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Merge pull request CamDavidsonPilon#248 from tomw2005/typo-fix
Fixed typo of the to they
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Chapter5_LossFunctions/Chapter5.ipynb

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"In Bayesian inference, we have a mindset that the unknown parameters are really random variables with prior and posterior distributions. Concerning the posterior distribution, a value drawn from it is a *possible* realization of what the true parameter could be. Given that realization, we can compute a loss associated with an estimate. As we have a whole distribution of what the unknown parameter could be (the posterior), we should be more interested in computing the *expected loss* given an estimate. This expected loss is a better estimate of the true loss than comparing the given loss from only a single sample from the posterior.\n",
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"First it will be useful to explain a *Bayesian point estimate*. The systems and machinery present in the modern world are not built to accept posterior distributions as input. It is also rude to hand someone over a distribution when all the asked for was an estimate. In the course of an individual's day, when faced with uncertainty we still act by distilling our uncertainty down to a single action. Similarly, we need to distill our posterior distribution down to a single value (or vector in the multivariate case). If the value is chosen intelligently, we can avoid the flaw of frequentist methodologies that mask the uncertainty and provide a more informative result.The value chosen, if from a Bayesian posterior, is a Bayesian point estimate. \n",
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"First it will be useful to explain a *Bayesian point estimate*. The systems and machinery present in the modern world are not built to accept posterior distributions as input. It is also rude to hand someone over a distribution when all they asked for was an estimate. In the course of an individual's day, when faced with uncertainty we still act by distilling our uncertainty down to a single action. Similarly, we need to distill our posterior distribution down to a single value (or vector in the multivariate case). If the value is chosen intelligently, we can avoid the flaw of frequentist methodologies that mask the uncertainty and provide a more informative result.The value chosen, if from a Bayesian posterior, is a Bayesian point estimate. \n",
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"Suppose $P(\\theta | X)$ is the posterior distribution of $\\theta$ after observing data $X$, then the following function is understandable as the *expected loss of choosing estimate $\\hat{\\theta}$ to estimate $\\theta$*:\n",
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