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  "body_markdown": "A probabilistic model can describe more than the data you observe. It can also include hidden variables that capture structure you cannot observe directly.\nBut defining that model is only the beginning. Once an observation\nx\nx\nx\nis available, the practical question changes:\nGiven this\nx\nx\nx\n, what does the model imply about the hidden variable\nz\nz\nz\n?\nThat is the central problem of\nPosterior Inference\n.\nThe notation is compact, but the computation is not always easy. High-dimensional latent spaces, complex posterior distributions, and interactions among hidden variables can make both the posterior itself and expectations under that posterior difficult to compute.\nStart with the Joint Distribution\nSuppose a probabilistic model contains an observed variable\nx\nx\nx\nand a hidden or latent variable\nz\nz\nz\n.\nThe model does not treat them as unrelated quantities. Instead, it represents their probabilistic relationship through a\nJoint Distribution\n:\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\nThis joint distribution describes how the observed data and the hidden variable fit together inside a single probability structure.\nOnce\nx\nx\nx\nis observed, however, the question becomes conditional. We are no longer asking only how\nx\nx\nx\nand\nz\nz\nz\nrelate in general. We want to know how the possible values of\nz\nz\nz\nare distributed given the particular observation\nx\nx\nx\n.\nThat conditional distribution is the posterior.\nPosterior Distribution: Conditioning on Observed Data\nThe\nPosterior Distribution\nis\np\n(\nz\n∣\nx\n)\n=\np\n(\nz\n,\nx\n)\np\n(\nx\n)\np(z\\mid x)=\\frac{p(z,x)}{p(x)}\np\n(\nz\n∣\nx\n)\n=\np\n(\nx\n)\np\n(\nz\n,\nx\n)\n​\nThe numerator\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\ncontains the probabilistic relationship between the latent variable and the observation. The denominator\np\n(\nx\n)\np(x)\np\n(\nx\n)\nnormalizes those values so that the result becomes a conditional probability distribution over\nz\nz\nz\n.\nThe distinction is important:\nThe joint distribution\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\ndescribes the probability structure of the model.\nThe posterior distribution\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\ntells us what that structure implies about\nz\nz\nz\nafter\nx\nx\nx\nhas been observed.\nIn that sense, the posterior connects the model with actual data.\nPosterior Inference Has Two Closely Related Tasks\nPosterior Inference is not limited to obtaining the posterior distribution itself.\nOne task is to compute\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\nwhich answers the question: how should probability be distributed across possible latent states after observing\nx\nx\nx\n?\nThe other task is to compute an\nexpectation under that posterior\n. In this case, the posterior is used to average a quantity required for later analysis or learning.\nThis distinction matters because some learning procedures use the posterior inside another computation. The EM Algorithm is a representative example.\nEM Shows Why Posterior Expectations Matter\nConsider a model with a latent variable\nz\nz\nz\n.\nBecause\nz\nz\nz\nis unobserved, the Complete-Data Log-Likelihood cannot be used directly as though both\nx\nx\nx\nand\nz\nz\nz\nwere known. The\nExpectation-Maximization (EM) Algorithm\ninstead uses the posterior distribution under the current model parameters to compute its expectation over the possible latent states:\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n≜\n∑\nz\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\nlog\n⁡\np\n(\nx\n,\nz\n;\nθ\n)\nQ(\\theta,\\theta^{old})\\triangleq\\sum_z p(z\\mid x;\\theta^{old})\\log p(x,z;\\theta)\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n≜\nz\n∑\n​\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\nlo\ng\np\n(\nx\n,\nz\n;\nθ\n)\nHere,\nθ\no\nl\nd\n\\theta^{old}\nθ\no\nl\nd\ndenotes the current model parameters, while\nθ\n\\theta\nθ\ndenotes the parameters being evaluated.\nThe calculation can be understood in three steps:\nUse\nθ\no\nl\nd\n\\theta^{old}\nθ\no\nl\nd\nto obtain the posterior\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\np(z\\mid x;\\theta^{old})\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\n.\nUse those posterior probabilities as weights for\nlog\n⁡\np\n(\nx\n,\nz\n;\nθ\n)\n\\log p(x,z;\\theta)\nlo\ng\np\n(\nx\n,\nz\n;\nθ\n)\n.\nSum over\nz\nz\nz\nto obtain\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\nQ(\\theta,\\theta^{old})\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n.\nFor each possible latent state, the Complete-Data Log-Likelihood is weighted by that state's posterior probability.\nSo\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\nQ(\\theta,\\theta^{old})\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\nis not merely an arbitrary weighted sum. It is the Complete-Data Log-Likelihood averaged under the posterior distribution of the latent variable.\nThis gives a useful implementation-level mental model:\nPosterior Inference determines how latent states are weighted, and learning uses those weights to construct the quantity it needs.\nPosterior Inference is therefore connected directly to model learning, not only to analyzing a probabilistic model.\nWhere Exact Inference Becomes Difficult\nThe conceptual process seems simple: define\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\n, observe\nx\nx\nx\n, obtain\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\n, and then compute any required posterior expectation.\nThe difficulty is that the last two computations are not always tractable.\nHigh-Dimensional Latent Spaces\nAs the dimensionality of the latent space increases, the space of possible latent states becomes harder to handle directly.