Write a paragraph similar to the one below, but change the topic and the content from Supervised Automatic Speech Recognition to Supervised Speech Processing, do not just change the word ASR to Speech Processing, but re-write the content for Speech Processing. For example, don't keep the WER part because we are talking about Speech Processing and not ASR. Write in latex, and do not render it; leave it raw:

Given a set of labeled training samples 𝒟𝑡𝑟 = {(x1, y1), · · · , (x𝑁 , y𝑁 )}, where x𝑖 ∈ X and y𝑖 ∈ Y correspond to speech and text of the 𝑖-th sample, respectively, ASR seeks for a function 𝑓 : X → Y from a set of hypotheses 𝑓 ∈ H that minimizes the loss ∑︀(x𝑗 ,y𝑗 )∈𝒟𝑡𝑡 𝐿(y𝑗 , 𝑓 (x𝑗 )) with respective to a predefined metric 𝐿 : Y × Y → R≥0 on an unseen test set 𝒟𝑡𝑡. A widely used evaluation metric 𝐿 is the word error rate
(WER), which can be computed as:
WER(y𝑟𝑒𝑓 , yh𝑦𝑝) = Levenshtein-Distance(y𝑟𝑒𝑓 , yh𝑦𝑝), (2.1)
Length(y𝑟𝑒𝑓 )
where y𝑟𝑒𝑓 and yh𝑦𝑝 are the reference transcript and the predicted transcript, re- spectively, and the Levenshtein distance (Levenshtein, 1966) (also known as the edit distance) is the minimum number of single-token edits required to change yh𝑦𝑝 into y𝑟𝑒𝑓 , which can be evaluated with dynamic programming efficiently. However, since it is not easy to directly optimize a model based on WER, a surrogate loss such as negative likelihood is often used as a replacement.
Modern speech recognition systems are built upon the noisy channel model proposed by NAME_1 (1976), which assumes that a text y is transmitted through a noisy channel that renders an audio x, and it is desired to recover the original text from the observed audio. The inference procedure can be formalized as determining the posterior probability of y for a given x. Therefore, instead of directly seeking for a deterministic
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function 𝑓, it is more common in modern ASR systems to estimate a distribution 𝑝(y | x), and obtain a deterministic function by setting 𝑓 (x) = arg maxy 𝑝(y | x).
Depending on how the posterior distribution 𝑝(y | x) is parameterized, ASR systems can be roughly classified into two categories: modularized and end-to-end, each of which has several different training objectives. In the following sections, we will introduce the two categories and discuss their properties.