="t680laSeUJRLdpmfI5xsd9lLGFY=">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</latexit><latexit sha1_base64="t680laSeUJRLdpmfI5xsd9lLGFY=">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</latexit><latexit sha1_base64="t680laSeUJRLdpmfI5xsd9lLGFY=">AAAC0nicfVJNbxMxEPUuXyV8BThysYhAqQirdUmV5oIquHAsommL4hB5nUlixd5d7FlEut0D4spf4wfwJ/gNONsUQdV2JEuj92bmjZ+d5Fo5jONfQXjt+o2btzZuN+7cvXf/QfPhowOXFVbCQGY6s0eJcKBVCgNUqOEotyBMouEwWbxd8YdfwDqVpfu4zGFkxCxVUyUFemjc/MlRFPQ5d8pQjvAVrSk/LEBXXMMU21z7URPxqeRG4NwZoTXYsv1is6o69BLypSe5VbM5bnLeMOeGvwHn/o4/qdVPOryzRbn7bLG8QvAKuTO9cbMVR9vdLuv3aRzFrNdnO6ukv81Yj7IorqNF1rE3bv7mk0wWBlKUWjg3ZHGOo1JYVFJD1eCFg1zIhZjB0KepMOBGZW17RZ95ZEKnmfUnRVqj/3aUwji3NImvrDc+z63ACzkL2qljuIgbFjjdGZUqzQuEVJ4uMS00xYyu3pdOlAWJeukTIa3y96ByLqyQ6H9Bwxt05gK9PDnYilgcsffd1u7rtVUb5Al5StqEkR7ZJe/IHhkQGbwKPgZJIMP98Dj8Fn4/LQ2Ddc9j8l+EP/4AxDrlFA==</latexit><latexit sha1_base64="t680laSeUJRLdpmfI5xsd9lLGFY=">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</latexit>

Four equivalent ways to generate the noised data:

Solution: If we sample an auxiliary Bessel, we can “recover” the underlying (sensitive) data y:

Bessel(m; v, a) =(a/2)2m+v

I|v|(a)m!�(m+ v + 1)<latexit sha1_base64="QfRcaX+OjFBYqSYJVumBPovbLeg=">AAACPXicbVBNb9NAEF23fJTwFcqRy0IEctQotSOkVipIFRyAW5GatlIcovFm3K66a1u7Y6vR1n+MC/+BGzcuHECIK1c2aQ7Q8qSRnt6bmd15aamkpSj6EqysXrt+4+bardbtO3fv3W8/WD+wRWUEDkWhCnOUgkUlcxySJIVHpUHQqcLD9PT13D+s0VhZ5Ps0K3Gs4TiXmRRAXpq09xPCMzLavULrtzSh3uF1j0OXP3vJk8yAcCFsDrof3EBv1E3j3k3ceX3ehNBNelw/5kkveQNaQ6j5Bq99xd1m0u5E/WgBfpXES9JhS+xN2p+TaSEqjTkJBdaO4qiksQNDUihsWkllsQRxCsc48jQHjXbsFtc3/KlXpjwrjK+c+EL9e8KBtnamU9+pgU7sZW8u/s8bVZRtj53My4owFxcPZZXiVPB5lHwqDQpSM09AGOn/ysUJ+MjIB97yIcSXT75KDgb9OOrH7593dl8s41hjj9gTFrKYbbFd9pbtsSET7CP7yr6zH8Gn4FvwM/h10boSLGcesn8Q/P4DGtuqyw==</latexit><latexit sha1_base64="QfRcaX+OjFBYqSYJVumBPovbLeg=">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</latexit><latexit sha1_base64="QfRcaX+OjFBYqSYJVumBPovbLeg=">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</latexit><latexit sha1_base64="QfRcaX+OjFBYqSYJVumBPovbLeg=">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</latexit>

Synthetic data: Ours is better than the naïve method at recovering the true topics.

Enron corpus data: Our method’s reconstruction error is comparable to the non-private method and its inferred topics are more coherent!

Synthetic data: Ours is better than the naïve method at recovering block structure.

Enron corpus data: Our method obtains lower heldout and reconstruction error than both the naïve and non-private method!

▪ Conditions on noised data▪ Truncates any negative values to 0▪ Standard non-private inference

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▪ Conditions on unnoised data▪ Standard non-private inference