\nThe challenge is not merely that\nz\nz\nz\nis hidden. A high-dimensional latent variable can create a large state space that must be considered during inference.\nComplex Posterior Distributions\nThe posterior itself may also have a complicated form.\nWhen that happens, required expectations under\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\nmay not be available through an analytical calculation.\nThis creates two related computational questions:\nCan\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\nbe computed exactly?\nCan the required expectation under\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\nbe computed exactly?\nDifficulty in either calculation can make inference a bottleneck.\nWhy Deep Models Add More Difficulty\nDeep Learning can add another source of difficulty through interactions among hidden variables.\nWhen hidden variables appear across multiple layers, the state of one variable can be affected by relationships involving other variables and layers. The resulting posterior structure must reflect those interactions.\nThe inference problem therefore becomes difficult for several reasons at once:\nthe latent space may be high-dimensional,\nthe posterior distribution may have a complex form,\nposterior expectations may not have an analytical solution,\ninteractions among hidden variables across layers can further complicate the posterior.\nThis is why the\nChallenge of Inference\ncannot be explained simply by saying that latent variables are unobserved.\nThe real difficulty comes from the structure of the inference problem. As the latent structure becomes more complicated, exact computation of the posterior and its expectations can become difficult enough to limit model learning and use.\nModeling and Inference Are Different Problems\nA useful mental model is to separate the probability model from the inference problem.\nThe modeling question is:\nWhat joint probability structure describes the observed and hidden variables?\nThat structure is represented by\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\nThe inference question is:\nGiven an actual observation\nx\nx\nx\n, what does that model imply about the hidden variable\nz\nz\nz\n?\nThat target is\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\nA later learning or analysis step may then require a quantity computed under that posterior, such as\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n≜\n∑\nz\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\nlog\n⁡\np\n(\nx\n,\nz\n;\nθ\n)\nQ(\\theta,\\theta^{old})\\triangleq\\sum_z p(z\\mid x;\\theta^{old})\\log p(x,z;\\theta)\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n≜\nz\n∑\n​\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\nlo\ng\np\n(\nx\n,\nz\n;\nθ\n)\nDefining a probabilistic model does not automatically mean that every quantity implied by that model can be computed easily.\nThe probability structure may be well defined while exact inference remains computationally difficult.\nWhy Approximate Posterior Inference Becomes Necessary\nIf exact Posterior Inference were always tractable, no additional solution would be needed.\nBut complex probabilistic models cannot always rely on exact computation of the posterior distribution or the required posterior expectations.\nThat is where\nApproximate Posterior Inference\nbecomes necessary.\nThe important point here is not a particular approximation method. It is the reason approximation is needed in the first place.\nAs latent structure becomes more complicated, exact posterior computation becomes harder. A complex posterior can also make the expectations required for learning difficult to evaluate. When those computations become impractical, inference itself becomes a computational limitation.\nApproximate Posterior Inference is the next step for dealing with that limitation.\nWhat to Look for in a Probabilistic Model\nWhen reading a latent-variable model, separate three pieces.\nFirst, identify the joint model:\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\nThen identify the posterior inference target:\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\nFinally, determine whether the algorithm needs the posterior itself or a quantity computed under it, such as the posterior expectation used by EM:\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n≜\n∑\nz\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\nlog\n⁡\np\n(\nx\n,\nz\n;\nθ\n)\nQ(\\theta,\\theta^{old})\\triangleq\\sum_z p(z\\mid x;\\theta^{old})\\log p(x,z;\\theta)\nQ\n(\nθ\n,\nθ\no\nl\nd\n)\n≜\nz\n∑\n​\np\n(\nz\n∣\nx\n;\nθ\no\nl\nd\n)\nlo\ng\np\n(\nx\n,\nz\n;\nθ\n)\nThat makes the practical question much clearer:\nCan the required posterior computation be performed exactly, or has inference itself become the computational problem?\nTakeaway\nPosterior Inference connects a probabilistic model with observed data and asks what the model implies about its hidden variables after that data is available.\nThe conceptual progression is straightforward: start from the joint distribution\np\n(\nz\n,\nx\n)\np(z,x)\np\n(\nz\n,\nx\n)\n, condition on the observed\nx\nx\nx\nto obtain\np\n(\nz\n∣\nx\n)\np(z\\mid x)\np\n(\nz\n∣\nx\n)\n, and then use that posterior directly or compute expectations under it when learning requires them.\nThe difficulty is computational. High-dimensional latent spaces, complex posterior distributions, and interactions among hidden variables can make exact Posterior Inference difficult.\nThat computational limitation is what motivates Approximate Posterior Inference.\nOriginally published at zeromathai.com.\nOriginal article:\nhttps://zeromathai.com/en/posterior-inference-course-en/",
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