sample = {2.0,  4.0 , -6.0 ,  -7.0};

pixlik =  
      Product[ 1/(1  +  (th - sample[[i]])^2), {i, 1, 
          4}]

marg  =  
      NIntegrate[pixlik, {th, -15, 15}]
pix = pixlik/marg;

NSolve[pix == 0.2, {th}]

NIntegrate[pix, {th, -6.8925580, -5.66730567}]

NSolve[pix == 0.1, {th}]

NIntegrate[pix, {th, -7.3278, -5.1240}] + 
      NIntegrate[pix, {th, 1.591, 3.12029}]

NSolve[pix == 0.0159, {th}]

NIntegrate[pix, {th, -8.1887, -3.02186}] + 
      NIntegrate[pix, {th, -0.615385, 4.75507}]

a = NSolve[pix == 0.008475, {th}]

NIntegrate[pix, {th, -8.49841, 5.07724}]

Plot[{pix, 0.1, 0.2,  0.0159,  0.008475}, {th, -10, 
        10}]